A circle is a fundamental shape in geometry. We can find circles in various aspects of our daily lives, from the shape of a button or coin to the shape of the sun. A circle is composed of a collection of all the points at a fixed distance from a fixed point called the centre of a circle. This fixed distance is called the radius of the circle. Formally, we can say that the radius of a circle is the distance from the centre of the circle to any point on its circumference.

In this article, we will discuss five different ways of calculating the radius of a circle. But before that, let’s familiarize yourself with some of the properties of a circle.

What is Radius of a Circle?

The radius of a circle is the distance from the centre of the circle to any point on its circumference. It is a constant length that defines the size of the circle and is typically denoted by the letter ‘r‘. The radius is one of the key parameters used to describe a circle, along with the diameter and the circumference.

Properties of a Circle

Various properties of a circle are:

• Radius is the distance from the centre of a circle to any point on the circumference of a circle.
• Ratio of the circumference of any circle to its diameter is always equal to π. Therefore, the circumference of a circle is the product of π and its diameter, or we can also say that the circumference of a circle is the product of π and two times its radius.
• Chord of a circle is a line segment obtained by joining any two points on the circumference. The length of the longest chord of a circle is called the diameter of a circle.
• Diameter of a circle is twice as long as the radius of a circle.
• Arc length of a circle is the distance along the arc (curved line)between two points on the circumference of a circle.
• Angle subtended by the arc to the centre of the circle is the angular distance formed by extending the radii from the centre of the circle to the endpoints of the arc on the circumference of the circle.

The image of circle with centre O and radius ‘r’ is added below:

Radius Calculation

Different Ways to Find Radius of a Circle

Various ways to find the radius of a circle include:

• Calculating Radius of a Circle Using Diameter
• Calculating Radius of a Circle Using Area
• Calculating Radius of a Circle using Circumference
• Calculating Radius of a Circle using Chord Length
• Calculating Radius of a Circle using Arc Length

Method 1: Calculating Radius of a Circle Using Diameter

We know that the diameter of a circle is twice the length of the radius of a circle.

Mathematically we can represent it as,

d = 2×r

r = d/2

where,

• r is Radius of a Circle
• d is Diameter of a Circle

Method 2: Calculating Radius of a Circle Using Area

We know that the area of a circle A is given by, the product of π and the square of the radius of a circle.

Mathematically we can write it as, A = πr²

r² = A / π

r = (A / π)^1/2

where,

• r is Radius of a Circle
• d is Diameter of a Circle
• A is Area of a Circle

Method 3: Calculating Radius of a Circle using Circumference

We know that the perimeter or the circumference of a circle is given by, the product of π and two times the radius of a circle.

Mathematically we can write it as, C = 2πr

r = C/2π

where,

• C is Circumference of Circle
• r is Radius of Circle

Method 4: Calculating Radius of a Circle using Chord Length

We can use chord length formula to find the radius of a circle. For this, we must know the chord length and the angle subtended by the chord at the centre of the circle.

Mathematically by chord length formula,

c = 2r sin(θ/2)

r = c/(2×sin(θ/2))

where,

• r is Radius of Circle
• c is Chord Length of Circle
• θ(in radian) is Central Angle inscribed by Chord

Method 5: Calculating Radius of a Circle using Arc Length

Given the arc length and the central angle of the arc, we can find radius of the circle using the arc length formula.

Mathematically by arc length formula,

a = 2πr(θ/360)

r = (a/2π)(360/θ)

where,

• r is Radius of Circle
• a is Arc Length of Circle
• θ(in degree) is Central Angle of Arc

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Practice Problems with Solutions on Radius of a Circle

Problem 1: If the circumference of a circle is 260cm, then find the radius of the circle.

Solution:

Given, circumference of circle = 260cm

2πr = 260

r = 260/ 2π

r = 41.38cm (approx.)

Therefore, the radius of the circle is 41.38cm (approx.).

Problem 2: The area of a circle is 320cm², find the radius of the circle.

Solution:

Area of the circle = πr²

320 = πr²

r² = 360/π

r = (360/π)^1/2

r = 10.09cm

Therefore, the radius of the circle is 10.09cm.

Problem 3: In a 2D plane the coordinates of the centre of a circle is (4, 5) and there lies a point (1, 2) on the circumference of the circle. What is the radius of this circle?

Solution:

Let the P(1, 2) be the point on the circumference of the circle and O(4, 5) be the point representing the centre of the circle. So, by distance formula,

PO = ((1 – 4)² + (2 – 5)²)^1/2

= ((-3)² + (-3)²)^1/2

= (9 + 9)^1/2

= 18^1/2

= 4.42cm (approx.)

Therefore the radius PO of circle is 4.42cm (approx.).

Problem 4: A circle has an arc of length 10π/3 cm and this arc subtends a central angle of 120^o degree. Calculate the radius of the circle.

Solution:

By the arc length formula,

Arc length = 2πr (θ/360)

10π/3 = 2πr(120 / 360)

Divide by π on both sides,

10/3 = 2r(1/3)

or, 2r/3 = 10/3

r = 5cm

Therefore, the radius of circle is 5cm.

Problem 5: If a circle has a chord of length 8cm and that chord subtends a central angle of 60^o then, what is the radius of the circle?

Solution:

From the chord length formula, we know that,

c = 2r sin(θ/2)—– equation(i)

So, before proceeding further with this formula, we need to convert 60^o into radian,

θ_radian = (π/180) × θ_degree

θ_radian = (π/180)×60

θ_radian = π/3

So, 60^o is π/3 in radian.

Now, using the chord length formula from equation(i), we get,

8 = 2r sin((π/3)/2)

8 = 2r sin(π/6)

or, r = 8/(2 × sin(π/6))

r = 8 / (2 × (1/2))

r = 8/1

r = 8 cm

Therefore, the radius of the circle is 8 cm.

Frequently Asked Questions

What is Relationship Between Radius and Diameter of a Circle?

Diameter d of a circle is twice the length of the radius r of the circle. So, mathematically we can represent this relationship as,

d = 2 × r

What is Equation of a Circle?

Let the centre of the circle be O(h, k) and radius be r, then the equation of circle is given by,

(x – h)² + (y – k)² = r²

This equation represents all the points (x, y) on the circumference of the circle that are at a distance r from the centre(h, k) of the circle.

What is the Longest Chord of a Circle?

The longest chord of a circle is the diameter of a circle.

Can two Circles with Different Radii ever have Same Circumference?

No, two circles with different values of radii will always have different circumference. Because circumference of a circle, by formula 2πr, is dependent on the radius of a circle, which means that if you change the value of radius then the value of circumference will change accordingly.

Is it Possible for a Circle to have Negative Radius?

No, the value of radius of a circle is always non-negative, because radius of a circle is the distance from the centre of a circle to any point on its circumference and therefore it can’t be negative.