lesson 6 problem solving practice surface area of prisms

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Surface area of a prism

Here you will learn about the surface area of a prism, including what it is and how to calculate it.

Students will first learn about the surface area of a prism as part of geometry in 6 th grade.

What is the surface area of a prism?

The surface area of a prism is the total area of all of the faces of the prism. Prisms are 3D shapes made up of flat faces, including polygonal bases and rectangle faces.

One type of prism is a rectangular prism .

A rectangular prism is made up of 6 rectangular faces, including 2 rectangular bases and 4 rectangular lateral faces.

For example,

Here is a rectangular prism and its net.

It has 3 pairs of congruent faces, since the opposite faces are the same.

To calculate the surface area of the rectangular prism, calculate the area of each face and then add them together.

The surface area of the prism is the sum of the areas. Add each area twice, since each rectangle appears twice in the prism:

40+40+24+24+15+15=158 \mathrm{~cm}^2

The surface area of the rectangular prism is equal to 158 \mathrm{~cm}^2 .

Note: Surface area is measured in square units. For example, \mathrm{mm}^2, \mathrm{~cm}^2, \mathrm{~m}^2 , etc.

Another type of prism is a triangular prism .

A triangular prism is made up of 5 faces, including triangular bases and 3 rectangular lateral faces.

Here is a triangular prism and its net.

The base of this triangular prism is an isosceles triangle – two of the side lengths are equal.

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 6.5 \mathrm{~mm} side of the triangle would combine to form an edge with each corresponding rectangle – making their lengths equal.

To calculate the surface area of the triangular prism, calculate the area of each face and then add them together.

The surface area of the prism is the sum of the areas. Add the area of the triangular base twice, since it appears twice in the prism:

32.5+50+32.5+21+21=157 \mathrm{~mm}^2

The surface area of the triangular prism is equal to 157 \mathrm{~mm}^2.

Surface area is measured in square units. For example, \mathrm{mm}^2, \mathrm{~cm}^2, \mathrm{~m}^2 , etc.

Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Geometry (6.G.A.4) Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

How to calculate the surface area of a prism

In order to work out the surface area of a prism:

Calculate the area of each face.

Add the area of each face together.

Include the units.

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Surface area of a prism examples

Example 1: surface area of a triangular prism with a right triangle.

Calculate the surface area of the triangular prism.

The area of the front of the prism is \cfrac{1}{2} \, \times 4 \times 3= 6 \mathrm{~cm}^{2}.

The back face is the same as the front face so the area of the back face is also 6 \mathrm{~cm}^{2}.

The area of the bottom is 4 \times 2= 8 \mathrm{~cm}^2.

The area of the left side is 2 \times 3=6 \mathrm{~cm}^2.

The area of the top side is 2 \times 5=10 \mathrm{~cm}^2.

It will make our working clearer if we use a table:

2 Add the area of each face together.

Total surface area: 6+6+8+6+10= 36

3 Include the units.

The measurements on this prism are in cm , so the total surface area of the prism is 36 \mathrm{~cm}^2 .

Example 2: surface area of a rectangular prism

Calculate the surface area of the rectangular prism.

A rectangular prism has 6 faces, with 3 pairs of identical faces.

Total surface area: 14+14+21+21+6+6= 82

The measurements on this prism are in m , so the total surface area of the prism is 82 \mathrm{~m}^2 .

Example 3: surface area of a triangular prism with an equilateral triangle – using a net

Calculate the surface area of the triangular prism. The base of the prism is an equilateral triangle with a perimeter of 16.5 \mathrm{~ft} .

First, use the perimeter of the base to find the length of each side. Since an equilateral triangle has all equal sides, s , the perimeter is s+s+s= 16.5 .

s= 5.5 \mathrm{~ft}

You can unfold the triangular prism, and use the net to find the area of each face:

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 5.5 \mathrm{~ft} side of the triangle would combine to form an edge with each corresponding rectangle – making their lengths equal.

The area of each triangular base: \cfrac{1}{2} \times 4.8 \times 5.5= 13.2

The area of each rectangular lateral face: 10 \times 5.5= 55

If you have trouble keeping track of all the calculations, use a net:

The area of the base is always equal to the opposite base, in this case the triangles. Notice, since the triangle is equilateral, all the rectangular faces are equal as well.

Total surface area: 13.2+13.2+55+55+55= 191.4

The measurements on this prism are in ft , so the total surface area of the prism is 191.4 \mathrm{~ft}^2 .

Example 4: surface area of a rectangular prism – using a net

Calculate the lateral surface area of the rectangular prism. The base of the prism is a square and one side of the base measures 3 \cfrac{2}{3} \text { inches} .

You can unfold the rectangular prism, and use the net to find the area of each face:

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 3 \cfrac{2}{3} \mathrm{~ft} side of the square would combine to form an edge with each corresponding rectangle – making their lengths equal.

The area of each rectangular lateral face:

\begin{aligned} & 9 \cfrac{4}{5} \times 3 \cfrac{2}{3} \\\\ & = \cfrac{49}{5} \times \cfrac{11}{3} \\\\ & = \cfrac{539}{15} \\\\ & = 35 \cfrac{14}{15} \end{aligned}

Remember, you are only finding the area of the lateral faces, so you do not need to calculate the area of the bases.

Notice, since the square has all equal sides, all the rectangular faces are equal as well.

Total lateral surface area:

\begin{aligned} & 35 \cfrac{14}{15}+35 \cfrac{14}{15}+35 \cfrac{14}{15}+35 \cfrac{14}{15} \\\\ & =140 \cfrac{56}{15} \\\\ & =143 \cfrac{11}{15} \end{aligned}

The measurements on this prism are in inches, so the total lateral surface area of the prism is 143 \cfrac{11}{15} \text { inches }^2 .

Example 5: surface area of a parallelogram prism with different units

Calculate the surface area of the parallelogram prism.

A parallelogram prism has 6 faces and, like a rectangular prism, it has 3 pairs of identical faces. The base is a parallelogram and all of the lateral faces are rectangular.

In this example, some of the measurements are in cm and some are in m . You must convert the units so that they are the same. Convert all the units to meters ( m ): 40 \mathrm{~cm}= 0.4 \mathrm{~m} and 50 \mathrm{~cm}= 0.5 \mathrm{~m}.

Total surface area: 0.48+0.48+1.8+1.8+0.75+0.75= 6.06

The measurements that we have used are in m , so the surface area of the prism is 6.06 \mathrm{~m}^2 .

Example 6: surface area word problem – missing height

Ginny painted an area of 264 \mathrm{~ft}^2 on the rectangular building shown above. She painted all sides, except the bottom. How many feet long is the building

The base is a rectangle that measures 6 \mathrm{~ft} by 7 \mathrm{~ft} . The area of each base is 42 \mathrm{~ft}^2 .

Total area of the bases: 42+42=84

Subtract the area of the bases from the total amount of paint Ginny used, to see how much was used on the lateral faces: 264-84=180

The total area of the faces left is 180 \mathrm{~ft}^2 .

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 7 \mathrm{~ft} and 6 \mathrm{~ft} sides of the rectangle would combine to form an edge with each corresponding rectangle – making their lengths equal.

Labeling the missing length as x and show area of 3 lateral faces as expressions:

• 6 \times x or 6 x
• 7 \times x or 7 x

Together the 3 faces equal 180 \mathrm{~ft}^2 , so the total of the missing areas can be written as:

6x + 7x + 7x= 180

Since 20 \times 9= 180 , the missing side length is 9 .

The missing measurement is not the area, it is a side length, which is measured in singular units.

The missing side length is 9 \mathrm{~ft} .

Teaching tips for the surface area of a prism

• Make sure that students have had time to work with physical 3D models and nets before asking them to solve with surface area.
• Choose worksheets that offer a variety of question types – a mixture of the prism shown or the net, a mixture of solving for the missing surface area or height/length/width, and some word problems.

Easy mistakes to make

• Confusing lateral area with total surface area Lateral area is the area of each of the sides and total surface area is the area of the bases plus the area of the sides. When asked to find lateral area, be sure to only add up the area of the sides – which are always rectangles in a prism.

Related surface area lessons

• Surface area
• Surface area of a rectangular prism
• Surface of a cylinder
• Surface area of a triangular prism
• Surface area of a pyramid
• Surface area of a cone
• Surface area of a sphere
• Surface area of a cube
• Surface area of a hemisphere

Practice surface area of a prism questions

1. Calculate the surface area of the triangular prism:

Total surface area = 24+24+96+72+120= 336 \mathrm{~cm}^{2}

2. Calculate the surface area of the rectangular prism:

Total surface area = 14+14+77+77+22+22= 226 \mathrm{~mm}^{2}

3. Calculate the surface area of the triangular prism:

You can unfold the triangular prism, and use the net to find the area of each face.

Remember that the edges in a prism always fold up together to form the prism – making their lengths equal.

The area of each triangular base:

\cfrac{1}{2} \times 4 \times 6= 12

12 \times 6.3= 75.6

12 \times 4= 48

Total surface area: 12+12+75.6+48+75.6= 223.2 \mathrm{~cm}^2

4. Calculate the surface area of the rectangular prism:

You can unfold the rectangular prism, and use the net to find the area of each face.

The area of each rectangular base:

\begin{aligned} & 1 \cfrac{1}{5} \times 7 \\\\ & =\cfrac{6}{5} \times \cfrac{7}{1} \\\\ & =\cfrac{42}{5} \\\\ & =8 \cfrac{2}{5} \end{aligned}

\begin{aligned} & 4 \cfrac{1}{3} \times 7 \\\\ & =\cfrac{13}{3} \times \cfrac{7}{1} \\\\ & =\cfrac{91}{3} \\\\ & =30 \cfrac{1}{3} \end{aligned}

\begin{aligned} & 4 \cfrac{1}{3} \times 1 \cfrac{1}{5} \\\\ & =\cfrac{13}{3} \times \cfrac{6}{5} \\\\ & =\cfrac{78}{15} \\\\ & =5 \cfrac{3}{15} \end{aligned}

Total surface area:

\begin{aligned} & 8 \cfrac{2}{5}+8 \cfrac{2}{5}+30 \cfrac{1}{3}+30 \cfrac{1}{3}+5 \cfrac{3}{15}+5 \cfrac{3}{15} \\\\ & =8 \cfrac{6}{15}+8 \cfrac{6}{15}+30 \cfrac{5}{15}+30 \cfrac{5}{15}+5 \cfrac{3}{15}+5 \cfrac{3}{15}\\\\ & =86 \cfrac{28}{15} \\\\ & =87 \cfrac{13}{15} \mathrm{~m}^2 \end{aligned}

5. Calculate the surface area of the prism:

Break the base up into a rectangle and a triangle and find the area of each:

24 \times 30= 720

\cfrac{1}{2} \times 24 \times 5= 60

Total area of the base: 720+60= 780

Calculate the area of each face:

Total surface area : 780+780+450+360+450+195+195= 3,210 \mathrm{~m}^{2}

6. Zahir was painting the pentagonal prism below. It took 820 \text { inches}^2 to cover the entire shape. If the area of the base is 140 \text { inches}^2 and each side of the pentagon is 9 \text { inches} , what is the height of the prism?

You can unfold the pentagonal prism, and use the net to find the area of each face:

Total area of the bases: 140+140= 280

Subtract the area of the bases from the total amount of paint Zahir used, to see how much was used on the lateral faces: 820-280= 540

The total area of the faces left is 540 \text { inches}^2.

Since the 5 faces are congruent, the total for each face can be found by dividing by 5 :

540 \div 5= 108

Labeling the missing length as x , means the area of each face can be written as 9 \times x or 9 x.

Since each face has an area of 108 \text { inches }^2 , the missing height can be found with the equation: 9 x=108.

Since 9 \times 12=108 , the missing height is 12 inches.

Surface area of a prism FAQs

The general formula is (S1+ S+ S3) L + bh , but it is not common for students to memorize this formula. It is typically easier for students to remember to find the area of the bases and the three rectangular faces, since these dimensions often vary. Finally, add the area of the faces all together to find the total surface area.

There are trapezoidal prisms, which have a trapezoid base. There are also hexagonal prisms, which have a hexagon as a base.

The next lessons are

• Pythagorean theorem
• Congruence and similarity
• Transformations

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9.10: Surface Area and Volume of Prisms

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3-D figures with 2 congruent bases in parallel planes and rectangles for their other faces.

A prism is a 3-dimensional figure with 2 congruent bases, in parallel planes, in which the other faces are rectangles.

The non-base faces are l ateral faces . The edges between the lateral faces are lateral edges .

This particular example is a pentagonal prism because its base is a pentagon. Prisms are named by the shape of their base. Prisms are classified as either right prisms (prisms where all the lateral faces are perpendicular to the bases) or oblique prisms (prisms that lean to one side, whose base is a parallelogram rather than a rectangle, and whose height is perpendicular to the base's plane), as shown below.

Surface Area

To find the surface area of a prism, find the sum of the areas of its faces. The lateral area is the sum of the areas of the lateral faces. The basic unit of area is the square unit.

\(Surface Area=B_{1}+B_{2}+L_{1}+L_{2}+L_{3}\)

\(Lateral Area=L_{1}+L_{2}+L_{3}\)

To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.

For prisms in particular, to find the volume you must find the area of the base and multiply it by the height.

Volume of a Prism: \(V=B\cdot h\), where \(B= area\: of\: base\).

If an oblique prism and a right prism have the same base area and height, then they will have the same volume. This is due to Cavalieri’s Principle , which states that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

What if you were given a solid three-dimensional figure with two congruent bases in which the other faces were rectangles? How could you determine how much two-dimensional and three-dimensional space that figure occupies?

Example \(\PageIndex{1}\)

The total surface area of the triangular prism is \(540\text{ units}^{2}\). What is \(x\)?

The total surface area is equal to:

\(A_{2\: triangles}+A_{3\: rectangles}=540\)

The hypotenuse of the triangle bases is 13, \(5^{2}+12^{2}\). Let’s fill in what we know.

\(\begin{aligned} A_{2\: triangles}=2(\dfrac{1}{2}\cdot 5\cdot 12)=60 \\ A_{3\: rectangles}&=5x+12x+13x=30x \\ 60+30x &=540 \\ 30x&=480 \\ x&=16\text{ units }\qquad \text{ The height is 16 units.}\end{aligned}\)

Example \(\PageIndex{2}\)

Find the volume of the right rectangular prism below.

The area of the base is \((5)(4)=20\) and the height is 3. So the total volume is \((20)(3)=60\text{ unit}^{3}\)

Example \(\PageIndex{3}\)

Find the surface area of the prism below.

To solve, draw the net of the prism so that we can make sure we find the area of ALL faces.

Using the net, we have:

\(\begin{aligned} SA_{prism}&=2(4)(10)+2(10)(17)+2(17)(4)\\ &=80+340+136 \\ &=556 \text{ cm}^{2}\end{aligned}\)

Example \(\PageIndex{4}\)

This is a right triangular prism. To find the surface area, we need to find the length of the hypotenuse of the base because it is the width of one of the lateral faces. We can use the Pythagorean Theorem to find this length.

\(\begin{aligned} 7^{2}+24^{2}&=c^{2} \\ 49+576&=c^{2} \\ 625&=c^{2} \qquad c=25\end{aligned}\)

Looking at the net, the surface area is:

\(\begin{aligned} SA&=28(7)+28(24)+28(25)+2(\dfrac{1}{2}\cdot 7\cdot 24) \\ SA&=196+672+700+168=1736 units^{2}\end{aligned}\)

Example \(\PageIndex{5}\)

You have a small, triangular prism-shaped tent. How much volume does it have once it is set up?

First, we need to find the area of the base.

\(\begin{aligned} B&=\dfrac{1}{2}(3)(4)=6 \text{ ft}^{2} \\ V&=Bh=6(7)=42 \text{ ft}^{3}\end{aligned}\)

Even though the height in this problem does not look like a “height,” it is because it is the perpendicular segment connecting the two bases.

• Draw the net of this prism.
• Find the area of the bases.
• Find the area of lateral faces, or the lateral surface area.
• Find the total surface area of the prism.
• How many one-inch cubes can fit into a box that is 8 inches wide, 10 inches long, and 12 inches tall? Is this the same as the volume of the box?
• A cereal box in 2 inches wide, 10 inches long and 14 inches tall. How much cereal does the box hold?
• A can of soda is 4 inches tall and has a diameter of 2 inches. How much soda does the can hold? Round your answer to the nearest hundredth.
• A cube holds \(216\text{ in}^{3}\). What is the length of each edge?
• A cube has sides that are 8 inches. What is the volume?

Use the right triangular prism to answer questions 11-15.

• Find the volume of the prism.
• What shape are the bases of this prism? What are their areas?
• What are the dimensions of each of the lateral faces? What are their areas?
• Find the lateral surface area of the prism.
• Describe the difference between lateral surface area and total surface area.

• What is the volume and surface area of one die?
• What is the volume and surface area of both dice?

Find the volume of the following solids. Round your answers to the nearest hundredth.

Find the value of \(x\), given the surface area.

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.3.

Additional Resources

Interactive Element

Video: Prisms Principles - Basic

Activities: Prisms Discussion Questions

Study Aids: Prisms and Cylinders Study Guide

Practice: Surface Area and Volume of Prism

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Grade 6 - Geometry

Standard 6.G.1.4 - Calculate the surface area of cubes and rectangular prisms.

Included Skills:

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

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Surface Area of Prisms and Pyramids Lesson Plan

In this lesson, students are introduced to the concept of surface area of prisms and pyramids for `6`th graders with an engaging exploratory activity where students unfold various real life or household prisms and pyramids to find the surface area. By now, students are familiar with prisms and pyramids and the nets of these `3D` solids.

ByteLearn gives students targeted feedback and hints based on their specific mistakes

Preview step-by-step-help

Students will be able to solve for surface area of prisms and pyramids given the net.

• Teacher slideshow
• Online Practice
• Surface Area of Prisms and Pyramids Activity Worksheet (One per student)
• Square Pyramid (Make these ahead of time)
• Tissue boxes
• Cereal boxes
• Triangular prism candy bar boxes

How to Teach Surface Area of Prisms and Pyramids

Start out the lesson by sharing slide one of the teacher slideshow and ask students to write down a bunch of real life or household items they can come up with that are prisms and pyramids.

Copy these Google Slides for free

Give students a few minutes to jot down their ideas and then have them share with a partner. Once the discussion comes to an end, have students share with the class. Students may share tissue boxes, cereal boxes, tents, dice, storage boxes, pyramids in Egypt, etc. Write these on the board for all students to see. Share any other examples that students did not get to share.

Explore the concept of surface area

Present a picture of the box and ask students how would they figure out how much cardboard was used to make this box.

Students are likely to come up with interesting ideas for finding how much cardboard was used. Some might want to measure the edges of the box. Some might suggest that we open up the box and measure the sides of each face - they are likely to make a connection to the idea of nets that learned in the earlier class.

Show them how the box looks like when it is opened. Of course there are flaps that are used to tuck in the edges. Tell them that for now we will ignore them or measure them separately. Introduce the concept of surface area and tell that today we will find the  surface area of prism and pyramids .   

Introduction to the activity

Explain to students that we’re going to find the surface area of some of these objects. We’ll use tissue boxes, cereal boxes, toblerone candy bars, and a square pyramid. We’ll unfold the boxes to be able to see the shapes in `2D`. Then we’ll find the area of each part to find the total surface area of the object!

Surface area of household objects activity

Hand out the  Surface Area of Prisms and Pyramids Activity Worksheet , rulers, scissors, and one tissue box, cereal box, toblerone box, and square pyramid to each group of four students. 

Remind students of the steps for finding surface area:

• Step `1`: Number each face of the objects
• Remind them that with triangles, they will need the height and the base of the triangle
• They should stick to one unit while measuring the dimensions
• You might have to do a demonstration of how to measure using the ruler (starting at `0` and not at `1`, etc)
• Step `3`: Find the area of each face (be careful about what formula to use based on the shape!)
• Step `4`: Add all of the areas together

The second part is where they do the actual measurements and calculations. There is an organized sheet where they can record their measurements and the areas.

Class presentation

Invite a few groups to present their work. Ask them to talk about what kind of object they were working with and how they found the area. Ask them to describe some of the difficulties they faced when doing this activity.

You could ask students what kind of people would want to know about how much cardboard is needed to make these objects. Ask them to estimate what it might cost to make the packaging that they were working with. 

Surface Area of Prisms and Pyramids Practice

After students have completed their activity, it’s time for some independent practice! ByteLearn gives you access to tons of surface area of prisms and pyramids activities. Check out the online practice and assign to your students for classwork and/or homework!

View this practice

Area Of A Parallelogram Lesson Plan Area Of A Triangle Lesson Plan Nets Of Solids Lesson Plan

Fun teaching resources & tips to help you teach math with confidence

Exploring Surface Area: Hands-On Lesson!

One topic that I believe is too often rushed through, whether in pre-algebra or geometry, is the surface area of three dimensional objects . Because it is often seen as an “easy” topic where students are expected to simply “plug the numbers into the formula,” it can be easy to say, “Here’s the formula, GO.” And while this may work for some students, especially students who have a strong foundation in geometry vocabulary and a strong understanding of area in general, simply “plugging numbers into the formula” doesn’t come as easily to others. This hands-on investigation for exploring surface area of prisms and cylinders will help students understand surface area in a conceptual way, and requires NO memorizing of formulas!

One reason the surface area formulas cause problems for students is that they don’t know what all the variables mean. For example, when I taught surface area and volume to a group of eighth graders, the majority of whom did not speak English as their first language, all the symbols in the formulas did not make sense to them.

It can also be confusing for students (even if they speak English!) to understand the ‘B’ in surface area formulas (standing for ‘ area of the base ‘) because they want to input a single number rather than finding the area of the base.

Exploring Surface Area of Prisms and Cylinders:

Because of all the ways the technical formulas can be confusing for students, I prefer to teach students based solely on the understanding that surface area simply means the sum of the areas of all surfaces .

This allows students to think logically through solving any surface area problem, rather than relying on memorizing a formula or trying to correctly “plug in the numbers.”

This also helps minimize careless mistakes . The following lesson is meant to work as an introduction, focusing only on prisms and cylinders .

(But don’t worry, there’s  another lesson that covers pyramids and cones ! 🙂 )

{Click HERE to download the complete Surface Area of Prisms and Cylinders Lesson!}

I hope you find this lesson useful, and if so, grab the next surface area lesson, as well my volume exploration !

Happy Teaching!

Looks great!

I tried downloading the Surface Area Investigation multiple times but I only get a practice worksheet and its answer key. I would really like the Teaching Tips, Student handout and discussion questions, Discussion questions answer key and the Printable nets for a pyramid and cone…

Does anybody have any ideas on how to download the rest of this investigation?

Sorry for the delay in my response. You are correct, the product was linked to the wrong pdf, but the issue has been resolved. You can find this lesson here: https://mathgeekmama.com/product/surface-area-of-prisms-and-cylinders-exploration/

And the second part of surface area (pyramids and cones) here: https://mathgeekmama.com/product/surface-area-of-pyramids-and-cones-exploration/

You should have everything you need in those two downloads. Thank you for your patience!

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Lesson 6: Surface Area and Volume Problem Solving

Table of contents.

© Jose Marines/shutterstock

How would you describe your favourite place to someone who has never been there? That place might be your bedroom at home, a cabin at the lake, or an ancient castle from your imagination. You could definitely use photos and drawings; but you also need a way to describe the size or dimensions of your place.

As you have been working on your math and spending time around your home and school, maybe you have noticed some basic geometric shapes in your environment. A single structure or space, such as the pictured castle, often includes a variety of geometrical shapes—prisms, pyramids, cones, cylinders, and spheres, for example.

If you have a structure that is made of more than one shape, how could you calculate its surface area and volume? What strategies can you develop to investigate the surface area and volume of complex shapes?

At the end of this lesson, you will be able to do the following:

• Solve problems that involve surface area and the volume of 3-D objects.
• Determine an unknown dimension of a right cone, cylinder, prism, pyramid, or sphere, given the object’s surface area or volume and the remaining dimensions.
• Solve problems that involve surface area or volume, given a diagram of the composite 3-D object.
• Describe the relationship between the volumes of right cones and cylinders with the same height and base and right pyramids and prisms with the same height and base.

Lesson Questions

• Why is visualization important to the study of the surface area and volume of 3-D objects?
• How does changing the dimensions of an object affect its surface area and volume?

As you work through each lesson, complete all the questions and learning activities in your binder using paper and pencil, clearly labeling your work (they refer to this as your course folder). These include the Are you Ready, Try This, Share and Self Check questions. Check your work if answers are provided. Remember that these questions provide you with the practice and feedback that you need to successfully complete this course. Once you have completed all of the learning activities, take the Lesson Quiz. This is the assessment for each lesson and is located under the Assess tab or by using the Quizzes link under the Activities block.

** Note – Share questions may have to be done on your own depending on your learning situation**

Are You Ready?

Complete these questions in your course folder (binder). If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

  1.   In your own words, define the following:

        a.   surface area

        b.   volume

 2.  a. For each of the following 3-D shapes, determine the surface area.

b. Determine the volume for each of the 3-D shapes in part a.

Once you have completed these exercises to the best of your ability, use the provided answer link to check your work.

If you feel comfortable with the concepts covered in the questions, move forward to Discover. If you experienced difficulties or want more practice, use the resources in Refresher to review these important concepts before continuing through the lesson or contact your teacher.

The interactive applet titled “Exploring Surface Area, Volume, and Nets—Use It” can help you review many aspects of surface area and volume. You will find the interactive piece on the right-hand side of the website.

Select the “Use It” tab at the bottom of the activity to review the nets of all of the 3-D objects that were previously explored. The activity tests your ability to visualize the net of a given 3-D object.

Select the “Explore It” tab to review the surface area and volume formulas for all of the 3-D objects previously introduced. This applet also enables you to do a side-by-side comparison of two formulas. For example, you can use this feature to compare

• the surface area formula of a sphere and the volume formula for a sphere
• the volume formulas of a cylinder and a cone

You can also access a video that demonstrates how math is used in designing large inflatable shapes. On the left-hand side of the website, choose “Exploring Surface Area and Volume (Video Interactive)” to view the video.

How does adjusting the dimensions of a 3-D object affect its surface area and volume? In the following Math Lab, you will investigate this question.

Go to Math Lab: Surface Area and Volume Analysis , print it (or copy by hand) and complete it. Save your work in your course folder (binder).

Take a look at the results from Math Lab: Surface Area and Volume Analysis.

It seems reasonable to assume that as the dimensions of an object are doubled, the surface area and volume are also doubled. Yet the results of the investigation proved otherwise. In fact, you may have found that as the dimensions are doubled, the surface area is quadrupled , or increases by a factor of four. At the same time, the volume undergoes an increase by a factor of eight! Where do these numbers come from?

Review your lab results and see if you can see any patterns in the ratios.

You may want to extend the investigation by quadrupling each dimension; then recalculate the surface area or volume.

Develop an explanation for how to predict the increase in surface area or volume.

Use your explanation to predict how the surface area and volume will change when the dimensions of an object are increased by a factor of 7.

If possible, work with other students to see if you can develop a formula or pattern for the change in surface area and volume.

To check your work, use the link below.

© Jim Parkin/shutterstock

As you look around your surroundings, you may find objects that resemble the 3-D objects studied in the previous two lessons. A soup can, a box, and a ball are examples of cylinders, prisms, and spheres, respectively. A pylon used to alert motorists of traffic obstructions resembles a cone, and some games use pyramid-shaped dice.

While there are many examples of prisms, pyramids, cylinders, spheres, and cones, you may notice that many other objects are actually composites of two or more of these basic 3-D objects.

The image of the industrial propane tank is an example of a composite figure. Can you tell what 3-D objects are used in the tank’s design?

Glossary Terms

Throughout Module 1, you have been adding and saving math terms to “Glossary Terms” in your binder.

In this lesson the suggested terms for your glossary are

• composite figure
• hemispherical
• surface area to volume ratio

Return your updated “Glossary Terms” to your course folder (binder).

----------------------------------------

In complex problems that have more than one three-dimensional shape, it is a good strategy to break the problem into parts so you are dealing with only one shape at a time.

In the case of the propane storage tank shown in the photo, you may want to determine its volume by first determining the volume of the cylinder and then determining the volume of the hemispheres (or half-spheres) on each end.

--------------------------------------

The steps below will help you to solve a surface area or volume problem:

Step 1: Decide which 3-D object or objects can be used to model the problem.

Step 2: Draw a rough sketch of this 3-D object, and label its dimensions.

Step 3: Decide which formulas you will use. You may need to select more than one formula in the case of composite figures. You might also find that you will need to modify the formula to fit the problem.

Step 4: Substitute given values into your formulas to solve the problem.

Work through the following textbook examples that show how problems involving composite figures are solved. In the solutions, pay attention to

• the importance of an accurate drawing
• any preliminary calculations that need to be made
• the sequence of steps

Read “Example 1: Determining the Volume of a Composite Object” on pages 56 and 57 in your textbook. Then read “Example 2: Determining the Surface Area of a Composite Object” on page 57. Finally, read “Example 3: Solving a Problem Related to a Composite Object” on page 58.

© Sue Smith/shutterstock

SC 1. A grain storage bin has a diameter of 4.8 m. The height of the straight side wall is 10 m . The cone top has an additional height of 1.5 m .

• What is the total volume of this bin?
• Suppose you want to paint the outside of one of the grain storage bins. If the slant height of the conical roof is 2.83 m , then what total surface area needs to be painted?

SC 2. Mr. Vanilla charges 0.5 cents/cm 3 for his ice cream. How much would you pay for one spherical-shaped scoop of ice cream if a scoop of ice cream has a radius of 3.5 cm?

SC 3. The right cylinder and right cone shown have the same radius and volume. The cylinder has a height of 12 in. What is h , the height of the cone?

SC 4. A glass containing water is in the shape of a right circular cylinder with a radius of 3 cm. The height of the water in the glass is 10 cm.

• What is the volume of the water in the glass? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
• Five spherical marbles of equal size are dropped into the glass. The water in the glass rises to a height of 11 cm. What is the increase in the volume of the glass contents? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
• What is the volume of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
• What is the radius of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.

Compare your answers.

How did you do on the Self-Check questions?Complete TT1 for more practice:

Foundations and Pre-calculus Mathematics 10 (Pearson)

TT 1. Complete “Exercises” questions 3,  and 9 on pages 59 and 60.

Use the link below to check your answers to Try This 1.

Possible TT1 Solutions

There are a few myths about surface area and volume you may hear. These are all INCORRECT .

• Surface area can be found only for two-dimensional objects.
• Surface area is a concept used only in mathematics classes.
• You cannot measure the volume of some objects because they do not have “regular” lengths, widths, or heights.
• An object’s volume is greater in water than in air.

Math 10C Module 1 Lesson 6

Complete the lesson quiz posted under the Assess tab or using the Quizzes link under the Activities block. Also ensure your work in your binder (course folder) is complete. 

Up to this point in the project, you have all been working with your own place. For this lesson, you will do a common problem. You will consider the dimensions of a woodshed.

Of course, it is still a collaborative approach—you can still work together with a partner.

Go to the Unit 1 Project and complete the Lesson 6 portion of the project.

The ratio between an organism’s surface area and volume has an enormous impact on its biology. For example, a cell that is shaped like a sphere has a low surface area to volume (SA : V) ratio.

© Nancy Nehring/iStockphoto

iStockphoto/Thinkstock

Cells that are extended (e.g., cylinder) have much more membrane per unit of cytoplasm, which means these cells have more surface area for each bit of goo inside of them. Extending the outer surface of a cell into fingers, like an amoeba, or indentations, like the red blood cells shown above, can greatly increase the total surface area.

Interestingly, scientists are identifying the ratio between the surface area and volume as a crucial factor in work with nanotechnology. Take a look at the nanobot in the second visual of red blood cells. What do you suppose its function is?

To find out more, perform an Internet search using the following keywords: “nanotechnology,” “surface area,” “volume,” “nanobot,” and “red blood cells.”

Math 10C Module 1 Lesson 6 In this lesson you examined the following questions:

In this lesson you solved problems involving the surface area and volume of 3-D objects. You learned how to approach problems involving composite figures. You learned that it is important to visualize each problem before putting pencil to paper.

You also learned that doubling the dimensions of a 3-D object, such as a prism or a cone, actually increases its surface area by a factor of 4 and its volume by a factor of 8.

You also looked at common myths regarding surface area and volume. Through research, you were able to find explanations that dispelled these myths.

You are now ready to proceed to Assignment 1 and complete it.

Please log in to save materials. Log in

• 6th Grade Mathematics
• Problem-Solving

Education Standards

Maryland college and career ready math standards.

Learning Domain: Geometry

Standard: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Standard: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Common Core State Standards Math

Cluster: Solve real-world and mathematical problems involving area, surface area, and volume

Diagrams & Problem Solving Strategies

Lesson Overview

Students revise their packing plans based on teacher feedback and then take a quiz.

Students will use their knowledge of volume, area, and linear measurements to solve problems. They will draw diagrams to help them solve a problem and track and review their choice of problem-solving strategies.

Key Concepts

Concepts from previous lessons are integrated into this assessment task: finding the volume of rectangular prisms. Students apply their knowledge, review their work, and make revisions based on feedback from the teacher and their peers. This process creates a deeper understanding of the concepts.

Goals and Learning Objectives

• Apply your knowledge of the volume of rectangular prisms.
• Track and review your choice of strategy when problem-solving.

Lesson Guide

Return students' solutions to the Self Check task. If you have not added questions to individual pieces of work, write your list of questions on the board now. Students can then select questions appropriate to their own work.

Give students a few minutes to read over the feedback.

SWD: Students with disabilities may have a more challenging time identifying areas of improvement to target from the Intervention questions. Model explicitly for students (using an example project or student sample) how to self-assess areas of growth by using the appropriate Interventions in order to plan for revisions.

Review your work on the Self Check problem and think about the following questions.

• Can you make a diagram of your packing plan?
• Do you think there’s a way you could pack the truck to get even more bananas in?

Math Mission

Discuss the Math Mission. Students will review their work and make revisions to find the best packing plan.

Review your work based on the feedback to find the best packing plan.

Revise Your Work

Put students in pairs to revise their work. Encourage students to incorporate ideas from their partners into their work.

Support student problem solving.

• Try not to make suggestions that move students toward a particular approach to this task. Instead, ask questions that help students to clarify their thinking.
• What questions were you asked in the feedback?
• How could you and your partner work together to address one of those feedback questions?

If several students in the class are struggling with the same issue, you could write a relevant question on the board. You might also ask a student who has performed well on a particular part of the task to help a struggling student.

SWD: Make sure that all students have access to and can comprehend the information in the Interventions so that they can accurately interpret your assessment of their work.

Students with disabilities may benefit from support in understanding the expectations. Allow multiple means of representing the information in the Interventions (visual presentation of text, TTS, visual supports, etc.).

Create and provide an enhanced version of the Interventions with embedded text structures (labels, highlights, words in bold) to cue students to pay closer attention to particular terms.

ELL: Student partner work provides students with opportunities to negotiate meaning, share ideas, and communicate with others during problem solving. This type of language-rich environment encourages students to work together and learn from one another, which is essential for ELLs' learning.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Students must make sense of the task. They must plan and carry out a solution strategy. After they complete the task, they must go back and check that their solution makes sense. Students also have the opportunity to work with another student and try to understand his or her approach to the task.

Mathematical Practice 6: Attend to precision.

Students are asked to present their work in a clear, precise way. They should try to use correct mathematical language and symbols and to label their work carefully so it is easy to follow.

Interventions

Student has difficulty getting started.

• What feedback did you get?
• How can you use the feedback to revise your work?

Student works unsystematically.

• How can you check that you addressed all the feedback?

Student presents his or her work poorly.

• Is your work clear?
• Have you given enough explanation?

Student has a correct solution.

• Can you find a different packing plan?

Possible Answers

• The crates inside the truck form a 2.4 m by 3 m by 9.8 m prism, which has a volume of 70.56 m 3 . The truck bed is a 2.5 m by 3 m by 10 m prism, which has a volume of 75 m 3 . Only 4.44 m 3 of the truck will be empty.
• Explanations will vary. Possible answer: This is the best plan because it minimizes the amount of unused space. Other ways of packing the crates (as listed below) will leave bigger gaps, and you will not be able to ship the maximum amount of product in each crate or truck:
• Put the 2.4 m sides of the crates parallel to the 2.5 m side of the truck and the 1.5 m sides of the crates parallel to the 10 m side of the truck. This would fit a 2 crate by 1 crate by 6 crate prism, for a total of only 12 crates.
• Put the 1.4 m sides of the crates parallel to the 2.5 m side of the truck. Only 1 crate would fit in this direction, leaving a 1.1 m by 10 m by 3 m gap, which is 33 m 3 .
• Put the 1.5 m sides of the crates parallel to the 2.5 m side of the truck. Only 1 crate will fit in this direction, leaving a 1 m by 10 m by 3 m gap, which is 30 m 3 , plus a 0.2 m by 1.5 m by 10 m gap on the top, which is 3 m 3 , for a total of 33 m 3 .

Work with your partner to revise your work on the Self Check problem based on the questions from the Opening and feedback from your partner and teacher.

Self Check Problem:

You are working for a trucking company that transports bananas. You want to transport the greatest number of bananas in each truckload. The bananas fill boxes, which are packed into crates. The crates are then packed into a truck.

• How can you pack each crate to fit the greatest number of boxes of bananas?
• How can you pack the truck to fit the greatest number of crates?
• Based on your packing plan, how much space in the truck will not be filled?

Prepare a Presentation

Preparing for ways of thinking.

While students work with their partners, note different student approaches to the task.

• How do they organize their work?
• Do they notice if they have chosen a strategy that does not seem to be productive? If so, what do they do?
• Describe the feedback that you received from your teacher and your partner.
• Explain how you revised your work based on that feedback.

Make Connections

Organize a whole class discussion to consider issues arising from the work students did to revise their work. You may not have time to address all these issues, so focus discussion on the issues most important to your students.

Mathematics

Have students give their presentations. Include students whose strategies did not work so they can talk about how and when they realized their strategy did not work and what they did about it. Have students share the feedback they received and how they addressed those comments. Have students ask questions and make observations as they view each other's presentations.

Ask questions such as the following:

• What was the most difficult part of this task? How did you approach finding a solution?
• Did you and your partner ever disagree about the best packing plan? Did you resolve your differences of opinion? How?

SWD: Some students with disabilities may struggle with self-assessment; use your knowledge of student strengths and vulnerabilities to inform and create interventions you will put into place for this period of class time.

Performance Task

Ways of thinking: make connections.

• Take notes as your classmates present their work and discuss how they revised it.

As your classmates present, ask questions such as:

• Can you explain your strategy for solving the problem?
• Did you revise your work or your explanation based on any of the questions in the Opening? How?
• What part of the problem was the most difficult for you? Why?
• Was it difficult for you to revise your work? In what ways?

Reflect On Your Work

Have each student write a brief reflection before the end of the class. Review the reflections to find out what students would do differently if they started the Self Check task over again.

ELL: Asking students to reflect on their learning provides opportunities for ELLs to develop literacy in English and proficiency in mathematics. Make sure students use both academic and specialized mathematical language when reflecting on their learning at the end of each session. In addition, make sure students use their rubric to help them identify areas of improvement.

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

What I would do differently if I started the Self Check problem now is …

Volume of Prisms

Solid geometry is concerned with three-dimensional shapes. In these lessons, we will learn

• what is a prism?
• how to find the volume of prisms.
• how to solve word problems about prisms.

Related Pages Volume Formula Volume Of Prism Worksheet Volume Formulas Explained More Geometry Lessons

A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the bases of the prism. A prism is named after the shape of its base.

The other faces are in the shape of parallelograms. They are called lateral faces.

A right prism is a prism that has its bases perpendicular to its lateral surfaces. If the bases are not perpendicular to its lateral bases then it is called an oblique prism.

When we cut a prism parallel to the base, we get a cross section of a prism. The cross section has the same size and shape as the base.

What is a prism and distinguishes between a right prism and an oblique prism?

How to label the parts of a prism and how to distinguish between an oblique and a right prism?

Volume of a Prism

The volume of a right prism is given by the formula:

Volume = Area of base × height = Ah

where A is the area of the base and h is the height or length of the prism.

Worksheet to calculate volume of prisms and pyramids.

Example: Find the volume of the following right prism.

Solution: Volume = Ah = 25 cm 2 × 9 cm = 225 cm 3

Example: Find the volume of the following right prism

Solution: First, we need to calculate the area of the triangular base.

We would need to use Pythagorean theorem to calculate the height of the triangle.

h 2 + 3 2 = 5 2

Volume of prism = Ah = 12 cm 2 × 8 cm = 96 cm 3

How to find the volume of a rectangular and a triangular prism? Step 1: Find the area of the base. Step 2: Multiply the area of the base times the height.

How to find the volume of any prism, right or oblique using a general formula?

Word problems about volume of prisms

The following video shows how to solve a word problem involving the volume of prisms.

Example: Find the volume and capacity of a swimming pool which is made up of a rectangular and trapezoidal prism.

Use the given net to determine the surface area and volume of a triangular prism

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Course: 5th grade   >   Unit 11

• Measuring volume as area times length
• Volume as area of base times height
• Volume of a rectangular prism

Volume of rectangular prisms

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍