The radius of a circle is the straight-line distance from the very center of the circle to any point on the circle. The nature of the radius makes it a powerful building block for understanding many other measurements about a circle, for example its diameter, its circumference, its area and even its volume (if you're dealing with a three-dimensional circle, also known as a sphere). If you know any of these other measurements, you can work backwards from standard formulas to figure out the circle or sphere's radius.

## Calculating Radius From Diameter

Figuring out a circle's radius based on its diameter is the easiest calculation possible: Just divide the diameter by 2, and you'll have the radius. So if the circle has a diameter of 8 inches, you calculate the radius like this:

8 inches ÷ 2 = 4 inches

The circle's radius is 4 inches. Note that if a unit of measurement is given, it's important to carry it all the way through your calculations.

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## Calculating Radius From Circumference

A circle's diameter and radius are both intimately tied to its circumference, or the distance all the way around the outside of the circle. (Circumference is just a fancy word for the perimeter of any round object). So if you know the circumference, you can calculate the circle's radius too. Imagine that you have a circle with a circumference of 31.4 centimeters:

Divide the circle's circumference by π, usually approximated as 3.14. The result will be the diameter of the circle. This gives you:

31.4 cm ÷ π = 10 cm

Note how you carry the units of measure all the way through your calculations.

Divide the result of Step 1 by 2 to get the circle's radius. So you have:

10 cm ÷ 2 = 5 cm

The circle's radius is 5 centimeters.

## Calculating Radius From Area

Extracting a circle's radius from its area is a little more complicated but still won't take many steps. Start by recalling that the standard formula for area of a circle is π_r_^{2}, where *r* is the radius. So your answer is right there in front of you. You just have to isolate it using appropriate mathematical operations. Imagine that you have a very big circle of area 50.24 ft^{2}. What is its radius?

Begin by dividing your area by π, usually approximated as 3.14:

50.24 ft^{2} ÷ 3.14 = 16 ft^{2}

You aren't quite done yet, but you're close. The result of this step represents *r*^{2} or the circle's radius squared.^{}

Calculate the square root of the result from Step 1. In this case, you have:

√16 ft^{2} = 4 ft

So the circle's radius, *r*, is 4 feet.

## Calculating Radius From Volume

The concept of radius applies to three-dimensional circles, which are really called spheres, too. The formula for finding a sphere's volume is a little more complicated – (4/3)π_r_^{3} –but, once again, the radius *r* is already right there, just waiting for you to isolate it from the other factors in the formula.

Multiply the volume of your sphere by 3/4. Imagine that you have a small sphere with volume 113.04 in^{3}. This would give you:

113.04 in^{3} × 3/4 = 84.78 in^{3}

Divide the result from Step 1 by π, which for most purposes is approximately 3.14. This yields the following:

84.78 in^{3} ÷ 3.14 = 27 in^{3}

This represents the cubed radius of the sphere, so you're almost done.

Conclude your calculations by taking the cube root of the result from Step 2; the result is the radius of your sphere. So you have:

^{3}√27 in^{3} = 3 inches

Your sphere has a radius of 3 inches; that would make it something like a super-sized marble, but still small enough to hold in your palm.