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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.

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 \text{ inches} ÷ 2 = 4 \text{ 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.

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 \text{ cm} ÷ π = 10 \text{ 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 \text{ cm} ÷ 2 = 5 \text{ cm}

The circle's radius is 5 centimeters.

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​r2, 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 ft2. What is its radius?

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

50.24 \text{ ft}^2 ÷ 3.14 = 16 \text{ ft}^2

You aren't quite done yet, but you're close. The result of this step represents ​r2 or the circle's radius squared.

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

\sqrt{16 \text{ ft}^2} = 4 \text{ ft}

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

The concept of radius applies to three-dimensional circles, which are really called spheres, too. The formula for finding a sphere's volume (​V​) is a little more complicated

V = \frac{4}{3}πr^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 in3. This would give you:

113.04 \text{ in}^3 × \frac{3}{4} = 84.78 \text{ in}^3

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

84.78 \text{ in}^3 ÷ 3.14 = 27 \text{ 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:

\sqrt[3]{27 \text{ in}^3} = 3 \text{ 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.