Circumference of a circle

Circumference of a circle
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What is a Circle?

A Circle is a round closed figure where all its boundary points are equidistant from a fixed point called the centre.

Circle shaped like a wheel

Try this simulation to observe how a circle is formed.

Click on the buttons RESET and SEE.

You can observe that the collection of points are all equidistant from the centre.


What is the Circumference of a Circle?

The circumference of a circle is its boundary or the length of the complete arc of a circle.

Let us understand this concept using an example.

Consider a circular park as shown below.

boy running around a park with starting point A

If a boy starts running from point 'A' and reaches the same point after taking one complete round of the park, a distance is covered by him.

This distance or boundary is called the circumference of the park which is in the shape of a circle.

Circumference is the Length of the Boundary

If we open a circle and measure the boundary just like we measure a straight line, we get the circumference of the circle in terms of units of length like centimetres, metres, or kilometres.

Click on "Go!" to observe the circumference measurements.


How to Find the Circumference of a Circle?

Although the circumference of a circle is its length, it cannot be calculated with the help of a ruler (scale) like it is usually done for other polygons.

This is because a circle is a curved figure.

Therefore, to calculate the circumference of a circle, we apply a formula which uses the radius or the diameter of the circle and the value of Pi (\(\pi\)).

Circle marked with radius, diameter, circumference

Radius

Radius is the distance from the centre of a circle to the boundary of the circle.

Diameter

Diameter is a straight line passing through the centre that connects two points on the boundary of the circle.

Pi

Pi is a special mathematical constant with a value approximated to 3.14159 or \(\pi  =  \frac{22}{7}\).

The value of \(\pi  =  \frac{22}{7}\) is used in various formulas.

 It is the ratio of circumference to diameter, where \(C = \pi D\)

Formula for the Circumference of a Circle

The circumference of a circle can be calculated using the radius  '\(r\)' of the circle and the value of \(\pi\).

Circumference of a circle \(= 2\pi r \)

While using this formula, if we do not have the value of the radius, we can find it using the diameter.

That is, if the diameter is known, it can be divided by 2 to obtain the value of the radius because:

Diameter of a circle \(= 2 \times r \)

Another way to calculate the circumference of a circle is by using the formula:

Circumference \(= \pi \times Diameter \)

If we need to calculate the radius or diameter, when the circumference of a circle is given, we use the formula:

\(\begin{align}Radius = \frac{Circumference}{2\pi }\end{align}\)
 
important notes to remember
Important Notes
  1. \(\pi\) (Pi) is a mathematical constant which is the ratio of the circumference of a circle to its diameter. It is approximated to \(\pi  =  \frac{22}{7}\) or \(3.14\)
  2. If the radius of a circle is extended further and touches the boundary of the circle, it becomes the diameter of a circle. Therefore, Diameter \(= 2 \times \) Radius
  3. Circumference is the distance around a circle or the length of a circle
  4. We can find the circumference of a circle using the radius or diameter
  5. Circumference \(= \pi \times Diameter \); Circumference \(=2 \pi r\)

Look at the following simulation.

Observe how the size of the circle changes when we change the length of the radius.

Pull and drag the green endpoint outwards on the circle.


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       Solved Examples

Example 1

 

 

A garland woven with flowers needs to be stuck around a circular ring to decorate it.

How long should the garland be so as to cover a ring of radius 28 cm?

Ring surrounded by a garland, radius marked as 28 cm

Solution:

Given, Radius of the ring = 28 cm

To find the length of the garland, we have to find the circumference of the ring.

Circumference of a circle is calculated using the formula

\[\begin{align} &= 2\pi r \\ & = 2 \times \left(\frac{22}{7}\right) \times \left({28}\right) \\ & = \text{176} \text{ cm} \end{align} \]

\(\therefore\) Length of the garland \(= 176 \text { cm}\)
Example 2

 

 

The circumference of a wheel is 440 cm.

Find its radius and diameter.

Circular Wheel of circumference 440 cm

Solution:

Given, Circumference of the wheel = 440 cm

Circumference of a circle \(= 2\pi r \)

Let the radius of the circle be R.

\[\begin{align} 440 &=2\pi R  \\ 440& = 2 \times \left(\frac{22}{7}\right) \times \left({R}\right) \\ R& = \text{} \text{ 70 cm} \end{align} \]

Diameter \(= 2\text{r}\)

Diameter \(= 2\times  {70}\)

\(\therefore\) Radius is 70 cm; Diameter is 140 cm
Example 3

 

 

The perimeter of a rectangular wire is 264 m.

The same wire is bent into the shape of a circle.

Find the radius of the circle formed.

Wire bent from a rectangle of perimeter 264 m to form a circle

Solution:

We know that, perimeter of the rectangle = Total length of the wire

Length of the wire used = Circumference of the circle formed

Hence, Circumference of the circle formed = 264 m

Circumference of a circle \( =2 \pi r\)

\[\begin{align} 2 \pi r &= 264 \text { m} \\  r&=42 \text { m} \end{align} \]

\(\therefore\) Radius is 42 m
Example 4

 

 

Tom runs three rounds at a circular stadium field every day before starting his cricket practice.

The diameter of the field is 28 m.

How much does he run around the field every day?

Circular cricket field of diameter 28 m.

Solution:

Since Tom runs around the circular field, we need to find the circumference of the field.

Given: \(Diameter = 28 \,\,m\)

\(\begin{align}Radius = \frac{28}{2}= 14 \, m\end{align}\); because \(Diameter = 2r\)

Circumference of a Circle \( =2 \pi r\)

\[\begin{align} &= 2\pi r \\ & = 2 \times \left(\frac{22}{7}\right) \times \left({14}\right) \\ & = \text{88} \text{ m} \end{align} \]

Hence, the circumference of the field is 88 m.

We know that Tom runs around the field thrice.

Hence, we multiply the circumference by 3

\[3 \times 88 =264\:\text{m}\]

\(\therefore\) Tom runs 264 metres.
Example 5

 

 

Find the radius of cake A whose circumference is 5 times that of the circumference of cake B with diameter 18 cm.

Find the radius of cake A whose circumference is 5 times that of the circumference of cake B with diameter 18 cm.

Solution:

Diameter of cake B = 18 cm

Therefore, Radius of cake B = 9 cm as \(Diameter = 2r\)

Circumference of cake B \( =2 \pi r\)

\[\begin{align} &= 2\pi r \\ & = 2 \times \left(\frac{22}{7}\right) \times \left({9}\right) \\ & =  \frac{396}{7} \end{align} \]

Since, the circumference of cake A is 5 times that of circumference of cake B, we get:

\[\begin{align} &= 5\times \frac{396}{7} \end{align} \]

Circumference of cake A \[\begin{align} &=  \frac{1980}{7} \end{align} \]

Now, let's find the radius of cake A.

Let the radius of cake A be R cm.

Circumference of cake A:

\[\begin{align}\frac{1980}{7} & =2 \pi R \\ \frac{1980}{7} &= 2 \times \left(\frac{22}{7}\right) \times \left({R}\right) \end{align} \]

Required radius = 45 cm

\(\therefore\) Radius of cake A is 45 cm.
 
Challenge your math skills
Challenging Question
  1. The diameter of the given semi-circular slice of pizza is 28 cm. What is the perimeter of this slice of pizza?

Semi-circular pizza with diameter 28 cm


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Practice Questions

Try the Questions for Circumference of a Circle in the worksheet given below.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What is the circumference of a circle?

The circumference of a circle is its boundary or the length of the complete arc of a circle.

2. How do you calculate the circumference of a circle?

The circumference of a circle is calculated with the help of a formula which needs the value of the radius and Pi.

Circumference of a circle \(= 2\pi r \)

3. How do you calculate the diameter from the circumference of a circle?

If we need to calculate the radius or diameter when the circumference of a circle is given, we use the formula: Circumference \(= \pi \times Diameter\)

  
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