Circumference of a Circle: A circle is a closed round shape with all the points equidistant from a fixed point known as the centre. We come across many shapes in our daily lives, but the circle is the most commonly used shape. A dinner plate, a ferris wheel, the dial of a clock, cricket ground, candy, and steering wheel are a few examples of circle shapes that we come across every day.

Let us understand the meaning of Circumference. The circumference of any object in mathematics specifies the route or border that surrounds the object. In other words, the circumference of a circle is also called as the perimeter of circle. The circle’s circumference is measured in units such as metres and centimetres. The ratio of circumference of a circle to its diameter is fixed for all circles drawn with any radius. This ratio is represented using the Greek letter (pi).

Circle’s Circumference

Circumference of a circle or perimeter of circle is the measure of the length of the boundary of the circle.

If the circle is cut at one end and opened to form a line segment, then its length is the circumference. It is generally measured in units like \( {\text{cm}}\) or \( {\text{m}}\)

It is essential to know the radius or the diameter of a circle in order to calculate its circumference.

Circumference of a Circle: Formula

If the radius of a circle is known, then the circumference of the circle can be calculated by using the formula,
Circumference \(= 2\pi r\)
where \(\;r\) represents the radius of the circle and \(\pi \) is the mathematical constant whose value is approximately equal to \(\frac{{22}}{7}\) or \(3.14\) (it is an irrational number with non-terminating decimal places)

Circumference Used in Real Life

Suppose if one wants to buy a pair of trousers, sweater, T-shirts etc., then the distance around the waist or chest must be calculated. Since our body is not a perfect circle so we have to measure its circumference by using a measuring tape. This technique is generally used by tailors to find the circumference of a dress.

Even clockmakers need to have good knowledge about circles as they have to give a uniform border while designing and decorating clocks.

Parts of Circumference of a Circle

When an arc is rotated around \(360^\circ \) then the circumference of a circle is formed.

Two types of arcs can be formed in a circle depending on its circumference. If the arc is more than half of the circumference, it is a major arc. If it is less than half of the circumference, it is a minor arc.

Circumference of a Circle with Diameter

We know that \(d = 2r.\)
Hence, the circumference of the circle in the form of diameter \( = \pi \left( {2r} \right) = \pi d\)
Where d represents the diameter of the circle

Circumference of a Semicircle

The circumference of a semicircle is equal to half of the circumference of a circle.
i.e., the circumference of a semicircle \( = \frac{1}{2}\left( {2\pi r} \right) + 2r = \pi r + d\)

How to find Width of Concentric Circles?

Let \({C_1}\) and \({C_2}\) be the circumference and \({R_1}\) and \({R_2}\) be the radius of the outer circle and the inner circle, respectively. Then,
Circumference of the outer circle \({C_1} = 2\pi {R_1}\)
\( \Rightarrow {R_1} = \frac{{{C_1}}}{{2\pi }}\)
Circumference of the inner circle \({C_2} = 2\pi {R_2}\)
\( \Rightarrow {R_2} = \frac{{{C_2}}}{{2\pi }}\)
Therefore, the width of the concentric circles \( = ({R_1} – {R_2}) = \frac{{{C_1}}}{{2\pi }} – \frac{{{C_2}}}{{2\pi }}\)

Solved Examples

Q.1. A walking track is in the form of a ring whose inner circumference is \({{220}}\,{\text{m}}\) and outer circumference is \({{330}}\,{\text{m}}\) Find the width of the track. Take \(\pi = \frac{{22}}{7}\)

Ans: Let \({R_1}\) and \({R_2}\) be the radii of the outer and inner ring.
Then, \(2\pi {R_1} = 330\)
\(2 \times \frac{{22}}{7} \times {R_1} = 330\)
\(\Rightarrow {R_1} = \frac{{330 \times 7}}{{2 \times 22}} \Rightarrow \;R = 52.5\;\text{m}\)
Now, \(2\pi R = 220\)
\(\Rightarrow \;2 \times \frac{{22}}{7} \times R = 220\)
\(\Rightarrow {R_2} = \frac{{220 \times 7}}{{2 \times 22}}\)
\(\Rightarrow {R_2} = \;35\;\text{m}\)
Therefore, the width of the walking track \( = \left( {52.5 – 35} \right)\;\text{m} = 17.5\;\text{m}\)

 

Q.2. The circumference of a circle exceeds the diameter by \(40\;\text{cm}.\) Find the radius of the circle. Take \(\pi = \frac{{22}}{7}\)

Ans: Let the radius of the circle \(= r\;\text{m}.\)
Then, the circumference of a circle \(= \;2\pi r\)
Since circumference exceeds diameter by \(40\;\text{cm}.\) Therefore, according to the question,
\(2\pi r = d + 40\)
\(\Rightarrow \;2\pi r = 2r + 40\)
\(\Rightarrow \;2 \times \left( {\frac{{22}}{7}} \right) \times r = 2r + 40\)
\(\Rightarrow \frac{{\;44r}}{7} – 2r = 40\)
\(\Rightarrow \frac{{\;\left( {44r\; – \;14r} \right)}}{7} = 40\)
\(\Rightarrow \frac{{\;30r}}{7} = 40\)
\(\Rightarrow \;r = \frac{{7 \times 40}}{{30}}\)
\(\Rightarrow \;r = \frac{{28}}{3}\;\text{cm}\)
Hence, the radius of the circle is \(\frac{{28}}{3}\;\text{cm}.\)

 

Q.3. The radius of the wheels of a bicycle is \(50\;\text{cm}.\) How many revolutions will each wheel make to travel a distance of \(157\;\text{metres}?\) Take \(\pi = 3.14.\)

Ans: Given, the radius of wheels of a bicycle \(\left( r \right) = \;50\;\text{cm}\)
We know that the circumference of a circle \(= 2\pi r\)
\(= \;2 \times 3.14 \times 50\;\text{cm}\)
\(= \;314\;\text{cm}\)
To find the number of revolutions of the wheel, divide the distance covered by the circumference of the wheel.
To convert \(157\;\text{metres}\) to \(\text{cm}\) multiply by \(100\)
Number of revolutions \(\; = \frac{{15700}}{{314}} = 50\;\text{revolutions}\)
Hence, the number of revolutions each wheel will make to travel a distance of \(157\;\text{metres}\) is \(50\) revolutions.

 

Q.4. A piece of a rectangular wire of length \(50\;\text{cm}\) and breadth \(100\;\text{cm}\) is cut and folded to make a circle. Find the circumference and radius of the circle so formed. Take \(\pi = 3.14.\)

Ans: The circumference of the circle formed \( = \) the perimeter of the rectangular wire
The perimeter of a rectangle \( = \;2\left( {L + W} \right)\)
\( = \;2\left( {50 + 100} \right)\;\text{cm}\)
\(= \;2 \times 150\;\text{cm}\)
\( = \;300\;\text{cm}\)
So, the circumference of the circle is \(300\;\text{cm}.\)
Now let’s calculate its radius.
Circumference \(= 2\pi r\)
\(300 = 2 \times \pi \times r\)
\(300 = 2 \times 3.14 \times r\)
\(300 = 6.28r\)
\(r = \frac{{300}}{{6.28}} = 47.77\;\)
Hence, the radius of the circle is \(47.77\;\text{cm}.\)

 

Q.5. A \(15\;inches\) (diameter) pizza is served to Tim and his friends. Calculate its circumference.

Ans: Diameter of pizza \(\left( d \right) = 15\;\text{inches}\)
The formula for the circumference of the circle in terms of diameter \(C = \pi d\)
\(C = \pi \times 15 = 15\pi \;\text{inches}\)
Hence, the circumference of pizza is \(15\pi \;\text{inches}.\)

 

Q.6. The wheel of a wheelchair has a diameter of \(14\;\text{m}.\) If the wheel rotates once how much distance does the wheelchair move?

Ans: If the wheel rotates once, the wheelchair will move by a distance equal to the circumference of the wheel.
Given, the diameter of the wheel \(\left( d \right) = 14\;\text{m}\)
We know that the circumference of a wheel \(C\; = \;\pi d\)
\(C = \frac{{22}}{7} \times 14\; = 44\;\text{m}\)
Thus, the wheelchair moves \(44\;\text{m}\) in one revolution of the wheel.

 

Q.7. Given the radius of a circle is \(r\;\text{cm}\) and if it is doubled then what will be the circumference of the new circle.

Ans: Given, the radius of the circle \( = r\;\text{cm}\)
Then the circumference of the circle \( = 2\pi r\)
If the radius of the circle is doubled then, the new radius \(R = 2r\;\text{cm}\)
Therefore, the circumference of the new circle \(= 2\pi R = 2\pi \times 2r = 4\pi r\)
Hence, the circumference of the new circle is \(4\pi r.\)

 

Q.8. Kelly works at a lab with a huge circular particle accelerator. It has a radius of \(6\,{\rm{metres}}\). What is the accelerator’s circumference?

Ans: Given, the radius of the accelerator \(= 6\;\text{m}\)
Then the accelerator’s circumference \(= 2\pi r\),
\(= 2\pi \times 6 = 12\pi \)
Hence, the accelerator’s circumference is \(12\pi .\)

 

Q.9. From a circular sheet of a radius \(5\;\text{cm},\) a circle of radius \(3\;\text{cm}\) is removed. Find the circumference of the remaining sheet.

Ans: Given, the radius of the outer sheet \({r_1} = 5\;\text{cm}\)
The radius of the inner sheet \({r_2} = 3\;\text{cm}\)
Circumference of the remaining part \(= 2\pi \left( {{r_1} – {r_2}} \right)\)
\(= 2\pi \left( {5 – 3} \right) = 2\pi \times 2 = 4\pi \)
Hence, the circumference of the remaining part is \(4\pi .\)

 

Q.10. Ratio of radii of two circles \(4:5.\) What is the ratio in their circumference?

Ans: Let the radius of the first the circle \({r_1} = 4x\)
The radius of the second circle \({r_2} = 5x\)
The ratio of the circumference of the two circles \(= \frac{{2\pi {r_1}}}{{2\pi {r_2}}} = \frac{{4x}}{{5x}} = \frac{4}{5}\)
Hence, the ratio of the circumference of the two circles is \(4:5.\)

FAQs on Circumference of a Circle

Below are the frequently asked questions on the Circumference of a Circle:

Q.1. What is the circumference of a circle?
Ans: The Circumference of a Circle is defined as the length around its boundary. In other words, if a circle is cut to form a straight line, then the length of that line will be the circumference of a circle.

Q.2. Why is the circumference of a circle \(2\pi r?\)

Ans: Consider a circle of diameter d and unroll the outer boundary or border of a circle, it will form a straight line segment whose length is equal to \(3\) times the diameter plus a little extra. The same result will be obtained using a circle with an extremely small radius or using a circle with an enormously large radius. It is to understand that this constant length is equal to the constant pi \(\left( {\pi = 3.14159…} \right).\)

 

Q.3. Write the two formulas for calculating the circumference of a circle?

Ans: There are two formulas for the circumference of a circle
(i) First formula: In terms of radius
If the radius of the circle is known, then the circumference of the circle is \(2\pi r.\)
(ii) Second formula: In terms of diameter
If the diameter of the circle is known, then the circumference of the circle is \(\pi d.\)
Since the diameter of the circle is two times the radius.

 

Q.4. What is the total circumference of a circle?
Ans: To find the total circumference of a circle, simply multiply the diameter of the circle with the constant \(\pi \) (pi). The total circumference can also be found by multiplying twice the radius with the constant pi \(\left( \pi \right).\)

 

Q.5. How to find diameter from the circumference of a circle?
Ans: Circumference of circle \(= \;\text{Diameter}\; \times \;\pi \)
Or, diameter \( = \frac{\text{Circumference}}{\pi }\)
So, the diameter of the circle in the form of the circumference is equal to the ratio of the circumference of the circle and the mathematical constant \(\pi .\)

 

Q.6. How to find a radius from the circumference of a circle?
Ans: Circumference of circle \(= \;2\pi r\)
Or, radius \( = \frac{\text{Circumference}}{{2\pi }}\)
So, the radius of the circle in the form of the circumference is equal to the ratio of the circumference of the circle and two times the mathematical constant \(\pi .\)

Q.7. How is the circumference of a circle different from the area of a circle?
Ans: Circumference is nothing but the distance around the boundary of the circle. But the area is the region inside that boundary of the circle.

Q.8. What is the formula for the area of a circle and the circumference of a circle?
Ans: The formula for the area of circle \(= \mathrm{πr}^2\)
And the circumference of the circle \(= 2\pi r\)
Therefore, to establish the relationship between the circumference of a circle and its area. The Area of a circle is half times the radius times the circumference of a circle.

 

Q.9. How to find the circumference of a circle?
Ans: First Method: Since the surface of the circle is curved, we can’t physically calculate the length of a circle by using a scale or ruler. But it can be done for polygons like triangles, squares, and rectangles. Instead, we can calculate the circumference of a circle using a thread. Trace the path of the circle by using a thread and mark the points on the thread. Now, calculate this length by using a ruler.

Second Method: A correct way of knowing the circumference of a circle is to calculate it if the radius or diameter is known. The figure given below shows a circle whose radius is \(r\) and the centre is \(O\). The diameter is twice the radius of the circle.

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Summary

In this article about the circumference of a circle, we learnt the definition, formulas, and ways to find out the circumference of a circle, along with several solved examples. We hope this detailed article on Circumference of a Circle has helped you. If you face any issue regarding the same do let us know about it in the comment section below and we will get back to you at the earliest. You can find more problems on circumference of a circle in ICSE Class 10 Sample Papers from Embibe.