Radius

Radius is defined as a line segment joining the center to the boundary of a circle or a sphere. The length of the radius remains the same from the center to any point on the circumference of the circle or sphere. It is half of the length of the diameter. Let us learn more about radius in this article.

In geometry, the radius is defined as a line segment joining the center of the circle or a sphere to its circumference or boundary. It is an important part of circles and spheres which is generally abbreviated as 'r'. The plural of radius is "radii" which is used when we talk about more than one radius at a time. The largest line segment in a circle or sphere joining any points lying on the opposite side of the center is the diameter, and 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. Look at the image of a circle given below showing the relationship between radius and diameter.

Now, let us learn the formulas of radius that will help you to calculate its length with the given information.

The radius of a circle and sphere can be calculated using some specific formulas that you are going to learn in this section. Here, we will talk about radius formulas for a circle. The radius of a sphere formula is discussed in the section below.

Radius Formula from 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 or D/2 units

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 units. 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 or C/2π units

Radius Formula with 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 square units. 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) units

Radius is one of the important parts of a circle. It is the distance 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 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. The size of the circle changes when the length of the radius varies.

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. Observe that OA = OB = OM = ON = OP = OQ = OX = OY.

The radius of a circle can be found using the three basic radius formulas i.e., 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 is Radius = Diameter/ 2.
• When the circumference is known, the formula is Radius = Circumference/2π.
• When the area is known, the formula for the radius is Radius = ⎷(Area of the circle/π).

For example, if the diameter is given as 24 units, then the radius is 24/2 = 12 units. 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. And, if the area of a circle is given as 616 square units, then the radius is ⎷(616×7)/22 = ⎷28×7 = ⎷196 = 14 units.

The radius of a circle equation on the cartesian plane with center (h, k) is given as (x − h)² + (y − k)² = r². 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.

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 units, where V represents the volume and the value of π is approximately 3.14.

Radius of Sphere using Surface Area = ⎷(A/4π) units, 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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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, or, 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 of a Circle with the 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 of a Curve?

The radius of a curve or an arc 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 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 for the radius is 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 that can be calculated using Cuemath's online calculator by simply entering any given value amongst, 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) units, where A is the given area.