Circumference of a Circle Definition, Formula with Solved Examples

Firstly, we must understand what a circle is. In simple geometrical words, a circle is a closed, round figure formed by joining all the points in a plane at a given distance from a particular point known as the center. The circumference of a circle is a significant element for a circle. Hence, we must know all the details regarding the circumference of a circle.

The circumference of a circle

The circumference of any shape in Mathematics determines the path or boundary that surrounds it. In other terms, circumference, also known as perimeter, is used to determine the boundary’s length of any shape. Therefore, the circumference of any shape plays a significant role in calculating its boundary dimensions.

The circumference is the measurement of the circle’s boundaries. When we cut open a circle and draw a straight line through it, this length is known as its circumference or the perimeter. It is commonly expressed in units like centimeters, meters, or other relevant units of length.

The circle’s radius is also taken into account to find the circle’s circumference. Thus, to calculate the perimeter of a circle, we must first determine the radius or diameter of that circle.

Circumference refers to the distance around a circle. It is a one-dimensional measurement of any two-dimensional circular surface’s boundary. So, finding the circumference of a circle is commonly known as calculating the circle’s perimeter since it follows the same principle as finding the perimeter of any polygon.

The circle is a simple, round geometrical shape, and the value of Pi(π) is approximately equal to 3.1415926535897………… we use a Greek letter to describe this value as non-terminating.

For a circle shown below, the circumference and diameter are:

In other terms, the circumference is the distance that surrounds the circle. The circle’s diameter is the distance across it from its center to the two points on its periphery. The ratio of a circle’s perimeter to its diameter is equal to π. As a result, we get a value close enough to the pi(π) value when we divide the circumference by the diameter of any circle. Thus, the following formula can be used to describe this relationship:

C/D = π, where C stands for circumference and D for diameter.

 C= πD is another way of writing this formula when we have to find the circumference of a circle and the circle’s diameter is specified.

Therefore, the circumference comprises various other factors. The three most important factors of a circle are center, diameter, and radius.

Center: The center is a location on the circumference at a set distance from any other point.

Diameter: The circle’s diameter is the distance from one end of the circle to a point on the other end of the circle, passing through the center.

Radius: The radius of a circle is the distance between the circle’s center and any point along its perimeter.

Circumference of a Circle Formula

The radius ‘r’ of the circle and the value of ‘pi’ can determine the circumference of a circle formula.

The circumference of the circle formula is = 2πR 

Where,

• The radius of the circle is R
• π is a mathematical constant having an approximate value of 3.14

Also, Pi (π) represents the circumference to the diameter ratio of any circle.

Therefore, C = πD

Where,

• The circumference of a circle is denoted by C
• The diameter is represented by D

Different formulas to find the circumference of a circle

To find the perimeter of a circle, we can use three distinct formulas

1. When the radius (R) of a circle is known, then the formula is:

     Circumference of a circle = 2πR

2. When the diameter (D) of a circle is known, then the formula is:

        Circumference = πD

3. If the area(A) of a circle is known, then the formula is:

         Circumference = 4πA, where A is the area of the circle.

 Circumference to Diameter

The radius of a circle is twice its diameter, which means D = 2R       

Also, the circumference ratio of a circle to its diameter is equal to Pi(π). Therefore, we can say that this is the definition of pi(π).

i.e., C = 2πR

=>   C = πD (since, D = 2R)                                                                                                    

Now, if we divide both sides with D (diameter), we will achieve a value that is very close to the approximate value of pi (π).

This means, C/D = π

How to find the circumference of a circle?           

Method 1:

We cannot physically measure the length of a circle with a scale because it is a curved surface. However, this is possible only for polygons such as squares, triangles, and rectangles. Instead, we can use a thread to measure the circumference of a circle. Using the thread, we can trace the curved path of the circle and mark places on the thread. A regular ruler can be used to measure this length.

Method 2:

Calculating the circumference of a circle is the most accurate way to determine it. The radius of the circle must be known for this method. The figure below shows a circle with radius R and center O. Its diameter is twice its radius.

Hence, we can conclude by saying that the circumference is an essential element to measure the dimensions of a circle. We are also now clear that the circumference of a circle is the product of the constant π and the diameter of the circle. So, now, if someone asks, what is the circumference of a circle? Or how to find the circumference of a circle? You can answer them without facing any difficulties.

 Here are some examples to help you understand the concept better

Example 1: What is the circle’s circumference with a diameter of 7 cm?Solution: From the question, the diameter is known to us, so the radius(R) = 7/2 cm = 3.5 cm Hence, the circumference of the circle = 2πR = 2 x 3.14 x 3.5 = 21.98 cm Example 2: Find the radius of the circle where C = 80 cm. Solution: So, the circumference is given = 80 cm The formula that we know is, C = 2πR                    Which implies, 80 = 2πR                                        =>80/2 = 2πR/2                                    =>40 = πR                                       =>R = 40/π Therefore, the radius of the circle is 40/ π, which is equal to 12.74 Example 3: Find the perimeter of a circle that has a radius of 9 cm? Solution: In the above question, the radius is given to us, R = 9 cm So, the formula for the circle’s circumference is known to us, and the perimeter is also known as the circumference. Therefore, the formula for the perimeter of a circle is, C= 2πR Now, by substituting the value of R, that is, 9 cm in this question, we get:                           C = (2 x 3.14 x 9) cm                               = 56.52 cm Example 4: Calculate the perimeter of a circle in terms of π, having a diameter of 20 cm. Solution: The circle’s diameter is given, which is equal to 20 cm. Therefore, the circle’s radius is, R = 20/2 cm = 10 cm. We know that the circumference formula of a circle is, C = 2πR So, the C for this circle is = (2 x π x 10) = 20π cm Example 5: What will be the diameter of a circle having a circumference of 8 cm? Solution: The circumference (C) is known in this question which is equal to 8 cm. We know that, C = 2πR                 =>C = πD (since, D=2R)                  =>D = C/π = (8/3.14) cm = 2.55 cm Hence, the diameter of the given circle with known circumference is equal to 2.55 cm. Example 6: If a circle has a circumference of 16cm, what will be the radius of that circle? Solution: Here, the given circumference is 16 cm The formula is known to us, which is C = 2πR               So, the radius, R = C/2π = (16)/ (2 x 3.14) cm = 2.547 cm