Radius of a Circle

What is the radius?

The radius of a circle or sphere is defined as the line segment that has one endpoint at the center of the circle and the second endpoint at the circumference. The outer boundary of a circle is known at the circumference.

When there is more than one radius, they are referred to as radii. When two radii are connected as a straight line through the center, that is known as the diameter.

A circle can have many different radii. Any line segment that connects the center of a circle with the outside boundary is a radius. Since a circle is defined as a shape that has all of the outer points, the circumference, equidistant from the center, all of the radius will have the same length. Let’s look at the example below.

How many radii do you see?

All of the radius of a circle are the same length. In the diagram below, there are four different radii. AO, BO, CO, and DO. Each of these radii are equal in length.
AO = BO = CO = DO

A line segment that connects the points on the circumference, for example, DA or AB, would be called a chord. The diameter is the longest chord possible in a circle.

The radius is proportional to the circle. As the radius gets longer, the circle gets bigger. If the circle gets smaller, the radius will become shorter.

Why is the radius important?

The radius is used to find many characteristics of a circle. If you wish to know the longest length across a circle, the diameter, you would multiply the radius by two. The circumference, or distance around the circle is found by multiplying two times the radius times pi.

$$\style{font-size:36px}{C\;=\;2\pi r}$$

The area of the circle can also be found by multiplying pi times the square of the radius

$$\style{font-size:36px}{A\;=\pi r^2}$$

Did you notice that symbol that looks like a table in both of the equations? That is called pi,  $$\pi$$.

What is pi?

The symbol pi is the ratio of the circumference of any circle to the diameter of that circle. If we were to look at the equation for circumference we can solve for pi. Two times the radius is also equal to the diameter.
$$\pi=\frac C{2r}$$                     $$\pi=\frac Cd$$

The ratio will always equal the same number, pi. Pi is an irrational number that continues indefinitely, 3.141592653589793238….but it is often rounded to 3.14.

How to find the radius

When you need to find the radius of a circle, you can use one of three different formulas, depending on the information that you are given. Below are the three formulas that can be used. They are given in their original formula and then solved for the radius, r.
Use the diameter formula when you are given the diameter.
$$d\;=\;2r$$     $$r\;=\;\frac d2$$

Example
Given a circle has a diameter of 30 centimeters, what is the radius of the circle?
To find the radius when given the diameter, you will want to divide the diameter by two.
$$30\;\div\;2\;=\;15$$
The radius of a circle that has a diameter of 30 cm is 15 cm.

Use the area formula when you are given the area of a circle.
$$A=\pi r^2$$              $$r\;=\;\sqrt{\frac A\pi}$$

Use the circumference formula when you are given the circumference of a circle.
$$C\;=2\pi r$$                $$r\;=\;\frac C{2\pi}$$

Example
If a circle has a circumference of $$34\pi$$, what is the radius of the circle?
Start with the formula of the circumference of a circle and substitute the known values in.
$$C\;=2\pi r$$              $$34\pi\;=\;2\pi r$$
Divide both sides by pi and then divide both sides by 2.
C = 17

Circles on the coordinate plane

When a circle is drawn on a coordinate plane you can use the equation of a circle. Given a circle with the center (h, k) and a point on the circumference (x, y)
$$r^2={(x\;-\;h)}^2\;+\;{(y\;-\;k)}^2$$
When the center of the circle is at the origin, (0, 0), the equation can be simplified to
$$r^2\;={\;(x)}^2\;+\;{(y)}^2$$

Let’s look at an example. We can see that there are many points on the circumference of the circle. We can use a point on the axis, (0, 2). We can substitute our point in the equation to find the length of the radius.
$$r^2\;={\;(x)}^2\;+\;{(y)}^2$$
$$r^2\;={\;(0)}^2\;+\;{(2)}^2$$
$$r^2\;=\;0\;+4\;=\;4$$
$$r = 2$$

Let’s try an example with the center at (3, 4) and the point of the circle at (3, 1).

We can start by substituting our known values.
h = 3, k = 4, x = 3, y = 1

$$r^2\;={\;(x\;-\;h\;)}^2\;+\;{(y\;-\;k)}^2$$
$$r^2\;={\;(3\;-3\;)}^2\;+\;{(1\;-\;4)}^2$$
$$r^2\;={\;(0)}^2\;\;{(-\;3)}^2$$
$$r^2\;=\;9$$
$$r = 3$$

If the equation of a circle seems familiar, it is due to your previous knowledge of the Pythagorean Theorem. If we look at what the equation of a circle is finding, we can see that it is based on the Pythagorean Theorem and the radius of the circle is the hypotenuse.

What about a sphere?

A sphere is a solid three-dimensional shape without any edges or vertices. It is defined by having an outer boundary that is equidistant from the center point. The line segment that connects the center to the outer boundary is, you guessed it, the radius!

The radius is used to find the volume of the sphere with the following equation.
$$V=\frac43\pi r^3$$
You can also use the radius to find the surface area of a sphere with this formula.
$$SA=4\pi r^2$$

Did you know?

• All points on the edge of a circle are the same distance to the center. This distance is known as the radius.
• An arc is part of the circumference of a circle.
• A line segment that connects two points on the circumference of the circle is called a chord.

Now that we can see how important the radius is to circles and spheres, let’s calculate some values using the radius!

Practice

Use 3.14 for the value of pi

1. If a circle has a diameter of 14 inches, what is the radius of that circle?
2. If a circle has a circumference of 27 inches, what is the radius of the circle to the nearest hundredth?
3. If a circle has a center point of (5, 2) and a point of the circumference is (4, 6), what is the radius of the circle to the nearest hundredth?

Solutions

1. If a circle has a diameter of 14 inches, what is the radius of that circle?
   Since the diameter of a circle is twice the measure of the radius, we can divide the diameter by two.
   $$14\;\div\;2\;=\;7\;inches$$
2. If a circle has a circumference of 27 inches, what is the radius of the circle to the nearest hundredth?
   First we will need to use the formula for the circumference of a circle
   $$C\;=\;2\pi r$$              $$r\;=\;\frac C{2\pi}$$
   Next we can substitute the known values and solve for the radius.
   $$r\;=\;\frac C{2\pi}\;=\;\frac{\;27}{2(3.14)}$$
   Now we can solve
   $$r=\frac{27}{2(3.14)}=\;\frac{27}{6.28}=\;4.30\;in$$
3. If a circle has a center point of (5, 2) and a point of the circumference is (4, 6), what is the radius of the circle to the nearest hundredth?
   We will start by finding the values of the known variables.
   h = 5, k = 2, x = 4, y = 6
   Now we can substitute the values into the equation, simplify and solve for the radius.
   $$r^2\;=\;{(x-h)}^2\;+\;{(y-k)}^2$$
   $$r^2\;=\;{(4\;-\;5)}^2\;+\;{(6\;-\;2)}^2$$
   $$r^2={(-1)}^2\;+\;{(4)}^2$$
   $$r^2\;=\;1\;+\;16$$
   $$r\;=\;\sqrt{17\;}=\;4.12$$