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 ft2. What is its radius?
Begin by dividing your area by π, usually approximated as 3.14:
50.24 ft2 ÷ 3.14 = 16 ft2
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:
√16 ft2 = 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 in3. This would give you:
113.04 in3 × 3/4 = 84.78 in3
Divide the result from Step 1 by π, which for most purposes is approximately 3.14. This yields the following:
84.78 in3 ÷ 3.14 = 27 in3
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 in3 = 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.