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# 11.2: Surface Area of Prisms and Cylinders

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Find the surface area of a prism and cylinder.

## Review Queue

1. Find the area of a rectangle with sides:
1. 6 and 9
2. 11 and 4
3. \begin{align*}5 \sqrt{2}\end{align*} and \begin{align*}8 \sqrt{6}\end{align*}
2. If the area of a square is \begin{align*}36 \ units^2\end{align*}, what are the lengths of the sides?
3. If the area of a square is \begin{align*}45 \ units^2\end{align*}, what are the lengths of the sides?

Know What? Your parents decide they want to put a pool in the backyard. The shallow end will be 4 ft. and the deep end will be 8 ft. The pool will be 10 ft. by 25 ft. How much siding do they need to cover the sides and bottom of the pool?

## Parts of a Prism

Prism: A 3-dimensional figure with 2 congruent bases, in parallel planes, and the other faces are rectangles.

The non-base faces are lateral faces.

The edges between the lateral faces are lateral edges.

This is a pentagonal prism.

Right Prism: A prism where all the lateral faces are perpendicular to the bases.

Oblique Prism: A prism that leans to one side and the height is outside the prism.

## Surface Area of a Prism

Surface Area: The sum of the areas of the faces.

\begin{align*}Surface \ Area &= B_1+B_2+L_1+L_2+L_3\\ Lateral \ Area &= L_1+L_2+L_3\end{align*}

Lateral Area: The sum of the areas of the lateral faces.

Example 1: Find the surface area of the prism below.

Solution: Draw the net of the prism.

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*}

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.

Example 2: Find the surface area of the prism below.

Solution: 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.

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

## Cylinders

Cylinder: A solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed.

A cylinder has a radius and a height.

A cylinder can also be oblique, like the one on the far right.

## Surface Area of a Right Cylinder

Let’s find the net of a right cylinder. One way to do this is to take the label off of a soup can. The label is a rectangle where the height is the height of the cylinder and the base is the circumference of the circle.

Surface Area of a Right Cylinder: \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}.

\begin{align*}& \ \underbrace{ 2 \pi r^2 } \ \ + \ \ \underbrace{ 2 \pi r} h\\ & \text{area of} \quad \ \ \text{length}\\ & \ \text{both} \qquad \ \ \ \text{of}\\ & \text{circles} \quad \ \ \text{rectangle}\end{align*}

To see an animation of the surface area, click http://www.rkm.com.au/ANIMATIONS/animation-Cylinder-Surface-Area-Derivation.html, by Russell Knightley.

Example 3: Find the surface area of the cylinder.

Solution: \begin{align*}r = 4\end{align*} and \begin{align*}h = 12\end{align*}.

\begin{align*}SA &= 2 \pi (4)^2+2 \pi (4)(12)\\ &= 32 \pi +96 \pi\\ &= 128 \pi \ units^2\end{align*}

Example 4: The circumference of the base of a cylinder is \begin{align*}16 \pi\end{align*} and the height is 21. Find the surface area of the cylinder.

Solution: We need to solve for the radius, using the circumference.

\begin{align*}2 \pi r &=16 \pi\\ r&=8\end{align*}

Now, we can find the surface area.

\begin{align*}SA &=2 \pi (8)^2+(16 \pi )(21)\\ &= 128 \pi +336 \pi\\ &= 464 \pi \ units^2\end{align*}

Example 5: Algebra Connection The total surface area of the triangular prism is \begin{align*}540 \ units^2\end{align*}. What is \begin{align*}x\end{align*}?

Solution: The total surface area is equal to:

\begin{align*}A_{2 \ triangles}+A_{3 \ rectangles}=540\end{align*}

The hypotenuse of the triangle bases is 13, \begin{align*}\sqrt{5^2+12^2}\end{align*}. Let’s fill in what we know.

\begin{align*}A_{2 \ triangles} &= 2 \left(\frac{1}{2} \cdot 5 \cdot 12\right)=60\\ A_{3 \ triangles} &= 5x+12x+13x=30x\\ 60+30x &= 540\\ 30x &= 480\\ x &= 16 \ units \qquad \text{The height is 16 units.}\end{align*}

Know What? Revisited To the right is the net of the pool (minus the top). From this, we can see that your parents would need 670 square feet of siding.

## Review Questions

• Questions 1-9 are similar to Examples 1 and 2.
• Question 10 uses the definition of lateral and total surface area.
• Questions 11-18 are similar to Examples 1-3.
• Questions 19-21 are similar to Example 5.
• Questions 22-24 are similar to Example 4.
• Questions 25-30 use the Pythagorean Theorem and are similar to Examples 1-3.
1. What type of prism is this?
2. Draw the net of this prism.
3. Find the area of the bases.
4. Find the area of lateral faces, or the lateral surface area.
5. Find the total surface area of the prism.

Use the right triangular prism to answer questions 6-9.

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. Writing Describe the difference between lateral surface area and total surface area.
6. 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?
7. A right cylinder has a 7 cm radius and a height of 18 cm. Find the surface area.

Find the surface area of the following solids. Round your answer to the nearest hundredth.

1. bases are isosceles trapezoids

Algebra Connection 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 = 1536 \pi \ units^2\end{align*}
3. \begin{align*}SA = 1568 \ units^2\end{align*}
4. The area of the base of a cylinder is \begin{align*}25 \pi \ in^2\end{align*} and the height is 6 in. Find the lateral surface area.
5. The circumference of the base of a cylinder is \begin{align*}80 \pi \ cm\end{align*} and the height is 36 cm. Find the total surface area.
6. The lateral surface area of a cylinder is \begin{align*}30 \pi \ m^2\end{align*} and the height is 5m. What is the radius?

Use the diagram below for questions 25-30. The barn is shaped like a pentagonal prism with dimensions shown in feet.

1. What is the width of the roof? (HINT: Use the Pythagorean Theorem)
2. What is the area of the roof? (Both sides)
3. What is the floor area of the barn?
4. What is the area of the rectangular sides of the barn?
5. What is the area of the two pentagon sides of the barn? (HINT: Find the area of two congruent trapezoids for each side)
6. Find the total surface area of the barn (Roof and sides).

1. 54
2. 44
3. \begin{align*}80 \sqrt{3}\end{align*}
1. \begin{align*}s = 6\end{align*}
2. \begin{align*}s = 3 \sqrt{5}\end{align*}

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Date Created:
Feb 22, 2012
Feb 03, 2016
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