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Radius or Diameter of a Circle Given Area

r = sqrt(A/π); d = 2sqrt(A/π)

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Radius or Diameter of a Circle Given Area
Credit: Irwin Scott
Source: https://www.flickr.com/photos/irwin-scott/5331896187/in/photolist-98amYB-7SZNHX-c9sKYb-rDpc1f-8F4usU-c9N3MY-c8EUJ7-c8EUE5-c9N5hC-c8WpQf-c8Uemo-c8EKaQ-c8EDgf-c8ELJj-c9N4r3-c8EGGL-c9N2mA-c8EFtG-c8ED6j-canxgy-c9N2qf-c8wvMU-c8UcKb-canmYW-c8wqZQ-c8EGE7-c9N6Lu-c8wrLf-c9sKvA-8F1geR-qJFZHN-nPQweg-rFAszn-6JMnsT-canoRq-c8wtRj-5nYb2q-c9N54o-c8x1YN-canvME-c8EJFU-canpSh-c8EUXj-c8EUvm-c8wuJC-c8Wrvf-c8EMny-c8EKE5-c8EKuC-2NrHe2
License: CC BY-NC 3.0

Richard and his classmate, Sid, are building a large wooden replica of a clock for their science class project. The clock’s face will be a perfect circle with an area of 530.66 square inches. Richard has to cut the long hand, or dial, which will extend from the center of the clock to the edge, but he isn’t sure how long it should be. How can Richard use what he knows about the area of the clock to determine how long the dial should be?

In this concept, you will learn to find the radius or diameter of a circle when you have been given the area.

Finding Radius or Diameter of a Circle

The equation for the area of a circle is \begin{align*}A = \pi r^2\end{align*}.

If you are given the area of a circle, you can use this equation and work backwards to find the unknown radius and diameter of the circle.

Here is an example.

The area of a circle is \begin{align*}153.86 \ in^2\end{align*}. Find the radius and diameter.

This problem requires you to figure out two different things. Let’s find the radius and then use that measure to figure out the diameter.

Start with the formula for finding the area of a circle.

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

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.

\begin{align*}153.86 = (3.14)r^2\end{align*}

Next, divide the area by pi. This will help to get one step closer to figuring out the radius.

\begin{align*}{3.14 \overline{ ) {153.86}}}\end{align*}

Remember, when you divide decimals, to move the decimal two places in the divisor and the dividend.

\begin{align*}{314 \overline{ ) {15386}}}\end{align*}

\begin{align*}& \overset{\quad \ \ 49}{314 \overline{ ) {15386}}}\\ & \quad \underline{ - \ 1256 \ \ }\\ & \qquad \ \ 2826\\ & \quad \ \ \underline{- \ 2826}\\ & \qquad \qquad 0\end{align*}

So far, the answer is 49, but that is not the radius because \begin{align*} r^2\end{align*} is on the right side of the equal sign.

\begin{align*}49 = r^2\end{align*}

Then, you need to figure out which number times itself is equal to 49. You can figure this out by finding the factors of 49.

\begin{align*}7 \times 7 = 49\end{align*}

Now, you know that the radius is 7 inches because \begin{align*}7 \times 7 = 49\end{align*}.

The radius is 7 inches.

Finally, to figure out the diameter, which is twice the radius, multiply the radius by 2.

\begin{align*}7 \times 2 = 14\end{align*}

The diameter is 14 inches.

Examples

Example 1

Earlier, you were given a problem about Richard and the clock he is making for his science class project.

Richard knows that the clock’s face is a circle and that the area of the circle is 530.66 square inches. Now, he needs to figure out the length of a dial that will extend from the center of the circle to the edge.

First, substitute the given information into the equation. You have the area and you know that pi is 3.14.

\begin{align*}\begin{array}{rcl} A & = & \pi r^ 2 \\ 530.66 & = & (3.14)r^2 \end{array}\end{align*}

Next, divide 530.66 by 3.14 to get 169 is equal to the radius squared.

\begin{align*}169 = r^2\end{align*}

Then, they asked themselves what number times itself is equal to 169. They found the answer by listing the factors of 169.

\begin{align*}13 \times 13 = 169\end{align*}

The radius of the circle is 13 inches.

So, the dial should be 13 inches long.

Example 2

Use the area of a circle formula to answer the following question.

If the area of a circle is 314 sq. cm, what is the radius of the circle?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.

\begin{align*}\begin{array}{rcl} A & = & \pi r^ 2 \\ 314 & = & (3.14)r^2 \end{array}\end{align*}

Next, divide 314 by 3.14, which will be 100.

\begin{align*}100 = r^2\end{align*}

Then, ask yourself what number times itself is equal to 100. You can also find the factors of 100.

\begin{align*}10 \times 10 = 100\end{align*}

The radius of the circle is 10 cm.

Example 3

The area of a circle is \begin{align*}12.56 \ cm^2\end{align*}. What is the radius? What is the diameter?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.

\begin{align*}\begin{array}{rcl} A &=& \pi r^2 \\ 12.56 &=& (3.14)r^2 \end{array}\end{align*}

Next, divide 12.56 by 3.14. You can move the decimal over two places and divide 1256 by 314, which is 4.

\begin{align*}4 = r^2\end{align*}

Then, ask yourself what number times itself is equal to 4. You can also find the factors of 4.

\begin{align*}2 \times 2 = 4\end{align*}

The radius of the circle is 2 cm.

Finally, to get the diameter, multiply the radius by 2.

\begin{align*}2 \times 2 = 4\end{align*}

The diameter is 4 cm.

The answer is the radius is 2 cm. and the diameter is 4 cm.

Example 4

The area of a circle is \begin{align*}200.96 \ m^2\end{align*}. What is the radius? What is the diameter?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.

\begin{align*}\begin{array}{rcl} A &=& \pi r^2 \\ 200.96 &=& (3.14)r^2 \end{array}\end{align*}

Next, divide 200.96 by 3.14. You can move the decimal over two places and divide 20096 by 314, which is 64.

\begin{align*}64 = r^2\end{align*}

Then, ask yourself what number times itself is equal to 64. You can also find the factors of 64.

\begin{align*}8 \times 8 = 64\end{align*}

The radius of the circle is 8 m.

Finally, to get the diameter, multiply the radius by 2.

\begin{align*}8 \times 2 = 16\end{align*}

The diameter is 16 m.

The answer is the radius is 8 m. and the diameter is 16 m.

Example 5

If the area of a circle is \begin{align*}379.94 \ m^2\end{align*}, what is the radius? What is the diameter?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.

\begin{align*}\begin{array}{rcl} A &=& \pi r^2 \\ 379.94 &=& (3.14)r^2 \end{array}\end{align*}

Next, divide 379.94 by 3.14. You can move the decimal over two places and divide 37994 by 314, which is 121.

\begin{align*}121 = r^2\end{align*}

Then, ask yourself what number times itself is equal to 121. You can also find the factors of 121.

\begin{align*}11 \times 11 = 121\end{align*}

The radius of the circle is 11 m.

Finally, to get the diameter, multiply the radius by 2.

\begin{align*}11 \times 2 = 22\end{align*}

The diameter is 22 m.

The answer is the radius is 11 m. and the diameter is 22 m.

Review

Use the given area to find the radius of each circle.

  1. \begin{align*}A = 12.56 \ sq. cm\end{align*}.
  2. \begin{align*}A = 28.26 \ sq. m\end{align*}.
  3. \begin{align*}A = 50.24 \ sq. cm\end{align*}.
  4. \begin{align*}A = 78.5 \ sq. ft\end{align*}.
  5. \begin{align*}A = 153.86 \ sq. m\end{align*}.
  6. \begin{align*}A = 200.96 \ sq. in\end{align*}.
  7. \begin{align*}A = 254.34 \ sq. ft\end{align*}.
  8. \begin{align*}A = 113.04 \ sq. mi\end{align*}.
  9. \begin{align*}A = 452.16 \ sq. m\end{align*}.
  10. \begin{align*}A = 615.44 \ sq. cm\end{align*}.
  11. \begin{align*}A = 803.84 \ sq. in\end{align*}.
  12. \begin{align*}A = 1017.36 \ sq. ft\end{align*}.
  13. \begin{align*}A = 1256 \ sq. ft\end{align*}.
  14. \begin{align*}A = 1384.74 \ sq. ft\end{align*}.
  15. \begin{align*}A = 1962.5 \ sq. ft\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.9. 

Resources

 

 

Vocabulary

\pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Area

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

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.

Pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.

Squaring

Squaring a number is multiplying the number by itself. The exponent 2 is used to show squaring.

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