Volume and Surface Area of Prisms - Geometry in Three Dimensions (Pre-Algebra)

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.

The house and the dog with two possible kennels. The dimensions of the triangular prism kennel: the height of the base is 1 in, the side length of the base is 1.2 in, the height of the prism is 2 in. The dimensions of the rectangular prism kennel: the base width is 0.8 in, the base length is 1 in, the height of the prism is 2 in

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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?

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.

A rectangular prism has a unique name if its bases are squares and its lateral edges are the same length as the square's sides.

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 $3 D$ 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 $m^{3} .$ The applet below illustrates the volume of some solids. Move the slider to fill the solids.

Discussion

Surface Area

The surface area $S A$ of a three-dimensional shape is the total area of all the surfaces of the shape. The lateral area $L A$ 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.

S A = L A + 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 $h .$

A prism with the area base B and height h

The volume of the prism is calculated by multiplying the base's area by its height.

$V = B h$

By Cavalieri's principle, this formula holds true for all types of prisms, including skewed ones.

Proof

Informal Justification

Recall 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 = h B + B + B + \dots + B + B ⇓ V = B \cdot 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.

$S A = 2 B + P h$

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 \cdot Base

The base area, often symbolized as

B,

can be substituted into the equation.

Surface Area = Lateral Area + 2 B

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 .

\underline{One Lateral Face} A = a h

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.

\underline{Lateral Area} n a h

Notice that

n a

is the perimeter of the base, which is often denoted by

P .

Then, the lateral area can be expressed as follows.

Lateral Area = P h

Therefore, the formula for the surface area is obtained.

Surface Area S A = = Lateral Area P h + + 2 \cdot Base 2 B

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 $h .$

A prism with the 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.

$V = w ℓ h$

Proof

The volume of a prism can found by multiplying the base area

B

and height

h .

V = B \cdot 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 = w ℓ

Substitute this expression for

B

in the first formula.

V = B \cdot h ⇓ V = w ℓ h

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 $P .$

A rectangular prism with the width w, length l, and height h

The surface area of the rectangular prism can be found using the following formula.

$S A = 2 (w ℓ + h ℓ + h w)$

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.

S A = P h + 2 B

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 = w ℓ

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.

S A = P h + 2 B

SubstituteII

$P = 2 (w + ℓ)$ , $B = w ℓ$

S A = 2 (w + ℓ) h + 2 w ℓ

Distr

Distribute $h$

S A = 2 (w h + ℓ h) + 2 w ℓ

FactorOut

Factor out $2$

S A = 2 (w h + ℓ h + w ℓ)

This process results in the formula for the surface area of a rectangular prism.

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.

$V = s^{3}$

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 a Rectangular Prism {V = B \cdot h B = ℓ \cdot w \Rightarrow V = ℓ \cdot w \cdot 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.

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.

V = ℓ \cdot w \cdot h

SubstituteValues

Substitute values

V = s \cdot s \cdot s

ProdToPowThreeFac

$a \cdot a \cdot a = a^{3}$

V = s^{3}

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.

$S A = 6 s^{2}$

Proof

The surface area of a prism is the sum of the lateral area and the combined areas of the two identical bases.

S = P h + 2 B

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 = 4 s

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.

S A = P h + 2 B

SubstituteValues

Substitute values

S A = 4 s (s) + 2 s^{2}

ProdToPowTwoFac

$a \cdot a = a^{2}$

S A = 4 s^{2} + 2 s^{2}

AddTerms

Add terms

S A = 6 s^{2}

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.

External credits: @pikisuperstar

a What is the volume of the bedroom?

b What is the surface area of the bedroom?

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 = w ℓ 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 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.

V = w ℓ h

SubstituteValues

Substitute values

V = (3.5) (4) (2.7)

Multiply

V = 37.8

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.

S A = 2 (w ℓ + h ℓ + h w)

Substitute

w

with

3.5,

ℓ

with

4,

and

h

with

2.7

into the formula and simplify.

S A = 2 (w ℓ + h ℓ + h w)

SubstituteValues

Substitute values

S A = 2 ((3.5) (4) + (2.7) (4) + (2.7) (3.5))

Multiply

S A = 2 (14 + 10.8 + 9.45)

AddTerms

Add terms

S A = 2 (34.25)

Multiply

S A = 68.5

The surface area of the bedroom is

68.5

square inches.

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.

External credits: @pikisuperstar

However, Mark remembers that the volume of the closet is

3.5

cubic inches. What is the surface area of the closet?

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.

S A = 2 (w ℓ + h ℓ + w h)

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 = w ℓ h

Substitute

V

with

3.5,

ℓ

with

1.4,

and

h

with

2

into the formula and solve it for

w .

V = w ℓ h

SubstituteValues

Substitute values

3.5 = w (1.4) (2)

Multiply

3.5 = w (2.8)

CommutativePropMult

Commutative Property of Multiplication

3.5 = 2.8 w

RearrangeEqn

Rearrange equation

2.8 w = 3.5

DivEqn

$LHS / 2.8 = RHS / 2.8$

w = \frac{3 . 5}{2 . 8}

UseCalc

Use a calculator

w = 1.25

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

S A .

S A = 2 (w ℓ + h ℓ + w h)

SubstituteValues

Substitute values

S A = 2 ((1.25) (1.4) + (2) (1.4) + (1.25) (2))

Multiply

S A = 2 (1.75 + 2.8 + 2.5)

AddTerms

Add terms

S A = 2 (7.05)

Multiply

S A = 14.1

The surface area of the closet is

14.1

square inches.

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.

External credits: @pikisuperstar

a What is the volume of the attic?

b What is the surface area of the attic?

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 = B h

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 = \frac{1}{2} b h

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 = \frac{1}{2} b h

SubstituteII

$b = 7$ , $h = 2.3$

A = \frac{1}{2} (7) (2.3)

Multiply

A = \frac{1}{2} (16.1)

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{1 6 . 1}{2}

CalcQuot

Calculate quotient

A = 8.05

The area of the triangle, and therefore the base area of the prism, is

8.05

square inches. Now that this value is known, substitute

B = 8.05

and

h = 9

in the formula for the volume of a prism.

V = B h

SubstituteII

$B = 8.05$ , $h = 9$

V = (8.05) (9)

Multiply

V = 72.45

The volume of the attic is

72.45

cubic inches.

b The surface area of the attic can be found by using the formula for the surface area of a prism.

S A = 2 B + P h

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

S A .

S A = 2 B + P h

SubstituteValues

Substitute values

S A = 2 (8.05) + (15.4) (9)

Multiply

S A = 16.1 + 138.6

AddTerms

Add terms

S A = 154.7

The surface area of the attic is

154.7

square inches.

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.

External credits: @pikisuperstar

What would the volume of this potential room be?

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.

S A = 2 B + P h

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 = \frac{1}{2} b h

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 = \frac{1}{2} b h

SubstituteII

$b = 4$ , $h = 2.7$

A = \frac{1}{2} (4) (2.7)

Multiply

A = \frac{1}{2} (10.8)

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{1 0 . 8}{2}

CalcQuot

Calculate quotient

A = 5.4

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 = 11.5 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

S A,

5.4

for

B,

and

11.5

for

P

into the formula and solve for

h .

S A = 2 B + P h

SubstituteValues

Substitute values

79.8 = 2 (5.4) + (11.5) h

Multiply

79.8 = 10.8 + 11.5 h

SubEqn

$LHS - 10.8 = RHS - 10.8$

69 = 11.5 h

RearrangeEqn

Rearrange equation

11.5 h = 69

DivEqn

$LHS / 11.5 = RHS / 11.5$

h = 6

The height of the prism is

6

inches. Finally, find the volume of the prism by using the following formula.

V = B h

Substitute

5.4

for

B

and

6

for

h,

then evaluate for

V .

V = B h

SubstituteII

$B = 5.4$ , $h = 6$

V = (5.4) (6)

Multiply

V = 32.4

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.

External credits: @pikisuperstar

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 = B h

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 = \frac{1}{2} b h

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 = \frac{1}{2} b h

SubstituteII

$h = 1$ , $b = 1.2$

A = \frac{1}{2} (1) (1.2)

OneMult

$1 \cdot a = a$

A = \frac{1}{2} (1.2)

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{1 . 2}{2}

CalcQuot

Calculate quotient

A = 0.6

The area of the base is

0.6

square inches. Now multiply this area by the height of the prism,

2

inches, to find the volume of the prism.

V_{t} = B h

SubstituteII

$B = 0.6$ , $h = 2$

V_{t} = (0.6) (2)

Multiply

V_{t} = 1.2

The volume of the triangular prism is

1.2

cubic inches.

Rectangular Prism

The volume of a rectangular prism can be calculated using the following formula.

V = w ℓ 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_{r} = w ℓ h

SubstituteValues

Substitute values

V_{r} = (0.8) (1) (2)

Multiply

V_{r} = 1.6

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.

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Catch-Up and Review

Proof

Proof

Proof

Proof

Proof

Proof

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Triangular Prism

Rectangular Prism

Comparison