Radius

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14 Feb 2024

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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.

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 = πr2. 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 = r2. 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 x2 + y2 = r2. 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 = r2 ⇒ x2 + (y − b)2 = r2.

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.
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