The **radius of a circle** is a line segment with one end at the center of the circle, and the other end on the circle itself. The radius is identified with a lower-case, italic $r$.

The formula you use depends on what is known about the circle. Below are three different formulas you can use to find the radius, $r$, of a circle.

If given... | Then... |
---|---|

Circumference (C) | $r=\frac{C}{2\pi}$ |

Area (A) | $r=\sqrt{\frac{A}{\pi}}$ |

Diameter (d) | $r=\frac{d}{2}$ |

The relationship between radius and diameter is an important one to know when learning to how to calculate the radius.

Since the radius is a line segment from the center to the circle, and the diameter, $d$, is a line segment from on side of a circle through the center of a circle and out to the other side of the circle, it follows that a radius is $\frac{1}{2}$ a diameter.

To find the radius of a circle, plug the given information into the correct formula. You can find the radius of any circle if you are given the circumference, area, or diameter.

No matter the size of the circle, the relationship of the radius to the circumference, $C$, is a constant, giving us the formula to find radius from circumference:

$r=\frac{Circumference\left(C\right)}{2\pi}$

You can also calculate the circumference of a circle with a given radius by using algebra to isolate the $C$ in our formula.

The radius to circumference formula is: $C=2\pi r$

You can use the area to find the radius and the radius to find the area of a circle. The **area **of a circle is the space it occupies, measured in square units.

Given the area, $A$, of a circle, its radius is the square root of the area divided by pi:

$r=\sqrt{\frac{A}{\pi}}$

The formula for radius to area is: $A=\pi {r}^{2}$

If you know the diameter of a circle, you can find its radius by dividing the diameter by $2$:

$r=\frac{d}{2}$

This is the simplest formula you can use to get the radius.

If you need to go from radius to diameter, multiply radius times $2$: $d=2r$

- A circle has a diameter of $6inches$. What is its radius?
- A circle has a radius of $98feet$. What is its diameter?
- A circle with a circumference of $365meters$ will have a radius of how many meters?
- A circle with a radius of $11inches$ will have a circumference of how many inches?
- With a radius of $41mm$, a circle will have an area equal to, what?
- What is the radius of the following circle?

We don't like to talk in circles, but make sure you work these out before checking the answers below!

- The radius of a $6inch$ circle is $3inches$.
- A circle has a radius of $98feet$, so its diameter must be $196feet$.
- A circle with a circumference of $365meters$ will have a radius of $58.4meters$.
- A circle with a radius of $11inches$ will have a circumference of $69.12inches$.
- With a radius of $41millimeters$, a circle will have an area of roughly $\mathrm{5,281.02}squaremillimeters$ or $m{m}^{2}$.
- The radius of the circle in the image above is $75m$.

After working your way through this lesson and video, you have learned:

- The definition of radius in geometry
- The radius of a circle formula
- How to find the radius of a circle based on the given information
- Hot to use the radius to calculate the diameter, circumference, and area of a circle

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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