Become a math whiz with AI Tutoring, Practice Questions & more.

HotmathMath Homework. Do It Faster, Learn It Better.

Understanding the radius is one of the most important elements of the geometry of the circle. The radius is a line segment with one endpoint at the center of the circle or sphere and the other at its boundary. The word may also refer to the length of such a line segment. Here is an example:

As you've previously learned, circles are defined as the set of all points in a plane at a given distance from the center. This means that all of the radii (the accepted plural form of radius) of a particular circle are the same length. Every circle has an infinite number of radii, and they can be named if specific points in the circle are labeled. Consider the following diagram:

In this example, PC, PB, and PA are all radii because they connect the origin point to a point on the outer boundary. Always put the origin point first when naming radii in this manner!

The formula $r=\frac{C}{2×\pi }$ may be used to calculate the radius of any circle or sdivhere, where C equals the circumference of the circle and π represents pi : an irrational number approximately equal to 3.14. Most problems will ask you to use 3.14 as the value of π unless you are permitted to use a calculator or asked to leave your answer in terms of π.

The diameter is also closely related to the radius in that the diameter is a line segment containing the center and both endpoints. If you know the diameter, the radius is simply half of that value. Likewise, you can calculate the diameter of any circle by doubling its radius.

If you know the area of a circle, you can use it to determine its radius as well. The procedure is dividing the area by π and then taking the square root of the result. This can be expressed mathematically as $r=\sqrt{\frac{A}{\pi }}$ .

The radius is important because it can be used to determine more information about a circle. For example, you can use it to calculate the area of a circle. The formula is $A=\pi {r}^{2}$ where A is the area, π is pi, and r is the radius.
Similarly, you can use the radius to calculate the circumference of a circle. The formula is $C=2\pi r$ where C is the circumference and r is the radius.

Again, you can also use the radius to find the diameter of a circle. Put simply, if you have the area, circumference, diameter, or radius of a circle, you can calculate all four values with a few formulas and a little bit of math.

a. If a circle's diameter is expressed as 40 units, what is its radius?

$\mathrm{radius}=\frac{\mathrm{Diameter}}{2}=\frac{\mathrm{40}}{2}=\mathrm{20}\mathrm{units}$

b. If a circle's radius is expressed as 100 units, what is its diameter?

$\mathrm{Diameter}=2×\mathrm{Radius}=2×100=200\mathrm{units}$

c. What is the radius of a circle with a circumference of 15 inches?

$\mathrm{Radius}=\frac{\mathrm{Circumference}}{2×\pi }=\frac{15}{2×3.14}\beta 2.39\mathrm{inches}$

d. If AC and AB are both radii on the same circle and AB measures seven inches, what is the length of AC?

Since both AC and AB are radii of the same circle, their lengths are equal. So, $\mathrm{AC}=7\mathrm{inches}$

e. What is the area of a circle with a radius of 4 cm?

$\mathrm{Area}=\pi ×{\left(\mathrm{radius}\right)}^{2}=3.14×{\left(4\right)}^{2}\beta 50.24{\mathrm{cm}}^{2}$

f. What is the circumference of a circle with a radius of 9 cm using 3.14 as the value of π?

$\mathrm{Circumference}=2×\pi ×\mathrm{Radius}=2×3.14×9\beta 56.52\mathrm{cm}$

g. What is the radius of a circle with an area of 379.94 cm2?

$\mathrm{Radius}=\sqrt{\frac{\mathrm{Area}}{\pi }}=\sqrt{\frac{379.94}{3.14}}\approx 11\mathrm{cm}$