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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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?
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{"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 3.1
Izabella made a figure by placing a cube with 11-centimeter sides on top of another cube with 13-centimeter sides.
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We are given the following figure and are asked to find the surface area. Let's start by analyzing the diagram.
The figure consists of two cubes. We can find their surface areas to get the desired area to be painted. However, notice that the smaller cube's lower base is adjacent to part of the larger cube's upper base. This means that the figure's surface area does not include the area of the smaller cube's lower base and an equally large part of the larger cube's upper base.
The surface area of the figure is the surface area of the smaller cube plus the surface area of the larger cube minus twice the area of the smaller cube's base. Let's find these areas one at a time, starting with the smaller cube. Recall that the surface area of a cube with a side length s can be calculated by using the following formula. S = 6s^2 Let's substitute 11 for s into the formula and find the surface area of the smaller cube.
The surface area of the smaller cube is 726 square centimeters. Next, let's calculate the surface area of the larger cube. This time, s will equal 13.
The surface area of the larger cube is 1014 square centimeters. Finally, we will calculate the area of the smaller cube's base, which is a square with the side length of 11 centimeters.
We can do this by squaring the side length of 11 centimeters. \begin{gathered} A_\text{square}={\color{#0000FF}{11}}^2={\color{#FF0000}{121}}\text{ cm}^2 \end{gathered} Now we can calculate the total surface area of the figure, which is the surface area of the smaller cube plus the surface area of the larger cube minus twice the area of the smaller cube's base. \begin{gathered} S_\text{figure} = {\color{#A800DD}{726}} + {\color{#FF00FF}{1014}} - 2 \times{\color{#FF0000}{121}} = 1498\text{ cm}^2 \end{gathered} The total surface area of the figure to be painted is 1498 square centimeters.
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A school locker at Harmony High School has the shape of a rectangular prism with a height of 26 inches, width of 12 inches, and length of 14 inches.
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We are given the dimensions of the school locker in a shape of a rectangular prism.
We want to determine how much the surface area of the locker increase if the height of the locker were increased by 2 inches. To do so, we can calculate the surface area of the locker both before and after increasing the height.
Original Locker
Remember that surface area is the total area of all the surfaces of a three-dimensional figure. Let's begin by recalling the formula for the surface area of a rectangular prism. S=2(wl+hl+wh) We can substitute w= 12, l= 14, and h= 26 into the formula and evaluate S. Let's do it!
We found that the surface area of the original locker is 1688 square inches.
Taller Locker
The height of the locker increases by 2 inches. Let's add 2 to the initial height to find the new one. h=26+2=28inches Now we can substitute w= 12, l= 14, and h= 28 into the surface area formula to find the surface area of the taller locker.
The surface area of the taller locker is 1792 square inches.
Comparison
Finally, we will find the difference between the increased surface area and the original surface area. 1792-1688=104 in^2 Therefore, the surface area of the locker increased by 104 square inches.
We are asked to find the decrease in volume of the locker after the decrease in its length. Let's begin by remembering the formula for the volume of a rectangular prism. V=wl h We can find the original volume of the locker by substituting w= 12, l= 14, and h= 26 into the formula. After the length of the locker is decreased, the new length would be 14-2= 12 inches. Let's use these values to find both the original and decreased volumes.
| Original Locker | Shallower Locker | |
|---|---|---|
| Dimensions | w= 12, l= 14, h= 26 | w= 12, l= 12, h= 26 |
| Substitute | V_\text{or}=({\color{#0000FF}{12}})({\color{#009600}{14}})({\color{#A800DD}{26}}) | V_\text{dec}=({\color{#0000FF}{12}})({\color{#FD9000}{12}})({\color{#A800DD}{26}}) |
| Multiply | V_\text{or}=4368 | V_\text{dec}=3744 |
Now that we know both volumes, we can calculate their difference. 4368-3744=624in^3 Therefore, the volume of the locker would decrease by 624 cubic inches.
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Volume and Surface Area of Prisms
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