How to Calculate Radius

How to Calculate Radius
••• Plus69/iStock/GettyImages

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 \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.

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 \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.

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​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.

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 (​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.

Related Articles

How to Calculate Round Area to Square Feet
How to Calculate Square Feet From the Perimeter
How to Calculate the Cubic Feet of a Circle
How to Figure Cubic Feet in a Cone Shape
How to Find the Volume of a Sphere in Terms of Pi
How to Calculate the Diameter of a Circle From a Linear...
How to Calculate Circumference in Feet
How to Calculate Area in Square Inches
How to Calculate a Circular Area
How to Calculate the Volume and Circumference of a...
How to Find the Area in Math
How to Calculate the Volume of an Octagon
How to Calculate the Tensile Capacity of a U-Bolt
How to Find Radius From Diameter
How to Calculate Acres in a Circle
How to Calculate the Area of a Curved Surface
How to Calculate the Area & Circumference of a Circle
How to Calculate the Apothem of a Polygon
How to Find the Radius of a Sphere When Given the Volume
How to Calculate Ellipse Dimensions