6.8: Radius Or Diameter Of A Circle Given Area - K12 LibreTexts

Finding the Radius or Diameter of a Circle Given Area

The formula for the area of a circle, \(A=\pi r^2\), can also be used to solve for the radius and diameter.

Let’s look an example.

The area of a circle is 113.04 square inches. What is its radius?

First, write the formula.

\(A=\pi r^2\)

Next, substitute in what you know.

\(113.04=(3.14)r^2\)

Next, begin isolating the r by dividing both sides of the equation by 3.14.

\(36=r^2\)

Then, take the square root of both sides.

\(6=r\)

The answer is r=6. The radius of the circle is 6 inches.

Example \(\PageIndex{1}\)

Earlier, you were given a problem about Clara and Grace, who were at the circular 113.04 sq. ft. fish pond.

Clara wondered if she could reach a penny in the middle without falling in.

Solution

First, write the formula.

\(A=\pi r^2\)

Next, substitute in what you know.

\(113.04=(3.14)r^2\)

Then, begin isolating the r by dividing both sides of the equation by 3.14.

\(36=r^2\)

Take the square root of both sides.

\(6=r\)

The answer is r=6. The radius of the circle is 6 feet. Unless Clara wants to swim with the goldfish, she’d best leave the penny where it is!

Example \(\PageIndex{2}\)

What is the diameter of a circle if its area is \(379.94 cm^2\)?

Solution

First, write the formula.

\(A=\pi r^2\)

Next, substitute in what you know.

\(379.94=(3.14)r^2\)

Next, begin isolating the r by dividing both sides of the equation by 3.14.

\(121=r^2\)

Then, take the square root of both sides.

\(11=r\)

Remember that you are solving for the diameter.

\(\begin{aligned} d&=2r \\ d&=2\times 11 \\ d&=22\end{aligned}\)

The answer is the diameter, d=22 cm.

Example \(\PageIndex{3}\)

Solve for the radius of a circle if area=153.86 sq. in.

Solution

First, write the formula.

\(A=\pi r^2\)

Next, substitute in what you know.

\(153.86=(3.14)r^2\)

Next, begin isolating the r by dividing both sides of the equation by 3.14.

\(49=r^2\)

Then, take the square root of both sides.

\(7=r\)

The answer is \(r=7\). The radius of the circle is 7 inches.

Example \(\PageIndex{4}\)

Find the radius of a circle with an area of 379.94 sq. ft.

Solution

First, write the formula.

\(A=\pi r^2\)

Next, substitute in what you know.

\(379.94=(3.14)r^2\)

Then, begin isolating the r by dividing both sides of the equation by 3.14.

\(121=r^2\)

Take the square root of both sides.

\(11=r\)

The answer is r=11. The radius of the circle is 11 feet.

Example \(\PageIndex{5}\)

The area of a circle is 452.16 sq. m. Find its radius.

Solution

First, write the formula.

\(A=\pi r^2\)

Next, substitute in what you know.

\(452.16=(3.14)r^2\)

Then, begin isolating the r by dividing both sides of the equation by 3.14.

\(144=r^2\)

Take the square root of both sides.

\(12=r\)

The answer is r=12. The radius of the circle is 12 meters.