Circles
Ever been to a race course as shown? A race track/course is seldom straight and often has semi-circular turns. But, now how to find the distance covered by an athlete (or) a horse if he takes, say two rounds of the track? In cases like these, we need to use methods to find the lengths around the circular shape. So, let's get started!
Circumference of a Circle
For an art project, Sneha needs to paste ribbon on circular cardboard cutouts. What length of ribbon is required to accomplish this task. Here, we need to find the circumference of the circular shapes. Here,
The distance around a circular region is known as its circumference
or in other words,
The length of the edge of a circular shaped region is called its circumference
Since, the edge is no longer “straight”, a ruler cannot be used for measurement. So, what can we do? (Unroll the wheel given below)
There is one way:
To find the length of ribbon required, we can use a marker to mark a point on the edge of the circle where it is touching the ground.
Now, slowly roll the circular shape on the table along a straight line (use a long ruler for precision) till the marked point again touches the surface. Mark this position on the surface as well.
Using the ruler, measure the distance between the two points on the surface.
This is the length of the ribbon required.
Say, now we need to make measurements on a bigger scale. For example: we need to find the distance that needs to be covered by an athlete during a race? Can this same method be used? While possible, it is still very difficult to find the distance around the track. Moreover, the chances of error occurring is very high.
Thus, we need some formula for such shapes, as we have for rectilinear figures or shapes.
Activity: Draw six circles of different radii and find their respective circumferences. Also, calculate the ratio of the circumference to the diameter.
| Circle | Circumference | Diameter | `C/d` |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 |
What do we infer from the above filled table? Do you observe something in the ratio column? Is the ratio (approximately) the same, for all the cases?
Can we say that, the circumference of a circle is always more than three times its diameter?
This ratio, that we have calculated, is a universal constant and is denoted by π (pronounced as pi). Its approximate value is or 3.14 (approx).
Note: The value of "3.14" is a rounded-off value and in reality, the value of π is infinite (it is a irrational number).
So, we can say that:
= π
where
‘C’ represents circumference of the circle and
‘d’ is the circle diameter.
(or)
C = πd
We know that diameter (d) of a circle is twice the radius (r) i.e.
d = 2r
So,
C = πd = π × 2r (or)
C = 2πr
Circumference of Circle = πd = 2πr
Looking at the figure below, can you tell:
(a) Which square has the larger perimeter- blue or red?
(b) Which is larger, perimeter of smaller square or the circumference of the circle?
Circumference of circle= πa = 3.14a
Blue Square area = 4a
Red Square area = 4s = 4(
Let's Solve
- The radius of a circular pipe is 10 cm. What length of a tape is required to wrap once around the pipe (π = 3.14)?
Find the perimeter of the given shape. (Take π = )
- Perimeter =
times of semicircle - The diameter of the semi-circle is 14 cm.
- Circumference of the semicircle =
- Calculating we get, perimeter =
cm - The total perimeter of the flower becomes
cm - Calculating perimeter
- Hence, perimeter of flower has been found.
Sudhanshu divides a circular disc of radius 7 cm in two equal parts. What is the perimeter of each semicircular shape disc? (Use π = )
- Perimeter of semi-circle =
where r is the radius. - Upon calculating, perimeter =
cm. - The length of the straight edge is
cm. - Length of straight edge is equal to the diameter
- Total perimeter of the disc =
cm - Hence, total perimeter of disc is 36 cm.