Learn what radius is by definition and discover four methods to calculate the radius of the circle—practice given examples and other questions.

1. Radius: Methods to Calculate the Radius
of a Circle
Almost all class teachers or private math tutors start the circle’s topic with the story of circular shape
technological invention. Do you know this? It was the wheel, Potter’s invention, in 3500 BC. It was
when people began to study circles, use circular-shaped objects, and start finding their
measurements using circumference, diameter, and radius.
So, let’s study the radius of a circle and how to calculate it using different methods and formulas.
Definition: Radius of a Circle
In geometry, by definition, a line segment from the midpoint or center of a circle to the outer edge
or boundary of the circle. It is also known as half of the length of the diameter. We use the word
‘radii’ when more than one radius is represented as it is the plural of radius.
The radius of a circle is usually represented by ‘R,’ or ‘r.’ Radius is also the dimension for a semi-
sphere, sphere, cones, and cylinders with circular bases.

2. Formula of Radius
As we know, the radius is half of the diameter of a circle therefore it is formula is:
Radius = Diameter / 2
Important Terms of a Circle
To master calculating the radius of the circle, you should know the following terms and their
measurements because without using them, you cannot find radii.
1. Area
The space occupied by the circle in a 2D or two-dimensional plane is known as the area of the circle. In
other words, the area of any circle is the space occupied within its circumference or boundary.
It is denoted by A, and the square unit is the unit of area. For instance, m2, cm2, in2, etc.

3. Formula of Area
The area of a circle is calculated by
Area = A = πr2
Or,
Area = A = πd2/4
(Value of Pi = π = 3.14 or 22/7)
Formula of Area and Central Angle of a Sector
Area sector = A sector = (θ/360°) × πr2
2. Midpoint or Center
The exact central point of the circle equidistant from the points on edge is known as the midpoint
or center of the circle.

4. 3. Diameter
A line segment starting from one point of the circle to another point is known as the diameter of
the circle. It is the longest chord that passes through the center of the circle, twice the length of
the circle’s radius.
The circle’s diameter is usually denoted by ‘D’ or ‘d’. You can find the circumference of a circle and
area with respect to the diameter.

5. Formula of Diameter
• Diameter = D = 2 x Radius
4. Circumference
The length of the circle’s boundary is equal to the circumference of the circle. Alternatively, the
circle’s circumference is the perimeter of a circle.
It is denoted by the uppercase letter ‘C.’

6. The Formula of Circumference
Following is the circle’s circumference formula
Circumference = C = π × d
Or
Circumference = C = 2 x r x π
Radius Formulas
You can calculate the radius of the circle by using some specific formulas. Below we have
mentioned the radius formulas:
1. From Area
The area is the total space occupied by the circle, and the radius can be calculated using its formula.
As the area of circle = πr2
Therefore, the radius formula using the area of a circle would be expressed as:

7. • Radius = r = √(Area/π) units
2. From Diameter
The longest chord or a line segment twice the length of the radius is known as the diameter of a
circle. With the diameter of a circle, you can easily find out the radius of the circle. Here is how:
As Diameter = D = 2 x radius
Therefore,
• Radius = D / 2 or Diameter / 2 units
3. From Circumference
A circle’s circumference is known as its perimeter or boundary. This is how you can express the
radius formula using the circumference of the circle’s formula:
As,
Circumference = C = 2 π r
Therefore,
Radius = r = C / 2 π or Circumference / 2 π units
Methods to Calculate the Radius of a Circle
In this section, you will learn how to calculate the radius of a circle by using the following formulas
and practice examples in all possible four ways.
1. By the Area
Set up the formula of the area of a circle before solving a question.
A = πr2
Therefore,
r2 = A / π
Take sqrt of both sides
⇒ r = √ A / π

8. Example Question:
If the Area of the circle is 153. 85 m2, then calculate its radius.
Solution:
As A = πr2
Given that,
A = 153. 85 m2
Pi = 3.14
Therefore,
r = √ A / π
putting values
⇒ r = √ 153.85 / 3.14
⇒ r = √ 49
⇒ r = 7
Hence, radius of the circle is 7 meters
2. By the Circumference
Set up the formula of the circumference of the circle before solving a question.

9. As
C = 2 π r
Therefore,
⇒ r = C / 2 π
Example Question:
Calculate the radius of the circle with 15 cm circumference. Write the complete formula.
Solution:
The formula of radius derived from the circumference of a circle formula:
r = C / 2 π
given values
C = 15 cm
Pi = 3.14
Putting values in formula
⇒ r = 15 / 2 x 3.14
⇒ r = 15 / 6.28
⇒ r = 2.39
Hence, the radius of the given circle is 2.39 cm.

10. 3. By the Diameter
Set up the formula of the circle’s diameter before solving a question.
As
D = 2 x r
Therefore,
⇒ r = D / 2
Example Question:
Find radius of the circle if its diameter is 6 centimeters.
Solution:
Given that
Diameter of the circle = D = 6 cm
Using formula
⇒ r = D / 2
⇒ r = 6 / 2
⇒ r = 3
Hence, the radius of the circle is 3 cm.
4. By Area and Central Angle of a Sector

11. Put given values in the formula of the area and central angle of a sector or A sector and after
calculating all the parts step by step, take the square root of both sides to find the radius of the
given circle. The formula of A sector is given below:
⇒ A sector = (θ/360°) × πr2
Example Question:
Calculate the radius if the central angle of a circle is 120 degrees with 100 cm2 area of the sector.
Solution:
Given that,
Area of sector = A sector = 100 cm2
Central angle of a circle = θ = 120°
Pi = π = 3.14
Using formula,
A sector = (θ/360°) × πr2

12. Putting values
⇒ 100 = (120 / 360) × πr2
⇒ 100 = 0.33 x πr2
⇒ 100 / 0.33 = πr2
⇒ 303.03 = πr2
⇒ 303.03 = (3.14) r2
⇒ 303.03 / 3.14 = r2
⇒ 96.51 = r2
Taking square root on both sides
⇒ √96.51 = √r2
⇒ 9.82 = r
Hence, the radius of the circle is about 9.82 cm.
Frequently Asked Questions
Question: What is half the diameter called?
Answer: Half the diameter of a circle is known as Radius. To summarize, a line segment joining the
center to the edge of the circle is called the radius of the circle. Half the length of the diameter,
easy to drill it into your mind.

13. Question: How do you drive the radius formula from the diameter of the
circle?
Answer: The diameter splits the circle in half, and as you know, the radius of the circle is half of the
diameter. Therefore, you can easily calculate the radius of a circle through its diameter. Here is
how:
Formula of diameter of circle = D = 2 x radius
Thus,
Radius = D / 2
Question: Do we use circumference in regular or everyday life?
Answer: Yes, look around, and you will see trillions of real-life examples of circles and applications
of circumference. Sometimes to pick the right size of an object, we should know the distance
around something, for instance, to find the right wheel for your vehicle. Moreover, to estimate the
wood in a tree, measuring its circumference is easier and quicker than directly measuring the
diameter. If you keep an eye, the list of examples and usage of circumference in everyday life would
be endless.
Question: What is the formula for the circumference of a circle?

14. Answer: The path or boundary that defines or surrounds the shape is called the circumference of
the circle. It is also known as the perimeter of the circle as it helps to identify the length of any
shape’s outline.
Therefore, the formula of circumference = 2πr
Question: What is the difference between the chord and diameter of the
circle?
Answer: A straight line joining two points on the circle is called a chord, and the longest possible
line that passes through the center of the circle or twice the radius is called its diameter. However,
the diameter is an example of a chord.

15. Question: What are the longest, longer, and shortest circle measures?
Answer: Following are the longest, longer and shortest measure of a circle:
Circumference = longest
Diameter= longer
Radius= shortest