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# Surface Area and Volume of Prisms

## 3-D figures with 2 congruent bases in parallel planes and rectangles for their other faces.

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Prisms

What if your family were ready to fill the new pool with water and they didn't know how much water would be needed? The shallow end is 4 ft. and the deep end is 8 ft. The pool is 10 ft. wide by 25 ft. long. How many gallons of water will it take to fill the pool? There are approximately 7.48 gallons in a cubic foot.

### Prisms

A prism is a 3-dimensional figure with 2 congruent bases, in parallel planes with rectangular lateral faces. The edges between the lateral faces are called lateral edges. All prisms are named by their bases, so the prism below is a pentagonal prism.

This particular prism is called a right prism because the lateral faces are perpendicular to the bases. Oblique prisms lean to one side or the other and the height is outside the prism.

#### Surface Area

Surface area is the sum of the areas of the faces of a solid.

Surface Area of a Right Prism: The surface area of a right prism is the sum of the area of the bases and the area of each rectangular lateral face.

#### Volume

Volume is the measure of how much space a three-dimensional figure occupies. The basic unit of volume is the cubic unit: cubic centimeter \begin{align*}(cm^3)\end{align*}, cubic inch \begin{align*}(in^3)\end{align*}, cubic meter \begin{align*}(m^3)\end{align*}, cubic foot \begin{align*}(ft^3)\end{align*}, etc. Each basic cubic unit has a measure of one for each: length, width, and height.

Volume of a Rectangular Prism: If a rectangular prism is \begin{align*}h\end{align*} units high, \begin{align*}w\end{align*} units wide, and \begin{align*}l\end{align*} units long, then its volume is \begin{align*}V=l \cdot w \cdot h\end{align*}.

If we further analyze the formula for the volume of a rectangular prism, we would see that \begin{align*}l \cdot w\end{align*} is equal to the area of the base of the prism, a rectangle. If the bases are not rectangles, this would still be true, however we would have to rewrite the equation a little.

Volume of a Prism: If the area of the base of a prism is \begin{align*}B\end{align*} and the height is \begin{align*}h\end{align*}, then the volume is \begin{align*}V=B \cdot h\end{align*}.

Recall that earlier in this Concept we talked about oblique prisms. These are prisms that lean to one side and the height is outside the prism. What would be the area of an oblique prism? To answer this question, we need to introduce Cavalieri’s Principle.

Cavalieri’s Principle: If two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

Basically, if an oblique prism and a right prism have the same base area and height, then they will have the same volume.

#### Finding the Surface Area

1. Find the surface area of the prism below.

Open up the prism and draw the net. Determine the measurements for each rectangle in the net.

Using the net, we have:

\begin{align*}SA_{prism} & = 2(4)(10)+2(10)(17)+2(17)(4)\\ & =80+340+136\\ & =556 \ cm^2\end{align*}

Because this is still area, the units are squared.

2. Find the surface area of the prism below.

This is a right triangular prism. To find the surface area, we need to find the length of the hypotenuse of the base because it is the width of one of the lateral faces. Using the Pythagorean Theorem, the hypotenuse is

\begin{align*}7^2+24^2&=c^2\\ 49+576&=c^2\\ 625&=c^2\\ c&=25\end{align*}

Looking at the net, the surface area is:

\begin{align*}SA& =28(7)+28(24)+28(25)+2\left ( \frac{1}{2} \cdot 7 \cdot 24 \right )\\ SA& =196+672+700+168=1736\end{align*}

#### Finding the Volume

1. A typical shoe box is 8 in by 14 in by 6 in. What is the volume of the box?

We can assume that a shoe box is a rectangular prism. Therefore, we can use the formula above.

\begin{align*}V & =(8)(14)(6)=672 \ in^2\end{align*}

2. You have a small, triangular prism shaped tent. How much volume does it have, once it is set up?

First, we need to find the area of the base. That is going to be \begin{align*}B=\frac{1}{2} (3)(4)=6 \ ft^2\end{align*}. Multiplying this by 7 we would get the entire volume. The volume is \begin{align*}42 \ ft^3\end{align*}.

Even though the height in this problem does not look like a “height,” it is, when referencing the formula. Usually, the height of a prism is going to be the last length you need to use.

#### Pool Problem Revisited

Even though it doesn’t look like it, the trapezoid is considered the base of this prism. The area of the trapezoids are \begin{align*}\frac{1}{2} (4+8)25=150 \ ft^2\end{align*}. Multiply this by the height, 10 ft, and we have that the volume is \begin{align*}1500 \ ft^3\end{align*}. To determine the number of gallons that are needed, divide 1500 by 7.48. \begin{align*}\frac{1500}{7.48} \approx 200.53\end{align*} gallons are needed to fill the pool.

### Examples

#### Example 1

Find the surface area of the regular pentagonal prism.

For this prism, each lateral face has an area of \begin{align*}160 \ units^2\end{align*}. Then, we need to find the area of the regular pentagonal bases. Recall that the area of a regular polygon is \begin{align*}\frac{1}{2} asn\end{align*}. \begin{align*}s = 8\end{align*} and \begin{align*}n = 5\end{align*}, so we need to find \begin{align*}a\end{align*}, the apothem.

\begin{align*}\tan 36^\circ & = \frac{4}{a}\\ a & =\frac{4}{\tan 36^\circ} \approx 5.51\\ SA & = 5(160)+2\left ( \frac{1}{2} \cdot 5.51 \cdot 8 \cdot 5 \right )=1020.4\end{align*}

#### Example 2

Find the volume of the right rectangular prism below.

A rectangular prism can be made from any square cubes. To find the volume, we would simply count the cubes. The bottom layer has 20 cubes, or 4 times 5, and there are 3 layers, or the same as the height. Therefore there are 60 cubes in this prism and the volume would be \begin{align*}60 \ units^3\end{align*}.

#### Example 3

Find the volume of the regular hexagonal prism below.

Recall that a regular hexagon is divided up into six equilateral triangles. The height of one of those triangles would be the apothem. If each side is 6, then half of that is 3 and half of an equilateral triangle is a 30-60-90 triangle. Therefore, the apothem is going to be \begin{align*}3 \sqrt{3}\end{align*}. The area of the base is:

\begin{align*}B=\frac{1}{2} \left ( 3 \sqrt{3} \right )(6)(6)=54 \sqrt{3} \ units^2\end{align*}

And the volume will be:

\begin{align*}V = Bh = \left ( 54 \sqrt{3} \right )(15)=810 \sqrt{3} \ units^3\end{align*}

#### Example 4

Find the area of the oblique prism below.

This is an oblique right trapezoidal prism. First, find the area of the trapezoid.

\begin{align*}B=\frac{1}{2} (9)(8+4)=9(6)=54 \ cm^2\end{align*}

Then, multiply this by the height.

\begin{align*}V=54(15)=810 \ cm^3\end{align*}

### Review

Use the right triangular prism to answer questions 1-5.

1. What shape are the bases of this prism? What are their areas?
2. What are the dimensions of each of the lateral faces? What are their areas?
3. Find the lateral surface area of the prism.
4. Find the total surface area of the prism.
5. Find the total volume of the prism.
6. Writing Describe the difference between lateral surface area and total surface area.
7. Fuzzy dice are cubes with 4 inch sides.
1. What is the surface area of one die?
2. Typically, the dice are sold in pairs. What is the surface area of two dice?
3. What is the volume of both dice?

Find the surface area and volume of the following solids.

1. Note: bases are isosceles trapezoids

Find the value of \begin{align*}x\end{align*}, given the surface area.

1. \begin{align*}SA = 432 \ units^2\end{align*}
2. \begin{align*}SA = 1568 \ units^2\end{align*}

Use the diagram below for questions 13-16. The barn is shaped like a pentagonal prism with dimensions shown in feet.

1. What is the area of the roof? (Both sides)
2. What is the floor area of the barn?
3. What is the area of the sides of the barn?
4. What is the total volume of the barn?
5. An open top box is made by cutting out 2 in by 2 in squares from the corners of a large square piece of cardboard. Using the picture as a guide, find an expression for the surface area of the box. If the surface area is \begin{align*}609 \ in^2\end{align*}, find the length of \begin{align*}x\end{align*}. Remember, there is no top.
1. How many one-inch cubes can fit into a box that is 8 inches wide, 10 inches long, and 12 inches tall? Is this the same as the volume of the box?
2. A cereal box in 2 inches wide, 10 inches long and 14 inches tall. How much cereal does the box hold?
3. A cube holds \begin{align*}216 \ in^3\end{align*}. What is the length of each edge?

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Color Highlighted Text Notes

### Vocabulary Language: English Spanish

lateral edges

Edges between the lateral faces of a prism.

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

Lateral Faces

All faces of a prism aside from the base are known as lateral faces.