Radius
Circles are one of the most commonly found shapes in the world. The fascinating properties of circles make it an important topic in geometry. Radius is one of the important parts of a circle. It is the distance between the centre of the circle to 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.
A circle can have more than one radius because there are infinite points on its circumference. This means that a circle has an infinite number of radii and all the radii of the circle are equidistant from the center of the circle. In this lesson, we will learn more about the radius of the circle.
What is Radius?
A circle is a collection of points on a twodimensional plane, which are equidistant from the center point 'O'. The distance from the center point to any endpoint on the circle is called the radius of a circle. It can also be defined as the length of the line segment from the center of a circle to a point on the circumference of the circle. A circle can have many radii (the plural form of radius) and they measure the same. The size of the circle changes when the length of the radius varies. The radius of a circle is generally abbreviated as ‘r’.
In the figure given below, the points A, B, M, N, P, Q, X, and Y lie on the boundary of the circle. Observe that these points are equidistant from the center O. So, all the line segments OA, OB, OM, ON, OY, OX, OP, and OQ can be termed as the radii of the circle. Observe that OA = OB = OM = ON = OP = OQ = OX = OY.
Diameter of a Circle
The diameter of a circle is a straight line, passing through the center and joining a point from one end of the circle to a point on the other end of the circle. The diameter is twice the length of the radius. Mathematically, it is written as: Diameter = 2 × radius. It is also the longest chord of a circle. In the above figure, AB is the diameter of the circle.
Circumference of a Circle
The perimeter of a circle is called its circumference. It is the boundary of a circle and can be expressed by the formula: C = 2 π r. Here, C is the circumference, r is the radius of the circle, and π is the constant which is equal to 3.14159.
Area of a Circle
The area of a circle is the space occupied by the circle. The relationship between 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.
How to Find the Radius of a Circle?
The radius of a circle can be found when the diameter, the area, or the circumference is known. Let us use 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)
Radius of a Circle Equation
Observe the diagram of a circle in the cartesian plane shown below. 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 are 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}
Radius of a Curve
The arc of the circle is the distance between two points along a section of a curve. The radius of the curve is the radius of the circle of which it is a part. 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)
Solved Examples

Example 1:
Find the radius of the circle whose center is O (2, 1), and the point P (5, 5) lies on the circumference.
Solution:
The equation of a circle in the cartesian plane is given by (x − h)^{2} + (y − k)^{2} = r^{2}. Substituting the value of (x, y) as (5, 5) and (h, k) as (2, 1) we get:
(5−2)^{2} + (51)^{2} = r^{2}
3^{2} + 4^{2}= r^{2}
9 + 16 = r^{2}
r^{2} = 25
r = 5Therefore, the radius of the given circle is 5 units.

Example 2:
The dimensions of the segment of a circle are as shown in the figure. Find the radius of the curve.
Solution:
The radius of the curve is calculated using the formula (H / 2) + (W^{2} / 8H). Given, W = 8 and H = 4. Substituting these values in the radius of a curve formula, we get:
(H / 2) + (W^{2} / 8H) = (4 / 2) + (8^{2} / 8(4))
= 2 + 64/32 = 2 + 2 = 4. Therefore, the radius of the curve is 4 yd.
FAQs on Radius
What is the Radius of a Circle?
The radius of a circle is the length of the line segment from the center of a circle to a point on the circumference of the circle. It is generally abbreviated as ‘r’.
How is Diameter Related to the Radius of the Circle?
The diameter of a circle is twice the radius, or, the radius is half the diameter. The relation between radius and diameter can be expressed in the formula: Diameter = 2 × radius.
How to Find the Radius of a Circle if Circumference is given?
The circumference of a circle and radius are related to each other and their relation can be expressed as Circumference = 2πR, where R is the radius. So, when the circumference is known, the formula used to calculate the radius of a circle is: Radius = Circumference / 2π
What is the Radius of a Curve?
The radius of a curve is the radius of the circle of which it is a part. When the length of the chord defining the base (W) and the height measured at the midpoint of the arc's base (H) is given, the formula to find the radius is: Radius = (H / 2) + (W² / 8H)
What is the Formula to Find the Radius of a Circle?
The radius of a circle can be calculated through various formulas. Observe the following formulas to calculate the radius:
 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)