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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

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?

External credits: @pikisuperstar

{"type":"choice","form":{"alts":["Triangular Prism","Rectangular Prism"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

Discussion

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

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

Determine whether the given solid is a triangular prism, a rectangular prism, or neither.

Discussion

There are two important characteristics that give information about $3D$ objects.

Concept

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

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

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.

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.

$V=hB+B+B+…+B+B ⇓V=B⋅h $

Discussion

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.

$SA=2B+Ph$

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. $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

It is possible to derive two specific formulas for the volume and surface area of a rectangular prism.

Rule

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.

$V=wℓh$

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

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.

$SA=2(wℓ+hℓ+hw)$

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

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.

$V=s_{3}$

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.
*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.

$Volume of aRectangular 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, Discussion

Consider a cube with a side length $s.$

The surface area of the cube is given by the following formula.

$SA=6s_{2}$

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.
$SA=Ph+2B$

SubstituteValues

Substitute values

$SA=4s(s)+2s_{2}$

ProdToPowTwoFac

$a⋅a=a_{2}$

$SA=4s_{2}+2s_{2}$

AddTerms

Add terms

$SA=6s_{2}$

Example

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.
### Hint

### Solution

The surface area of the bedroom is $68.5$ square inches.

External credits: @pikisuperstar

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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a Use the formula for the volume of a rectangular prism.

b Use the formula for the surface area of a rectangular prism.

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

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? ### Hint

### Solution

External credits: @pikisuperstar

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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.

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=3.5in_{3} $

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

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.
### Hint

### Solution

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.
The volume of the attic is $72.45$ cubic inches.
The surface area of the attic is $154.7$ square inches.

External credits: @pikisuperstar

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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a Use the formula for the volume of a prism.

b Use the formula for the surface area of a prism.

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=21 bh $

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 $A.$
$A=21 bh$

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=15.4in. $

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

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? ### Hint

### Solution

External credits: @pikisuperstar

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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.

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=21 bh $

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.
$A=21 bh$

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=11.5in. $

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 $h.$
$SA=2B+Ph$

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

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

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.