## How to Find the Radius of a Circle? Formula & Example

How to find the radius of a circle? A circle’s radius is one of its most important components. It is the distance between the center of a circle and any point on its boundary. In other words, when we connect the center of a circle to any point on its circumference using a straight line, that line is the radius of that circle. Circles are among the most common shapes in nature. Their fascinating properties make them a very important topic in geometry.

There are infinite points on a circle’s circumference, so a circle can have more than one radius. A circle has an infinite number of radii, and all of the radii are equally distant from the center. We will learn more about the radius in this lesson.

## What is the Radius of a Circle?

Circles are made up of points arranged equidistant from the center point ‘O’ on a two-dimensional plane. Circles are measured by the distance between the center and any point on their circumference. A line segment from a point on a circle’s circumference to the center of a circle can also be defined as its length. There can be any number of radii on a circle and they all measure the same. If the radius varies, the size of the circle varies as well. Circular radiuses are generally abbreviated with ‘r’.

On the boundary of the circle are the points A, B, M, N, P, Q, X, and Y. Each of these points is equidistant from center O. As a result, all the line segments OA, OB, OM, ON, OY, OX, OP, and OQ can be considered the radii of the circle. You can observe that OA = OB = OM = ON = OP = OQ = OX = OY.

## Radius Formulas

**Radius Formula Using Diameter of a Circle: **A circle’s diameter is a straight line passing through the center and joining a point at one end to a point at the other end. A circle’s diameter is twice as long as its radius. Mathematically, it is formulated as Diameter = 2 × radius. It is also the longest chord in a circle. A is the circle’s diamete. When the diameter of a circle is given, the radius formula is expressed as:

Radius Formula = Diameter/2 or D/2 units

**Radius Formula Using Circumference of Circle: **The circumference of a circle is its perimeter. It is the boundary of a circle and can be expressed by the formula: C = 2 π r. C is the circumference, r is the radius of the circle, and * is the constant 3.14159.** **Radians are calculated by dividing circumference by 2π. Radii are calculated using the circumference of a circle, as follows:

Radius Formula = Circumference/2π or C/2π units

**Radius Formula Using Area of Circle: **A circle has an area which is the space it occupies. The relationship among the radius and area is given by the formula, Area of the circle = π r^{2}. Here, r is the radius and π is the constant which is equal to 3.14159. The radius formula using the area of a circle is as follows:

Radius Formula = √(Area/π) units

## How to Find the Radius of a Circle?

There are three basic formulas that can be used to calculate the radius of a circle, i.e., when the diameter, area, or circumference of the circle is known. Let’s apply these formulas to find the radius of a circle.

- When the diameter is known, the formula for the radius of a circle is: Radius = Diameter / 2
- When the circumference is known, the formula for the radius is: Radius = Circumference / 2π
- When the area is known, the formula for the radius is: Radius = ⎷(Area of the circle / π)

## The radius of a Circle Equation

Observe the a circle in the cartesian plane. The coordinates of the center are (h, k). The radius of a circle equation in the cartesian coordinate plane is given by (x − h)^{2} + (y − k)^{2} = r^{2}. Here, (x, y) are the points on the circumference of the circle that is at a distance ‘r’ (radius) from the center (h, k). When the center of the circle is at origin (0,0), the equation of the circle reduces to x^{2} + y^{2} = r^{2}

## The radius of a Curve

The arc of a circle is the distance between two points along a section of a curve. Curves have a radius determined by the radius of the circle they are a part of. When the length of the chord defining the base (W) and the height (H) measured at the midpoint of the arc’s base is given, then the formula to find the radius is: Radius = (H / 2) + (W² / 8H)

**What is the Radius of a Circle in Geometry?**

The radius of a circle is the length of the line segment connecting the center of the circle to a point on its circumference. It is generally abbreviated as ‘r’.

**Diameter and Radius: What’s the Relationship?**

The diameter of a circle is twice its radius, or, the radius is half the diameter. The relation among radius and diameter can be expressed in the formula: Diameter = 2 × radius.

**What is the radius of a circle if the circumference is known?**

Circumference and radius of a circle are related, and their relationship can be expressed as Circumference = 2πR, where R is the radius. If you know the circumference of a circle, you can calculate its radius using this formula: Radius = Circumference / 2π

**How to Calculate Radius of Circle Using a Calculator?**

The radius is equal to half the diameter, and can be calculated using Cuemath’s online tool simply by entering a diameter, circumference, or area of the circle.