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Area of a Circle
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Introduction

In this lesson, you will learn how to calculate the area of a circle.


This video illustrates the lesson material below. Watching the video is optional.


Calculating the Area of a Circle

The formula for area of a circle is:
\begin{align*} Area=\pi(r)^2\end{align*}

  • The symbol \(\pi\) represents pi, or the number 3.1459265359…
  • The r stands for radius. The radius is the straight line from the center of a circle to the circumference of a circle.
  • The \(r^2\) means radius squared; whatever it is attached to must be doubled.

If area is how many unit squares fit within a shape, how many squares fit in a circle? Because squares don’t fit evenly into circles, you don’t usually get an even answer when looking for the area of a circle. You must also use \( \pi \) (pi).

Example 1
Find the area of a circle whose radius is 25 centimeters (cm).

A circle with a radius of 25 centimeters and a diameter of 50 centimeters.

Figure 1

Write the problem like this: \(Area=\pi(25\space cm)^2\). Notice that the unit of measurement is in the parentheses, which means it will also be squared. Area is measured in units that are squared, so you must square both the number and the unit.

\begin{align*} & Area=\pi(r)^2 & \color{red}\small\text{Formula for area of a circle}\\\\ & Area=\pi(25\space cm)^2 & \color{red}\small\text{Substitute given terms}\\\\ & Area=\pi(625\space cm^2) & \color{red}\small\text{Solve exponents}\\\\ & Area=1963.5\space cm^2 & \color{red}\small\text{Multiply}\\\\ \end{align*}

Remember, most calculators will have a button that will automatically input \(pi\) for you. You may also multiply by 3.14 instead. Then, simplify the answer by rounding. The answer above is rounded to the nearest tenth.

Example 2
Find the area of a circle whose radius is 12.5 cm.

a circle with a radius of 12.5 centimeters and a diameter of 25 centimeters.

Figure 2

\begin{align*} & Area=\pi(r)^2 & \color{red}\small\text{Formula for area of a circle}\\\\ & Area=\pi(12.5\space cm)^2 & \color{red}\small\text{Substitute given terms}\\\\ & Area=\pi(156.25\space cm^2) & \color{red}\small\text{Solve exponents}\\\\ & Area=490.9\space cm^2 & \color{red}\small\text{Multiply}\\\\ \end{align*}

Rounded to the nearest tenth, the area of this circle is \(490.9cm^2\).

Conclusion

It might be interesting to note that even though the circle in Example 1 has a radius that is twice the size of the circle in Example 2, their areas do not function the same way. The bigger circle has an area that is almost four times as big as the smaller circle.

This figure shows the two circles from figure 1 and figure 2.

Figure 3


Things to Remember


  • The formula for the area of a circle is \( A=\pi(r)^2\).
  • The radius of a circle is found by measuring from the center of the circle to the circumference. (It is also half the size of the diameter.)

Standard Mathematical Formats with Pi

When pi is part of a solution there are two ways you can display the solution. The first way is to write the number part of the solution multiplied to pi such as \( 13\pi ft \) or \( 5.3\pi cm\). You generally write the number then pi and then the units.

The second way to show your solution is to multiply the number portion of the solution to pi and then round to an appropriate place value. (Example: \( 13\pi ft = 40.84 ft \) rounded to the nearest hundredth.)

In this course, you will always multiply pi into the solution and round to an appropriate place value. Just know that the other way is commonly used, and you may see it in textbooks or other classes as a standard way to write solutions when pi is involved.

Practice Problems

  1. A circle has a radius of 21.5. Find the area of the circle. Round to the nearest tenth. (
    Solution
    x
    Solution: \(1452.2\) (when using the pi button on the calculator)
    )
  2. A button is in the shape of a circle. If the radius of the button is 7 mm, find the surface area of the top of the button. Round to the nearest hundredth. (
    Solution
    x
    Solution: \(153.94 \text{ mm}^{2}\) (when using the pi button on the calculator)
    )
  3. A helicopter landing pad has a diameter of 28 meters or a radius of 14 meters. Find the surface area of the landing pad. Round your answer to the nearest whole number. (
    Solution
    x
    Solution: \(616\:{\text{m}}^{2}\) (when using the pi button on the calculator)
    Details:
    This is a picture of a circle with a line extending from the center to the edge of the circle, representing the radius of the helicopter pad. The radius measures 14 meters.
    You are trying to find the area of the helicopter pad, which is shaped like a circle. Since the pad is circular, you can use the formula for the area of a circle:

    \({\text{A}} = {\text{π}} {\text{ r}}^{2}\)

    The first thing to do is replace ‘r’ with 14m:

    \({\text{A}} = {\text{π}} (14{\text{m}})^{2}\)

    Next, square 14 m (or multiply \(14 \times 14\)) to get \(196 {\text{m}}^{2}\):

    \({\text{A}} = {\text{π}} 196{\text{m}}^{2}\)

    Since you can multiply in any order, you can switch the order of 196 and \({\text{π}}\) and write the area like this:

    \({\text{A}} = 196 {\text{π}} {\text{ m}}^{2}\)

    To write the area as a decimal, multiply 196 by \({\text{π}}\) to get:

    \({\text{A}} = 615.7521601... {\text{m}}^{2}\)

    The area of the helicopter landing pad is about \(616 {\text{m}}^{2}\) when rounded to the nearest whole number.
    | )
  4. A rock is thrown into a pond and creates circular ripples that radiate away from the center. At one point, the largest ripple has a radius of 7.25 feet. Find the surface area of the ripple. Round to the nearest tenth. (
    Solution
    x
    Solution: \(165.1 \text{ ft}^{2}\) (when using the pi button on the calculator)
    Details:
    This is a picture of a circle with a line extending from the center to the edge of the circle, representing the radius of the ripple. The radius measures 7.25 ft.
    You are trying to find the area inside the largest ripple. Since the ripples are circular, you can use the formula for the area of a circle:

    \({\text{A}} = {\text{π}} {\text{ r}}^{2}\)

    First, replace ‘r’ with 7.25 ft

    \({\text{A}} = {\text{π}} (7.25 {\text{ft}})^{2}\)

    Next, square 7.25 ft to get:

    \({\text{A}} = {\text{π}} 52.5625 {\text{ft}}^{2}\)

    You can switch the order of 52.5625 and \({\text{π}}\) since you can multiply in any order.

    \({\text{A}} = 52.5625 {\text{π}} {\text{ ft}}^{2}\) (This is in a standard mathematical format.)

    If needed, you can multiply \({\text{π}}\) by 52.5625 to get:

    \({\text{A}} = 165.129963854... {\text{ft}}^{2}\)

    The area of the circular area inside the ripple is about \(165.1 {\text{ft}}^{2}\) when rounded to the nearest tenth.
    )
  5. A frisbee is a circular disk tossed back and forth between players in a game. If the radius of the frisbee is 5 inches, find the surface area of the top of the frisbee. Round to the nearest tenth. (
    Solution
    x
    Solution: \(78.5 \text{ in}^{2}\) (when using the pi button on the calculator)
    )
  6. A cellphone tower has the ability to provide service to a circular region with a radius of 40.3 miles. Find the total surface area of the coverage zone. Round to the nearest hundredth. (
    Video Solution
    x
    Solution: \(5102.23 \text{ mi}^{2}\) (when using the pi button on the calculator)
    Details:

    (Area of a Circle #6 (02:18 mins) | Transcript)
    )

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