from a handpicked tutor in LIVE 1to1 classes
Radius
The radius of a circle is defined as a line segment that joins the center to the boundary of a circle. The length of the radius remains the same from the center to any point on the circumference of the circle. The radius is half the length of the diameter. Let us learn more about the meaning of radius, the radius formula, and how to find the radius of a circle.
1.  What is Radius? 
2.  Radius of Circle Formulas 
3.  Radius of Circle 
4.  How to Find the Radius of a Circle? 
5.  Radius of a Circle Equation 
6.  Radius of a Sphere 
7.  FAQs on Radius 
What is Radius?
Radius is defined as a line segment that connects the center of a circle or a sphere to its circumference or boundary. It is an important part of circles and spheres and is generally abbreviated as 'r'. The plural of radius is 'radi' which is used when we talk about more than one radius at a time.
Meaning of Radius
The radius of a circle is the distance from the center to any point on the boundary of the circle. It should be noted that the length of the radius is half of the length of the diameter. It can be expressed as d/2, where 'd' is the diameter of the circle or sphere. Observe the figure of a circle given below which shows the relationship between radius and diameter.
Now, let us learn about the radius formulas that are used when the other parameters are given.
Radius of Circle Formulas
The radius of a circle can be calculated using some specific formulas that depend on the known quantities and parameters.
Radius Formula with Diameter
The diameter is a straight line passing through the center and joining a point from one end 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. When the diameter of a circle is given, then the radius formula is expressed as:
Radius = Diameter ÷ 2
Radius Formula from Circumference
The perimeter of a circle is called its circumference. It is the boundary of a circle and can be expressed by the formula: C = 2πr. Here, C is the circumference, r is the radius of the circle, and π is the constant which is equal to 3.14159. The radius is the ratio of circumference to 2π. The radius formula using the circumference of a circle is expressed as:
Radius = Circumference/2π
Radius Formula using Area
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 = πr^{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 expressed as:
Radius = √(Area/π)
Radius of Circle
Radius is an important part of a circle. It is the length between the center 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 segment is the radius of that particular circle. A circle can have multiple radii because there are infinite points on the circumference of a circle. This means that a circle has an infinite number of radii and all these radii are equidistant from the center of the circle. The size of the circle changes as soon as the length of the radius changes.
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 are termed as the radii of the circle. It is to be noted that OA = OB = OM = ON = OP = OQ = OX = OY.
How to Find the Radius of a Circle?
The radius of a circle can be found using the three basic radius formulas. These formulas are formed using the diameter, the area, and the circumference. Let us use these formulas to find the radius of a circle.
 When the diameter of a circle is known, the formula is, Radius = Diameter/ 2. For example, if the diameter is given as 24 units, then the radius is 24/2 = 12 units.
 When the circumference of a circle is known, the formula is, Radius = Circumference/2π. If the circumference of a circle is given as 44 units, then its radius can be calculated as 44/2π. This implies, (44×7)/(2×22) = 7 units.
 When the area of a circle is known, the formula for the radius is, Radius = ⎷(Area of the circle/π). if the area of a circle is given as 616 square units, then the radius is ⎷(616×7)/22 = ⎷28×7 = ⎷196 = 14 units.
Radius of Circle Equation
The radius of a circle equation on the cartesian plane with center (h, k) is given as (x − h)^{2} + (y − k)^{2} = r^{2}. This is known as the equation of a circle when the radius is known. 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}. Observe the diagram of a circle on the cartesian plane shown below. Here, the coordinates of the center are (0, b) and the radius of the circle is represented by 'r' joining the center to the point (x, y) on the circle. So, we just need to substitute these values in the above equation to get the radius of the circle equation. The equation to find the radius of this circle is (x − 0)^{2} + (y − b)^{2} = r^{2} ⇒ x^{2} + (y − b)^{2} = r^{2}.
Radius of a Sphere
A sphere is a 3D solid figure. The radius of the sphere is the segment from the center to any point on the boundary of the sphere. It is a determining factor while drawing a sphere as its size depends on its radius. Like a circle, there can be infinite radii drawn inside a sphere and all those radii will be equal in length. To calculate the sphere's volume and surface area, we need to know its radius. And we can easily calculate the radius of the sphere from its volume and surface area formulas.
Radius of Sphere from Volume = ^{3}⎷(3V/4π), where V represents the volume and the value of π is approximately 3.14
Radius of Sphere using Surface Area = ⎷(A/4π), where A represents the surface area.
Use our free online radius of sphere calculator to calculate the radius with the given volume, surface area, or diameter of a sphere.
☛ Related Articles
Check these interesting articles related to the radius and its formulas.
Radius of Circle 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 radius of circle equation 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} + (5 1)^{2} = r^{2}
3^{2} + 4^{2}= r^{2}
9 + 16 = r^{2}
r^{2} = 25
r = 5
Therefore, the radius of the given circle is 5 units.

Example 2: What is the radius of a circle with a circumference of 15 inches?
Solution:
The radius of the circle with a circumference of 15 inches can be calculated by using the formula, r = C/2π
r = 15/2π
r = (15×7)/(2×22)
r = 105/44
r = 2.39 inches
Therefore, the radius of the given circle is 2.39 inches approximately.

Example 3: What is the radius of the circle if the area is 36 m^{2}?
Solution:
Given, area of circle, A = 36 m^{2}.
The radius of the circle using area can be calculated by the formula, r = ⎷(A/π)
r = ⎷(36/π)
r = ⎷(36×7)/(22)
r = ⎷252/22
r = ⎷11.45 m
r = 3.38 m
Therefore, the radius of the given circle is 3.38 meters approximately.
FAQs on Radius of Circle
What is the Radius of a Circle in Geometry?
The radius of a circle is the length of the line segment from the center to a point on the circumference of the circle. It is generally abbreviated as ‘r’. There can be infinite radii drawn in a circle and the length of all those radii will be the same. It is half of the diameter of the circle.
How is Diameter Related to the Radius of the Circle?
The diameter of a circle is twice the radius. It can also be said that the length of the radius is half the diameter. The relation between radius and diameter can be expressed in the formula: Diameter = 2 × radius. Use a free online radius calculator to calculate the radius with the given diameter.
How to Find the Radius from Circumference?
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 Formula?
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 is, Radius = Diameter / 2
 When the circumference is known, the formula is, Radius = Circumference / 2π
 When the area is known, the formula is, Radius = ⎷(Area of the circle / π)
How to Calculate Radius of Circle Using Calculator?
The length of the radius is equal to half the length of the diameter which can be calculated using Cuemath's online radius calculator by simply entering any given value such as the diameter, circumference, or area of a circle.
How to Find the Radius of a Circle with the Area?
If the area of a circle is given, then the formula to find the radius is given as Radius = ⎷(A/π), where A is the given area.
How to Calculate Radius from Diameter?
The radius of a circle can be calculated when the diameter is given. The relationship between the radius and the diameter of a circle is expressed as, Radius = Diameter ÷ 2. So, if the diameter of a circle is given as 6 units, then the radius will be, 6 ÷ 2 = 3 units.
visual curriculum