2. Volume and Surface Area of Prisms
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Chapter 10
2.
Volume and Surface Area of Prisms
The volume and surface area of prisms are essential concepts in geometry, applied in many real-life situations like construction and design. Rectangular and triangular prisms, as well as cubes, are common shapes for which these calculations are necessary. Volume shows how much space is inside a solid, while surface area helps determine the outer surface area, which is useful for estimating material needs. These calculations are crucial for anyone working with space, structure, or design, helping to make accurate decisions. Understanding how to calculate the volume and surface area of prisms makes it easier to apply these concepts in everyday tasks, from building structures to designing products.
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Lesson Settings & Tools
| | 19 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Volume and Surface Area of Prisms
Slide of 19
Three-dimensional figures like cubes, prisms, and pyramids, appear in different forms everywhere in the real world. Some numeric characteristics about these objects can be very useful for various aspects of daily life. This lesson will show how to calculate the volume and surface area of different kinds of prisms.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Choosing the Shape of a Dog House
Mark is building a model of the house of his dreams. He enjoys constructing different rooms and furniture from various materials. He also wants to build a dog house for his toy dog Bubbles. 
However, he cannot decide on the shape of the dog house. He knows it will be a prism, but should it be rectangular like a box or triangular like a tent? Given their dimensions, which dog house offers the most space for Bubbles?
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Discussion
Rectangular and Triangular Prisms
Prisms can be categorized by the shape of their bases. For example, a prism with a triangular base is called a triangular prism. A rectangular prism, on the other hand, has a rectangular base.
Discussion
Cube
A cube is a three-dimensional solid object bounded by six square faces with three meeting at each vertex. All square faces have the same side length. A cube is a specific type of rectangular prism.
Cubes have 6 faces, 12 edges, and 8 vertices.
Pop Quiz
Identifying the Type of a Prism
Determine whether the given solid is a triangular prism, a rectangular prism, or neither.
Discussion
Volume and Surface Area
There are two important characteristics that give information about 3D objects.
Concept
Volume
The volume of a solid is the measure of the amount of space inside the solid. It is the three-dimensional equivalent of the area of the figure. Volume is measured using cubic units, such as cubic meters, or m3. The applet below illustrates the volume of some solids. Move the slider to fill the solids.
Discussion
Surface Area
The surface area SA of a three-dimensional shape is the total area of all the surfaces of the shape. The lateral area LA is the total area of all lateral surfaces, the surfaces of the shape that are not bases. Lateral area can be defined for polyhedrons, cones, and cylinders. In these instances, the surface area of the shape can be calculated by the formula below.
SA=LA+Area of the Bases
The net of a solid can be helpful to visualize the surfaces of a three-dimensional figure.
Discussion
Volume of a Prism
Consider a prism with a base area B and height h.
The volume of the prism is calculated by multiplying the base's area by its height.
V=Bh
By Cavalieri's principle, this formula holds true for all types of prisms, including skewed ones.
Proof
Informal JustificationRecall that the volume of a solid is the measure of the amount of space inside the solid. Note that the top and bottom faces of the prism are always the same and parallel to each other.
Additionally, the prism is, so to speak, filled
with identical base areas that are stacked on top of each other to the height of the prism.
V=hB+B+B+…+B+B⇓V=B⋅h
Discussion
Surface Area of a Prism
Consider a prism with a height h, base area B, and base perimeter P.
The surface area of the prism is the sum of the two base areas and the lateral area, which can be calculated as the product of the base perimeter and the height of the prism.
SA=2B+Ph
Proof
The surface area of the prism can be seen as the sum of two separate parts: the lateral area and the combined area of the two identical bases. 
Notice that the lateral surface consists of rectangles equal to the number of sides in the base. The pentagonal prism shown here has five lateral faces because a pentagon has five sides. The area of each rectangular lateral face is the product of its sides a and h.
Surface Area=Lateral Area+2⋅Base
The base area, often symbolized as B, can be substituted into the equation. Surface Area=Lateral Area+2B
To determine the lateral area of the prism, consider a net of the given prism. Let a be the length of the side of the base, assuming that it is a regular polygon. One Lateral FaceA=ah
If there are n rectangular lateral faces in a prism, then the total lateral area is the product of n and the area of one lateral face.
Lateral Areanah
Notice that na is the perimeter of the base, which is often denoted by P. Then, the lateral area can be expressed as follows.
Lateral Area=Ph
Therefore, the formula for the surface area is obtained. Surface AreaSA==Lateral AreaPh++2⋅Base2B
Note that although this proof is written for a regular prism, it is also true for a non-regular prism.
Discussion
Volume and Surface Area of a Rectangular Prism
It is possible to derive two specific formulas for the volume and surface area of a rectangular prism.
Rule
Volume of a Rectangular Prism
Consider a rectangular prism with a width w, length ℓ, and height h.
The volume of the prism is calculated by multiplying the area of the base by the height of the prism.
V=wℓh
Proof
The volume of a prism can found by multiplying the base area B and height h.
V=B⋅h
The base of a rectangular prism is a rectangle. The area of a rectangle is the product of its width w and length ℓ.
B=wℓ
Substitute this expression for B in the first formula.
V=B⋅h⇓V=wℓh
This process produces the formula for the volume of a rectangular prism.
Discussion
Surface Area of a Rectangular Prism
Consider a rectangular prism with a height h, base area B, and base perimeter P.
The surface area of the rectangular prism can be found using the following formula.
SA=2(wℓ+hℓ+hw)
Proof
Recall that the surface area of a prism is the sum of the lateral area and the combined areas of the two identical bases.
This process results in the formula for the surface area of a rectangular prism.
SA=Ph+2B
Here P is the perimeter of the base, h is the height of the prism, and B is area of one base. The base of a rectangular prism is a rectangle, so its area is the product of its width and length.
B=wℓ
The perimeter of a rectangle can be found using the following formula.
P=2(w+ℓ)
Now the expressions for P and B can be substituted into the formula presented in the beginning and the whole expression simplified.
SA=Ph+2B
SubstituteII
P=2(w+ℓ), B=wℓ
SA=2(w+ℓ)h+2wℓ
Distr
Distribute h
SA=2(wh+ℓh)+2wℓ
FactorOut
Factor out 2
SA=2(wh+ℓh+wℓ)
Discussion
Volume of a Cube
Consider a cube with a side length s.
The volume V of the cube can be calculated by raising the side length s to the power of 3, or cubing
it.
V=s3
Proof
The volume of a prism is calculated by multiplying the area of one base by the height. For rectangular prisms, this value is found by multiplying the area of the rectangular base by the height of the prism.
The formula for the volume of a cube can therefore be derived by substituting s into the standard equation for the volume of a rectangular prism.
Volume of a Rectangular Prism{V=B⋅hB=ℓ⋅w⇒V=ℓ⋅w⋅h
A square is a special type of rectangle whose length and width are equal, so the area of a square is found by calculating the square of the side length. Similarly, a cube is a special type of rectangular prism whose length, width, and height are equal. Discussion
Surface Area of a Cube
Consider a cube with a side length s.
The surface area of the cube is given by the following formula.
SA=6s2
Proof
The surface area of a prism is the sum of the lateral area and the combined areas of the two identical bases.
S=Ph+2B
Since the base of a cube is a square with a side length s, its area is equal to the square of s.
B=s2
The perimeter of the square base can be calculated as the product of 4 and s.
P=4s
Finally, the height of the cube is also equal to the side length s. h=s
Now these values will be substituted into the formula and the resulting equation simplified.
SA=Ph+2B
SubstituteValues
Substitute values
SA=4s(s)+2s2
ProdToPowTwoFac
a⋅a=a2
SA=4s2+2s2
AddTerms
Add terms
SA=6s2
Example
Renovations to the Bedroom
Mark wants to start painting the walls in the model bedroom and fill it with furniture. To understand how much paint he needs and how much space is available, Mark has to determine the surface area and the volume of the room. He starts by measuring the dimensions of the room.
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a What is the volume of the bedroom?
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b What is the surface area of the bedroom?
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Hint
a Use the formula for the volume of a rectangular prism.
b Use the formula for the surface area of a rectangular prism.
Solution
a The bedroom has the shape of a rectangular prism, so its volume can be calculated by using the formula for the volume of a rectangular prism.
V=wℓh
Here, w is the width, ℓ is the length, and h is the height of the prism. The diagram shows that the width of the room is 3.5 inches, the length is 4 inches, and the height is 2.7 inches. Substitute these values into the formula and evaluate.
The volume of the room is 37.8 cubic inches.
b The surface area of the bedroom can be found by using the formula for the surface area of a rectangular prism.
SA=2(wℓ+hℓ+hw)
Substitute w with 3.5, ℓ with 4, and h with 2.7 into the formula and simplify.
SA=2(wℓ+hℓ+hw)
SubstituteValues
Substitute values
SA=2((3.5)(4)+(2.7)(4)+(2.7)(3.5))
Multiply
Multiply
SA=2(14+10.8+9.45)
AddTerms
Add terms
SA=2(34.25)
Multiply
Multiply
SA=68.5
Example
Working on the Closet
Later in the day, Mark decides to work on the bedroom closet, which also has a rectangular prism shape. His measurements show that the length of the closet is 1.4 inches and its height is 2 inches, but he is not able to measure the width since the closet was already installed in the wall.
However, Mark remembers that the volume of the closet is 3.5 cubic inches. What is the surface area of the closet?
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Hint
Find the width of the prism by using the known volume of the prism and the formula for the volume of a rectangular prism. Then, use the formula for the surface area of a rectangular prism.
Solution
The surface area of a rectangular prism can be found by using the following formula.
The width w of the closet is 1.25 inches. Now there is enough information to calculate the surface area of the closet. Substitute the values into the formula for the surface area of a rectangular prism and evaluate SA.
The surface area of the closet is 14.1 square inches.
SA=2(wℓ+hℓ+wh)
Here, w is the width, ℓ is the length, and h is the height of the prism. The length and the height of the prism are known but the width is not. To find the width of the prism, use the fact that the volume of the prism is 3.5 cubic inches.
V=3.5 in3
Recall the formula for the volume of a rectangular prism.
V=wℓh
Substitute V with 3.5, ℓ with 1.4, and h with 2 into the formula and solve it for w.
V=wℓh
SubstituteValues
Substitute values
3.5=w(1.4)(2)
Multiply
Multiply
3.5=w(2.8)
CommutativePropMult
Commutative Property of Multiplication
3.5=2.8w
RearrangeEqn
Rearrange equation
2.8w=3.5
DivEqn
LHS/2.8=RHS/2.8
w=2.83.5
UseCalc
Use a calculator
w=1.25
SA=2(wℓ+hℓ+wh)
SubstituteValues
Substitute values
SA=2((1.25)(1.4)+(2)(1.4)+(1.25)(2))
Multiply
Multiply
SA=2(1.75+2.8+2.5)
AddTerms
Add terms
SA=2(7.05)
Multiply
Multiply
SA=14.1
Example
Designing the Attic
After mostly finishing with the first floor, Mark begins brainstorming ideas for the design of the attic. He needs to determine how much space there is in the attic and the areas of the walls, floor, and ceiling.
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a What is the volume of the attic?
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b What is the surface area of the attic?
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Hint
a Use the formula for the volume of a prism.
b Use the formula for the surface area of a prism.
Solution
a The attic has the shape of a triangular prism, which means that its volume can be calculated by using the formula for the volume of a prism.
V=Bh
Here, B is the area of one base and h is the height of the prism. The base of a triangular prism is a triangle, so recall the formula for the area of a triangle.
A=21bh
The diagram shows that the base of the triangle is 7 inches and its height is 2.3 inches. Substitute 7 for b and 2.3 for h in the formula and solve for A.
A=21bh
SubstituteII
b=7, h=2.3
A=21(7)(2.3)
Multiply
Multiply
A=21(16.1)
MoveRightFacToNumOne
b1⋅a=ba
A=216.1
CalcQuot
Calculate quotient
A=8.05
b The surface area of the attic can be found by using the formula for the surface area of a prism.
SA=2B+Ph
In this formula, B is the area of one base, P is the perimeter of the base, and h is the height of the prism. The lengths of all the sides of the triangular base are given on the diagram. Find the perimeter of the triangle by adding all the side lengths together.
P=7+4.2+4.2=15.4 in.
In Part B it was found that the base area is 8.05 square inches. Substitute B=8.05, P=15.4, and h=9 into the formula for the surface area of a prism and evaluate SA.
SA=2B+Ph
SubstituteValues
Substitute values
SA=2(8.05)+(15.4)(9)
Multiply
Multiply
SA=16.1+138.6
AddTerms
Add terms
SA=154.7
Example
The Space Next to the Bathroom
Finally, Mark noticed that there is some empty space above the living room next to the upstairs bathroom. He wants to potentially turn in into another room. Mark knows that the surface area of that room would be 79.8 square inches.
What would the volume of this potential room be?
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Hint
Use the formula for the surface area of a prism and the given surface area to find the height of the prism. Then, find the volume of the prism.
Solution
The potential room has the shape of a triangular prism. Recall the formula for the surface area of a prism.
The area of the triangular base of the prism is 5.4 square inches. Next, add the side lengths of the triangle to calculate the perimeter of the base of the prism.
The height of the prism is 6 inches. Finally, find the volume of the prism by using the following formula.
SA=2B+Ph
Here, B is the area of one base, P is the perimeter of the base, and h is the height of the prism. The base of the prism is a right triangle, so recall the formula for the area of a triangle.
A=21bh
Here b is the base of the triangle and h is its height. Substitute 4 for b and 2.7 for h into the formula and find the area of the triangle.
A=21bh
SubstituteII
b=4, h=2.7
A=21(4)(2.7)
Multiply
Multiply
A=21(10.8)
MoveRightFacToNumOne
b1⋅a=ba
A=210.8
CalcQuot
Calculate quotient
A=5.4
P=4+2.7+4.8=11.5 in.
Now every value in the formula for the surface area of a triangular prism is known except for its height h. Mark knows that the surface area of the potential room is 79.8 square inches. Substitute 79.8 for SA, 5.4 for B, and 11.5 for P into the formula and solve for h.
SA=2B+Ph
SubstituteValues
Substitute values
79.8=2(5.4)+(11.5)h
Multiply
Multiply
79.8=10.8+11.5h
SubEqn
LHS−10.8=RHS−10.8
69=11.5h
RearrangeEqn
Rearrange equation
11.5h=69
DivEqn
LHS/11.5=RHS/11.5
h=6
V=Bh
Substitute 5.4 for B and 6 for h, then evaluate for V.
The volume of the potential room is 32.4 cubic inches.
Pop Quiz
Calculating the Volume or Surface Area of a Prism
Calculate the volume or surface area of the triangular or rectangular prism given the base area B, the height, and the side length of the base. The base is always a regular polygon. Round the answer to one decimal place if necessary.
Closure
Calculating the Volume of the Dog House
As mentioned before, Mark is building a model of the house of his dreams. He also wants to build a dog house for his toy dog Bubbles. 
However, he cannot decide on the shape of the dog house. He knows it will be a prism, but should it be rectangular like a box or triangular like a tent? Given their dimensions, which dog house offers the most space for Bubbles?
External credits: @pikisuperstar
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Hint
Use the general formula for the volume of a prism to calculate the volume of the triangular prism. Then, use the formula for the volume of a rectangular prism to find the volume of the rectangular prism. Compare the volumes.
Solution
There are two possible shapes of the dog house for Bubbles. To determine which one offers the most space for the dog, each of their volumes should be calculated.
Triangular Prism
The volume of a triangular prism can be found by using the general formula for the volume of a prism.V=Bh
Here, B is the area of one base and h is the height of the prism. This means that the area of the base must be found first to apply this formula. The base of this prism is a triangle, which means that its area is half the product of its base b and height h.
A=21bh
The diagram shows that the triangle has a height of 1 inch and a base length of 1.2 inches. This information can be used in the formula to calculate the area.
A=21bh
SubstituteII
h=1, b=1.2
A=21(1)(1.2)
OneMult
1⋅a=a
A=21(1.2)
MoveRightFacToNumOne
b1⋅a=ba
A=21.2
CalcQuot
Calculate quotient
A=0.6
Rectangular Prism
The volume of a rectangular prism can be calculated using the following formula.V=wℓh
Here, w is the width, ℓ is the length, and h is the height of the prism. The diagram shows that the width of the prism is 0.8 inches, the length is 1 inch, and the height is 2 inches. Substitute these values into the formula.
The volume of the rectangular prism is 1.6 cubic inches. Comparison
Finally, the volumes of the prisms can be compared.
| Volume of the Triangular Prism | Volume of the Rectangular Prism |
|---|---|
| 1.2 in.3 | 1.6 in.3 |
The volume of the rectangular prism is greater than the volume of the triangular prism. This means that the dog house in the shape of the rectangular prism is more spacious, so Mark should choose that one.
Volume and Surface Area of Prisms
Exercise 2.1
The toy company Mini Beach Makers makes rectangular sandboxes that measure 7 feet by 6 feet by 1.2 feet. Ali's father buys a sandbox and 45 cubic feet of sand for his children.
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We need to figure out whether 45 cubic feet of sand will completely fill up a sandbox that measures 7 feet by 6 feet by 1.2 feet. Let's start by drawing the sandbox!
The volume of a three-dimensional figure is the measure of space that it occupies expressed in cubic units. Notice that the volume of the sandbox is equal to the volume of sand that will fill the sandbox. Since we know that the sandbox is rectangular, we can find its volume V by multiplying its length l, width w, and height h. V= l w h For this sandbox, l = 7, w= 6, and h= 1.2. Let's substitute these values into the formula to calculate the volume of the sandbox!
The volume of the sandbox is 50.4 cubic feet. Now we can compare this volume with the volume of sand purchased by Ali's father. ccc Volume of & & Volume of sand purchased the sandbox & & by the customer [0.5em] 50.4ft^3 & > & 45ft^3 The volume of sand purchased by Ali's father is less than the volume of the sandbox. This means that the sand will not completely fill up the sandbox.
We are asked to determine the surface area of the sandbox that should be painted yellow. Note that the sandbox does not have a top and the bottom does not have to be painted. This means that the only part of the sandbox that needs to be painted are the lateral surfaces. In other words, the area to be painted is equal to the lateral area of the rectangular prism.
\begin{gathered}
S_\text{sandbox}=Ph
\end{gathered}
The lateral area of a prism is the product of the perimeter of the base P and the height of the prism h. The base of the sandbox is a rectangle with a length of 7 feet and a width of 6 feet. Its perimeter can be found by the following formula.
P=2(w+l)
Let's substitute l= 7 and w= 6 into the formula to find P.
The perimeter of the sandbox is 26 feet. Let's multiply it by the height of 1.2 feet to calculate the surface area of the lateral faces of the sandbox.
We found that Ali's father needs to paint an area of 31.2 square feet.
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Jordan is mailing her aunt a package.
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We can determine the amount of cardboard used to create the shipping container by finding the surface area of the package. We can see that it has the shape of a triangular prism.
Let's recall the formula for the surface area of a prism. S=2B+Ph Here, B is the base area, P is the base perimeter, and h is the height of the prism. Let's start by calculating the base area B. Since the base is a triangle, we can use the formula for the area of a triangle. A=1/2bh From the diagram, we know that b= 6 and h= 4.5. Let's substitute these values into the formula and solve for A.
The area of the triangular base of the prism is 13.5 square inches. Next, let's find the perimeter of the base by adding all the side lengths together. P=5.4+5.4+6=16.8inches Now we have enough information to find the surface area of the prism. Let's substitute 13.5 for B, 16.8 for P, and 14 for h into the formula for the surface area of a prism and evaluate S.
We found that the surface area of the prism is 262.2 square inches. This means that Jordan used 262.2 square inches of cardboard to create the shipping container.
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Dylan is constructing a die. He wants the side of the die to be 4 centimeters long.
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We need to find how much empty space there will be inside the box. The die is in the shape of a cube. Let's recall that the volume of a cube with a side length s can be found by raising s to the power of 3. V=s^3 We know that the side length of the die Dylan is making is 4 centimeters. Let's substitute 4 for s and evaluate V.
The volume of the die — and the amount of empty space inside the box — is 64 cubic centimeters.
Now we can determine how much red paper material Dylan needs to make the die by calculating the surface area of the die. Let's remember the formula for the surface area of a cube. S=6s^2 Let's substitute 4 for s again and solve for S.
The surface area of the die is 96 square centimeters, which means that Dylan needs 96 square centimeters of red paper material.
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Find the surface area of the regular hexagonal prism.
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We want to find the surface area of the given regular hexagonal prism.
Remember that surface area is the total area of all the surfaces of a three-dimensional figure. The surface of a solid includes lateral surfaces and bases. Let's draw a net of the prism to see all surfaces of the figure.
The surface of the figure includes 6 identical rectangular lateral faces and 2 identical hexagons for its bases. The sum of the area of these rectangles and hexagons gives us the total surface area of the given figure. Let's calculate the area of each figure one at a time.
Rectangles
We can find the area of a rectangle by multiplying its length by its width. In this case, the length is 13 centimeters and the width is 5 centimeters.
Let's multiply these dimensions to find the area! A_r=13 * 5 ⇓ A_r= 65cm^2 The area of each identical rectangle is 65 square centimeters.
Hexagons
Notice that we can divide a regular hexagon into 6 identical triangles.
This allows us to calculate the area of the hexagonal base by multiplying the area of one triangle by 6. Let's use the formula for the area of a triangle! A_t=1/2bh In this formula, b is the base length and h is the height of the triangle. In our case, the base is 5 centimeters and the height is 4.3 centimeters.
The area of each triangle is 10.75 square centimeters. Now, we will multiply this number by 6 to determine the area of the hexagonal base B. B=6(10.75) ⇓ B= 64.5cm^2 The area of each bases is 64.5 square centimeters.
Surface Area
Now that we have the area of all the faces, we can calculate the surface area of the prism by adding all of them together. Since there are 6 identical lateral rectangular faces, we can multiply the area of the rectangle by 6. Similarly, we can multiply the area of the hexagon by 2 as there are 2 hexagonal bases. SA= & 6* 65+ 2 * 64.5 ⇓ & SA= & 519cm^2 The surface area of the regular hexagonal prism is 519 square centimeters.
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Volume and Surface Area of Prisms
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