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How to Find the Radius of a Circle With the Area in 3 Simple Ways

If you want to know how to find the radius of a circle with the area, you have to know the formula for the area of a circle. You must also familiarize yourself with concepts such as the circumference, sector, and diameter of a circle. To get started, let's define what the radius is.

What Is the Radius of a Circle?

The lowercase r in the center of the circle below represents its radius.

how to find the radius of a circle with the area: Diagram showing the radius of a circle

As you can see, the radius of the circle is marked by a line segment that goes from the edge of the circle to the center of the circle. If you were to draw a line from the top of the circle to the bottom intersecting the radius, you would form a semicircle:

how to find the radius of a circle with the area: Diagram of a semicircle with a line from top to bottom intersecting the radius

Diameter and Circumference of a Circle

how to find the radius of a circle with the area: Diagram showing the circumference and diameter of a circle

The circle above displays circumference and diameter. The circumference of the circle is the distance around the edge of the whole circle. The diameter of the circle is the length from one end of the circle to the other, passing through the center of the circle. Because the line segment of the diameter intersects the center of a circle, diameters are always twice the length of the radius.

Sector of a Circle

The shaded area of the circle below shows a sector, which is a sliced off section of a circle:

how to find the radius of a circle with the area: Diagram of a circle with a shaded sector

The inner line segments of the sector both equal the radius of the circle. The angle that these two measurements make is called a central angle.

How to Find the Radius of a Circle With the Area and Circumference Formulas

When the radius is unknown, you can use a circumference, area of a circle, or area of a sector formula to determine its length:

how to find the radius of a circle with the area: Formula for finding the radius using the area of the circle

how to find the radius of a circle with the area: Formula for finding the radius of a circle using the area of a sector

how to find the radius of a circle with the area: Formula for finding the radius using the circumference of the circle

Let’s use these formulas to solve for the radius of three different circles, starting with the area of a circle formula.

Let’s take the square root of a circle with a given area of 12 and divided by pi to determine the radius:

how to find the radius of a circle with the area: Given area of the circle

how to find the radius of a circle with the area: Formula for finding the radius using the area of the circle

how to find the radius of a circle with the area: Formula for finding the radius using the area of the circle

how to find the radius of a circle with the area: Formula for finding the radius using the area of the circle

Now, let's determine the radius of a circle with a sector angle measurement of 24° and an area of 60 using the area of a sector formula:

Angle of the sector

Area of the sector

Formula for finding the radius of a circle using the area of a sector

Formula for finding the radius of a circle using the area of a sector

Formula for finding the radius of a circle using the area of a sector

Formula for finding the radius of a circle using the area of a sector

Formula for finding the radius of a circle using the area of a sector

Finally, let's determine this radius of a with a circumference of 16:

Circumference of the circle

Formula for finding the radius using the circumference of the circle

Formula for finding the radius using the circumference of the circle

Formula for finding the radius using the circumference of the circle

Now You Know How to Find the Radius of a Circle With the Area

Finding the radius of a circle requires you to use formulas such as the area or sector area of a circle formulas. You can also use the diameter (the length from one end of a circle to the other) and the circumference (the length around the outside of a circle) to find the missing length of a radius.

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