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.
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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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{"type":"choice","form":{"alts":["Triangular Prism","Rectangular Prism"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
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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.
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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.
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Identifying the Type of a Prism
Determine whether the given solid is a triangular prism, a rectangular prism, or neither.
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Volume and Surface Area
There are two important characteristics that give information about 3D objects.
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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 m^3. The applet below illustrates the volume of some solids. Move the slider to fill the solids.
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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.
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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.
This means that the volume of the prism can be calculated as the sum of all these base areas. The number of bases is equal to the height of the prism. Therefore, the volume of a prism equals the product of its base area and height. V=B+B+B+...+B+B_h ⇓ V=B* h
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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. 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.
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. One Lateral Face A=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 Area nah 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.
ccccc Surface Area & = & Lateral Area & + & 2* Base [0.8em] SA & =& Ph & + & 2B
Note that although this proof is written for a regular prism, it is also true for a non-regular prism.
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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.
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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 = wl h
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= wl h This process produces the formula for the volume of a rectangular prism.
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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. 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.
This process results in the formula for the surface area of a rectangular prism.
SA=Ph+2B
SubstituteII
P= 2(w+l), B= wl
SA= 2(w+l)h+2 wl
Distr
Distribute h
SA=2(wh+l h)+2wl
FactorOut
Factor out 2
SA=2(wh+l h+wl)
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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=s^3
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. Volume of aRectangular Prism V= B* h B=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.
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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=6s^2
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=s^2
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= P h+2 B
SubstituteValues
Substitute values
SA= 4s( s)+2 s^2
ProdToPowTwoFac
a* a=a^2
SA=4s^2+2s^2
AddTerms
Add terms
SA=6s^2
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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.
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)
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
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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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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"square inches","answer":{"text":["14.1"]}}
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.
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 in^3
Recall the formula for the volume of a rectangular prism.
V=wl h
Substitute V with 3.5, l with 1.4, and h with 2 into the formula and solve it for w.
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.
V=wl h
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=3.5/2.8
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
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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.
A=1/2bh
SubstituteII
b= 7, h= 2.3
A=1/2( 7)( 2.3)
Multiply
Multiply
A=1/2(16.1)
MoveRightFacToNumOne
1/b* a = a/b
A=16.1/2
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
SubstituteValues
Substitute values
SA=2( 8.05)+( 15.4)( 9)
Multiply
Multiply
SA=16.1+138.6
AddTerms
Add terms
SA=154.7
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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.
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=1/2bh
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.
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.
P=4+2.7+4.8= 11.5in.
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.
The height of the prism is 6 inches. Finally, find the volume of the prism by using the following formula.
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.
A=1/2bh
SubstituteII
b= 4, h= 2.7
A=1/2( 4)( 2.7)
Multiply
Multiply
A=1/2(10.8)
MoveRightFacToNumOne
1/b* a = a/b
A=10.8/2
CalcQuot
Calculate quotient
A=5.4
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
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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.
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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
{"type":"choice","form":{"alts":["Triangular Prism","Rectangular Prism"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
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=1/2bh 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=1/2bh
SubstituteII
h= 1, b= 1.2
A=1/2( 1)( 1.2)
OneMult
1* a=a
A=1/2(1.2)
MoveRightFacToNumOne
1/b* a = a/b
A=1.2/2
CalcQuot
Calculate quotient
A=0.6
V_\text{t}=Bh
SubstituteII
B= 0.6, h= 2
V_\text{t}=({\color{#FF0000}{0.6}})({\color{#A800DD}{2}})
Multiply
Multiply
V_\text{t}=1.2
Rectangular Prism
The volume of a rectangular prism can be calculated using the following formula. V=wl h 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.V_\text{r}=w\ell h
SubstituteValues
Substitute values
V_\text{r}=({\color{#FF00FF}{0.8}})({\color{#FD9000}{1}})({\color{#A800DD}{2}})
Multiply
Multiply
V_\text{r}=1.6
Comparison
Finally, the volumes of the prisms can be compared.
| Volume of the Triangular Prism | Volume of the Rectangular Prism |
|---|---|
| 1.2in.^3 | 1.6in.^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 1.1
Match each name to a figure.
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We are given four different three-dimensional figures. Let's analyze each of them to determine its name.
Option A
Consider the first given figure.
We can see that this figure has a rectangular base and four triangular lateral faces that have a common vertex. This description matches the definition of a pyramid.
Option B
Let's take a look at the second given figure.
We can see that the figure has two rectangular parallel bases that are connected by four rectangular lateral faces. This description matches the definition of a rectangular prism.
Option C
Let's analyze the third given figure.
This figure has six identical square faces. This information corresponds to the definition of a cube.
Option D
Consider the last given figure.
We can see that it has two triangular parallel bases connected by three rectangular lateral faces. This matches the definition of a triangular prism.
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Consider a cube with a side length of 7 feet.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"cubic feet","answer":{"text":["343"]}}
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We are asked to calculate the volume of a cube. Let's start by recalling that the volume of a cube is calculated by raising the side length to the power of 3. V=s^3 We know that the side length of the cube is 7 feet. Let's substitute this value into the formula and evaluate V.
The volume of the cube is 343 cubic feet.
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Consider a cube with a side length of 9 inches.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"square inches","answer":{"text":["486"]}}
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We are asked to calculate the surface area of the cube. Let's start by recalling the formula for the surface area of a cube. S=6s^2 We know that the side length of the cube is 9 inches. Substitute this value into the formula and evaluate S.
The surface area of the cube is 486 square inches.
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Consider a rectangular prism with the given dimensions.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"cubic meters","answer":{"text":["60"]}}
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We are asked to find the volume of the given prism. Let's start by recalling that the volume of a rectangular prism is calculated as the product of the width w, the length l, and the height h of the prism. V=wl h From the diagram, we know that the width of the prism is 3 meters, the length is 4 meters, and the height is 5 meters. Let's substitute these values into the formula and evaluate V.
The volume of the prism is 60 cubic meters.
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Consider a triangular prism with the given dimensions.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"cubic inches","answer":{"text":["2160"]}}
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We are asked to find the volume of the given prism. Let's start by recalling that the volume of a prism is calculated as the product of the base area B and the height of the prism h. V=Bh From the diagram, we can see that the base of the prism is a right triangle with a height h of 12 inches and a base b that is 15 inches long. We can calculate its area by using the following formula. A=1/2bh Substitute 15 for b and 12 for h and solve for A.
We found that the base area of the prism is 90 square inches. Now we have enough information to calculate the volume of the prism. Let's substitute 90 for B and 24 for h into the formula for the volume of a prism and evaluate V.
The volume of the prism is 2160 cubic inches.
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Consider a rectangular prism with the given dimensions.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"square centimeters","answer":{"text":["972"]}}
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We are asked to calculate the surface area of the prism. Let's start by recalling the formula for the surface area of a rectangular prism. S=2(wl+hl+wh) Here, w is the width, l is the length, and h is the height of the prism. From the diagram, we know that the width of the prism is 9 centimeters, the length is 18 centimeters, and the height is 12 centimeters. Let's substitute these values into the formula and evaluate S.
We found that the surface area of the prism is 972 square centimeters.
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Consider a triangular prism with the given dimensions.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"square millimeters","answer":{"text":["204"]}}
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We are asked to calculate the surface area of the prism. Let's begin by recalling the formula for the surface area of a prism. S=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. We can see that the base of the prism is a triangle.
The area of a triangle with a height h and a base length b is calculated as half the product of h and b. A=1/2bh Let's substitute 8 for b and 3 for h into this formula and solve for A.
The area of the triangle — and, consequently, the base area of the prism — is 12 square millimeters. Next,let's calculate the perimeter of the triangle by adding all the side lengths of the triangle together. P=8+5+5=18mm Finally, we will substitute 12 for B, 18 for P, and 10 for h into the formula for the surface area of a prism. Then we can evaluate S. Let's do it!
The surface area of the given triangular prism is 204 square millimeters.
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Volume and Surface Area of Prisms
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