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Area of a Circle

Pi times the radius squared.

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Area of a Circle
License: CC BY-NC 3.0

Angelica’s dad is buying a round swimming pool for the yard. The brochure says \begin{align*}\text{diameter} = 24 \ ft\end{align*}. What square footage of the yard will the pool cover?

In this concept, you will learn how to find the area of a circle.

Finding the Area of a Circle

A circle is a set of connected points equidistant from a center point. The diameter is the distance across the center of the circle and the radius is the distance from the center of the circle to the edge.

The number pi, \begin{align*}\pi\end{align*}, is the ratio of the diameter to the circumference. We use 3.14 to represent pi in operations. You can find the area of a circle by taking the measurement of the radius, squaring it and multiplying it by pi. Here is the formula: \begin{align*}A = \pi r^2\end{align*}.

Let’s look at an example.

What is the area of the circle below?

License: CC BY-NC 3.0

First, write the formula.

\begin{align*}A = \pi r^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(3.14)(12)^2\end{align*}

Then, following the order of operations, figure out the exponent first and then multiply.

\begin{align*}\begin{array}{rcl} A &=& (3.14)(144) \\ A &=& 452.16 \ sq \ cm \end{array}\end{align*}

The answer is \begin{align*}A = 452.16 \ sq. cm\end{align*}.

Sometimes, you will be given a problem with the diameter and not the radius. When this happens, you can divide the measurement of the diameter by two and then use the formula.

Examples

Example 1

Earlier, you were given a problem about Angelica and the 24-foot diameter pool.

She wants to know how many square feet of ground it would cover.

First, recognize that you have been given a diameter and divide by 2 to get the radius.

\begin{align*}\begin{array}{rcl} r &=& \frac{d}{2} \\ r &=& \frac{24}{2} \\ r &=& 12 \ \text{feet} \end{array}\end{align*}

Next, substitute this value, along with pi, into the formula for the area of a circle.

\begin{align*}A=\pi r^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(3.14)(12)^2\end{align*}

Then, following the order of operations, figure out the exponent first and then multiply.

\begin{align*}\begin{array}{rcl} A &=& (3.14)(144) \\ A &=& 452.16 \ sq.ft. \end{array}\end{align*}

The answer is \begin{align*}A = 452.16 \ sq. ft\end{align*}. The round swimming pool will cover \begin{align*}452.16 \ sq. ft.\end{align*} of Drayton’s backyard.

Example 2

Some students have formed a circle to play dodge ball. The radius of the circle is 21 feet. What is the area of their dodge ball circle?

License: CC BY-NC 3.0

First, write the formula.

\begin{align*}A = \pi r^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(3.14)(21)^2\end{align*}

Then, following the order of operations, figure out the exponent first and then multiply.

\begin{align*}\begin{array}{rcl} A &=& (3.14)(441) \\ A &=& 1,384.74 \ sq.ft. \\ \end{array}\end{align*}

The answer is \begin{align*}A = 1, 384.74 \ sq. ft\end{align*}.

Example 3

Find the area of a circle with a radius of 9 inches.

First, write the formula.

\begin{align*}A= \pi r^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(3.14)(9)^2\end{align*}

Then, following the order of operations, figure out the exponent first and then multiply.

\begin{align*}\begin{array}{rcl} A &=& (3.14)(81) \\ A &=& 254.34 \ sq.in \end{array}\end{align*}

The answer is \begin{align*}A = 254.34 \ sq. in\end{align*}.

Example 4

Find the area of a circle with a radius of 11 inches.

First, write the formula.

\begin{align*}A= \pi r^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(3.14)(11)^2\end{align*}

Then, following the order of operations, figure out the exponent first and then multiply.

\begin{align*}\begin{array}{rcl} A & = & (3.14)(121) \\ A & = & 379.94 \ sq.in. \end{array}\end{align*}

The answer is \begin{align*}A = 379.94 \ sq. in\end{align*}.

Example 5

Find the area of a circle that has a diameter of 8 feet.

First, recognize that you have been given a diameter and divide by 2 to get the radius.

\begin{align*}\begin{array}{rcl} r &=& \frac{d}{2} \\ r &=& \frac{8}{2} \\ r &=& 4 \ \text{feet} \end{array}\end{align*}

Next, substitute this value, along with pi, into the formula for the area of a circle.

\begin{align*}A=\pi r^2\end{align*}

Next, substitute in what you know.

\begin{align*}A=(3.14)(4)^2\end{align*}

Then, following the order of operations, figure out the exponent first and then multiply.

\begin{align*}\begin{array}{rcl} A &=& (3.14)(16) \\ A &=& 50.24 \ sq.ft. \end{array}\end{align*}

The answer is \begin{align*}A = 50.24 \ sq. ft\end{align*}.

Review

Find the area of each circle given the radius or diameter. Round to the nearest hundredth when necessary.

  1. \begin{align*}r = 3 \text{ } in\end{align*} 
  2. \begin{align*}r = 5 \text{ } in\end{align*} 
  3. \begin{align*}r = 4 \text{ } ft\end{align*} 
  4. \begin{align*}r = 7 \text{ } m\end{align*} 
  5. \begin{align*}r = 6 \text{ } cm\end{align*} 
  6. \begin{align*}r = 3.5 \text{ } in\end{align*} 
  7. \begin{align*}d = 16 \text{ } in\end{align*} 
  8. \begin{align*}d = 14 \text{ } cm\end{align*} 
  9. \begin{align*}d = 20 \text{ } in\end{align*} 
  10. \begin{align*}d = 15 \text{ } m\end{align*} 
  11. \begin{align*}d = 22 \text{ } cm\end{align*} 
  12. \begin{align*}d = 24 \text{ } mm\end{align*} 
  13. \begin{align*}d = 48 \text{ } in\end{align*} 
  14. \begin{align*}r = 16.5 \text{ } in\end{align*} 
  15. \begin{align*}r = 25.75 \text{ } in\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.17.

Resources

 

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Vocabulary

Area

Area is the space within the perimeter of a two-dimensional figure.

Circle

A circle is the set of all points at a specific distance from a given point in two dimensions.

Diameter

Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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