We know that circle is a closed figure which is the path described by the point which moves such that it is at a constant distance from a fixed point. We are already aware of the parts of a circle, which is the center, radius, diameter, chord and the properties of chords of a circle. Being a two dimensional we can find the area and circumference of a circle. We are familiar in finding the perimeter of a square or a rectangle, which is the sum of all the sides, which is the boundary of a closed figure. When we increase the number sides of a regular polygon, the perimeter or the length of the boundary is the sum of all the sides, called the perimeter of the closed figure. In this section let us see how to find the circumference of a circle.

What is circumference? It is the length of the boundary of a circle. It is the length of the arc of the closed curve. It is also called the perimeter of the circle.

Let us observe the following circles.
Circumference Definition
In the above figure, we observe that the length of the boundary of the circle increases as the radius increases.
Therefore, the boundary of the circle depends on the radius of the circle.

Circumference of a Circle Formula

We know that diameter of a circle d = 2 x radius = 2 r
The circumference is calculated using the formula C = 
2 $\pi$ r
It is found that the circumference of a circle bears a constant ratio with the diameter of the circle.

(i.e) $\frac{C}{d}$ = constant

= $\pi$ [ since the constant is an irrational number, it is denoted as $\pi$ ]
Therefore, we have the circumference of the circle C = $\pi$ d
= $\pi$ (2 r) [ by substituting d = 2 r ]
=> C = 2 $\pi$ r which is the formula to find the circumference of a circle.

Since the value of $\pi$ is approximately equal to the value of $\frac{22}{7}$, we can plug in $\frac{22}{7}$ in the place of $\pi$ while solving numerical problems.

As the diameter of a circle increases, the length of the arc of the circle increases.

The ratio of diameter to the circumference bears a constant ratio which is equal to the reciprocal of $\pi$

(i.e), $\frac{d}{C}$ = $\frac{1}{pi}$

=> $\pi$ d = C [by cross multiplication]
=> $\pi$ (2 r) = C
=> 2 $\pi$ r = C (Circumference of a circle of radius r or diameter d)


Solved Examples

Question 1: Calculate the circumference of a circle of diameter 10 cm.
Solution:
 
We have, the radius of the circle = 10 cm
Circumference of a circle for the given diameter = $\pi$ d
                                                                    = $\pi$ (10)
                                                                    = 10 $\pi$
Substituting $\pi$ = 3.1415,
                                   we get, circumference = 10 (3. 1415)
                                                                    = 31.415
                                                                    = 31.42 cm [ correct to two decimal places ]
 

Question 2: Calculate the circumference of a circle of radius 8 cm
Solution:
 
Method 1 :                      We have the radius = 8 cm
Using the above formula,          Circumference = 2 $\pi$ r
                                                                   = 2 $\pi$ (8)
                                                                   = 16 $\pi$
                                                                   = 16 x 3.1415
                                                                   = 50.254
                                   Circumference        = 50.26 cm [ correct to 2 decimal places ]

Method 2 :                     We have the radius = 8 cm
                    Therefore,                diameter d = 2 r
                                                                  = 2 (8)
                                                                  = 16 cm
Using the above formula,         Circumference = $\pi$ d
                                                                  = $\pi$  (16)
                                                                  = 16 $\pi$ 
                                                                  = 16 x 3.1415
                                                                  = 50.254
                                                                  = 50.26 cm [ correct to 2 decimal places ]
 

Question 3: The circumference of a circle is 123.2 cm. Calculate the radius of the circle.
Solution:
 
Circumference = 123. 2 cm

We have the formula,          Circumference C = 2 $\pi$  r
                                                                 = 123.2 cm
Method 1:
                                                2 $\pi$  r    = 123.2 cm

                                        =>                r    = $\frac{123.2}{2\pi}$
                                                               
                                                                 = 19.617
                                                  radius     = 19.62 cm [ correct to 2 decimal places ]
Method 2:  We have

         the circumference,                2 $\pi$ r = 123.2

substituting $\pi$ = $\frac{22}{7}$, we get,
                                              2 x $\frac{22}{7}$ r = 123.2

                           =>                    $\frac{44}{7}$ r = 123.2

                          =>                                r = 123.2 x $\frac{7}{44}$

                                                 radius      = 19.6 cm

 

Question 4: The diameter of a bicycle wheel is 28 cm. How many revolutions will it make in moving 13.2 km.?
Solution:
 
Distance traveled by the wheel in one revolution = circumference of the wheel
                                                                     = $\pi$ d

                                                                     = $\frac{22}{7}$ (28)

                                                                     = 88 cm
                 Total distance traveled by the wheel = 13.2 km
                                                                     = 13.2 x 1000 m
                                                                     = 13200 m
                                                                     = 13200 x 100 cm
                                                                     = 1320000 cm
        Number of revolution made to travel 88 cm   = 1

number of revolutions made to travel 1320000 cm = $\frac{1320000}{88}$

                                                                       = 15,000

                         Number of revolutions made = 15,000
 

Question 5: The radius of a wheel is 35 cm and it takes 5 minutes to make 250 rotations. Find the speed of the wheel in km per hour. [ take $\pi$ = $\frac{22}{7}$ ]
Solution:
 
Radius of the wheel = 35 cm
                             Circumference of the wheel = 2 $\pi$  r

                                                                     = 2 x $\frac{22}{7}$ x 35

                                                                     = 220 cm
                                                                     = 2.2 m
Therefore, the distance covered in one rotation = 2.2 m
                  Distance covered in 250 rotations = 250 x 2.2
                                                                   = 550 m
Therefore, the distance covered in 5 minutes   = 550 m

              => distance covered in 1 min           = $\frac{550}{5}$

                                                                   = 110 m
            Therefore, distance covered in 60 min = 110 x 60
                                                                   = 6,600 m
                                                                   = 6.6 km
                      ( i. e ) distance covered in 1 hr = 6.6 km
                      =>          speed of the wheel = 6.6 km / hr
 

Question 6: The shape of a park is a rectangle bounded by semi circles at the ends, each of diameter 35 m, as shown in the figure. Find
a. the perimeter of the park.
b. Cost of fencing it at the rate of $\$$ 3 per m.
Circumference of a Circle Example
Solution:
 
Diameter of the park = 35 m
                     Length of the rectangular portion = 50 m
The above park consists of the rectangle and the two semi circles.
             Therefore, the boundary of the park = circumference of the two semi circles + 2 (length of the rectangle)
Since the circumference of two semi circles = circumference of one full circle, we get
                        the boundary of the park = circumference one circle + 2 (length of the rectangle)
                                                               = $\pi$ d + 2 (50)

                                                               = $\frac{22}{7}$ (35) + 100
                                   
                                                               = 110 + 100 = 210 m
             ( i. e ) the perimeter of the park = 210 m    ------------------------------- ( a )
                 Cost of fencing @ $\$$ 3 per m = 210 x 3
                                                               = $\$$ 610
                                The cost of fencing = $\$$ 610   ----------------------------- ( b )