Circles

Introduction to Circle

Definition of a Circle

A circle is formed by the set of points which are at a constant or fixed distance (radius) from a fixed point (centre) in the plane.

The fixed point is called the origin or centre of the circle.

The fixed distance of the points from the origin is called the radius.

Shape of a Circle

Let's take a fixed point \(P\) and try to draw lines from it.

Let's draw several lines of length 3 cm away from point \(P\) in all possible directions.

When we join these ends together in a curve, it forms a circle.

Circle with multiple radii of length 3 cm

Try this simulation which shows how a circle is formed by the collection of points which are equidistant from the centre.

(Click on the buttons: Reset and See)

The circle we have drawn resembles the wheel of a cart.

Wheel of a cart

Have you ever ridden a ferris wheel at a carnival?

What is the shape of the ferris wheel?

You are right!

It is a circle!

Ferris wheel in an amusement park

Let us look at a few other examples.

Have you noticed the shape of the wheel of a bicycle?

A cycle wheel

Here is an example of a pizza.

A pizza 

Did you find anything common in these items?

Yes, it's their shape, which is a circle.

Circles are one of the most commonly found shapes in the world.

The fascinating properties of circles makes it an important topic in Maths and Geometry.


Properties of a Circle

Imagine a circular park in your neighbourhood.

Try to identify the various parts of a circle with the help of the figure and table given below.

A circular park labelled with the circumference, radius, diameter, chord, play area, entrance gate, exit gate, and fountain.

In a Circle In our park Named by the letter
Centre Fountain \(F\)
Circumference Boundary  
Chord Play area entrance \(PQ\)
Radius Distance from the fountain to the Entrance gate \(FA\)
Diameter Straight Line Distance between Entrance Gate and Exit Gate through the fountain \(AFB\)
Minor segment The smaller area of the park, which is shown as the Play area  
Major segment The bigger area of the park, other than the Play area  
Interior part of the circle The green area of the whole park  
Exterior part of the circle The area outside the boundary of the park  
Arc Any curved part on the circumference.  

Could you find these parts in the park?

Well done!


Parts of a Circle

Radius

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

Let's refer to the park that was shown earlier.

A circle can have multiple radii as they all start from the centre and touch the boundary of the circle at various points.

Multiple radii shown in the park

Now, let's look at how the size of a circle changes when you change the length of the radius.

Pull the green endpoint outwards, in the simulation below, to notice these changes.

Diameter

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

A circle marked with diameter, centre, boundary

Let's go back to the figure of the park.

We can see that \(AFB\) is the diameter.

We should note that there can be multiple diameters in that park, but they should:

  • pass through the fountain (centre) in a straight line
  • touch one boundary of the park to the other

Multiple diameters shown in the park

Chord

Any line segment that touches the circle at two different points is known as the chord of a circle

Referring to the park, we can see that \(PQ\) is the chord that shows where the play area starts.

Circle marked with chord

 

Circular park marked showing the chord

Circumference

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

Considering the same park, if you jog around the boundary of that park and complete one full circle without going anywhere else, that complete round is the circumference of a circle.

Circle showing the circumference

Observe how the radius and the circumference are connected to each other through this simulation.

Pull and drag the radius from 1 cm to 5 cm.

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

 
important notes to remember
Important Notes
  1. Diameter is the longest chord in a circle
  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 length of a circle

    We can find the circumference of a circle using the radius or diameter

Circle Formulas

Here are three basic formulas that are used in calculating the various dimensions of a circle.

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

If the radius is known, the diameter can be calculated by multiplying it by 2

If the diameter is known, the radius can be calculated by dividing it by 2

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

The circumference of a circle can be calculated using the radius of the circle and the value of pi.

Area of a circle \( = \pi {r^2} \)

The Area of a circle is the region occupied by the circle on a two-dimensional plane, which can be calculated using the radius and the value of pi.


Solved Examples

Let us see how the shape of a circle is seen as a part of our everyday lives.

Example 1

 

 

David was playing near a swimming pool.

He threw his ball which splashed in the water and floated away.

With reference to the exterior and interior of a circle, can you spot his location and the place where his ball is floating?

Swimming pool with a ball inside and a boy outside

Solution: 

Subject Position
David Exterior part of the circle
His ball Interior part of the circle

This one was simple!

Were you able to find his location?

Good job!

Example 2

 

 

Identify the parts of a circle hidden in this wheel.

Find the radius, the diameter and the circumference with the help of the given colours and clues.

A wheel with a coloured radius and diameter

Solution:

Parts hidden in the wheel Parts of a Circle
\(OC\) (red) Radius
\(HD\) (green) Diameter
The Black boundary Circumference
Example 3

 

 

Do you like cake?

Most people do.

David bought a circular chocolate cake from his favourite bakery.

A circular chocolate cake

He cut it into two parts, but the portions could not be cut equally.

Can you identify the major segment, the minor segment, the chord and the arc in it?

Solution:

The smaller part of the cake is the minor segment.

The bigger part of the cake is the major segment.

The curve is the Arc.

The line which separates the cake into two parts is the chord.

A chocolate cake cut into major and minor segments; arc and chord are labelled

Example 4

 

 

John went swimming in a circular swimming pool.

After swimming, he runs one round along the boundary of the pool.

If the radius of the pool is 35 metres, can you find the distance that John ran around the pool?

Swimming pool with a ball inside and a boy outside

Solution:

To find the distance that John ran, you need to know the circumference of the circle (pool).

For this, you need to know the value of \(\pi\) and \(r\), where ‘\(r\)’ is the radius of the pool.

Given:

  • \( r = 35 \) meters
  • \(\begin{align}\pi = {\frac{22}{7}} \end{align}\)

Using the formula, Circumference (C) \(= 2\pi r\)

\[\begin{align}C &= 2 \times {\frac{22}{7}} \times 35 \\\\ &= 220 \text{ meters}\end{align}\]

\(\therefore\) John ran \(220\) meters
Example 5

 

 

You want to decorate your table top, which is in the shape of a circle, with a colourful sticker.

If the radius of the table top is 21 cm, find the amount of paper you need to cover its top surface.

A circular wooden table

A circular sticker with designs

Solution:

Given,

  • Radius of the table top \(=\) 21 cm
  • \(\begin{align}\pi = {\frac{22}{7}} \end{align}\)

Using the formula, Area (A) \(= \pi r^2\)

\[\begin{align}A &= {\frac{22}{7}} \times 21 \times 21 \\\\ &= 1386 \text{ cm}^2 \end{align}\]

\(\therefore\) Area of table top \(= 1386 \text{ cm}^2 \)

How to draw a Circle?

First let us try to draw a circle for fun with our own tools!

A piece of wool is fixed to a thumb pin which is placed on a board. a pencil is fixed at the loose end. A circle is drawn using the pencil at the loose end.

Step 1: Take a thumb pin and press it on a board firmly.

Step 2: Take a piece of string/wool and make a loop; put it around the thumb pin.

Step 3: Now place a pencil on the free end of the loop in such a way that you can move the pencil along with the loop. This is how you can draw a circle!

Now let us try to construct a circle with some accurate measurements.

You will need a ruler (scale), a compass and a sharpened pencil to draw a circle which has a radius of 5 cm.

Step 1:

Mark point \(O\) as the centre in the middle of the page.

Step 2:

Since the radius is 5 cm, adjust the compass and pencil to measure 5 cm.

Measure 5 cm using a scale and compass.

Step 3:

Keep the pointed end of the compass on point \(O\) and bring the pencil side in a moveable position.

Now twist and rotate the pencil side all around point \(O\) in such a way that a circle is formed.

Using the compass, draw a circle measuring 5 cm radius.

 
tips and tricks
Tips and Tricks
  1. Make sure that the pointed end of the compass remains fixed on point \(O\) until the circle is drawn, otherwise the circle will not be an accurate one.
  2. After measuring the radius with the help of a ruler (scale), ensure that the adjusted length is not changed or moved.

Practice Questions 

Here are a few problems related to Circle.

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

 
 
 
 
 
 
 

Important Topics


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 a circle?

A circle is defined as a closed, two-dimensional curved shape which is formed by the set of all points which are at a constant or fixed distance (radius) from a fixed point (centre) in the plane.

For example, the wheel of a bicycle has the shape of a circle.

A wheel of a bicycle has the shape of a circle.

2. What are the different types of circles?

There are various types of circles in Geometry, one of them is concentric circles.

These are circles with a common centre.

Concentric circles

3. Why is a circle so important?

There are various uses of circles in every field of work because of its symmetrical properties.

It is used by architects and designers in designing athletic tracks, recreational parks, buildings, etc.

The wheel, which is also in the shape of a circle, has been one of the greatest inventions of mankind.

4. What are the various formulas for circles?

These are the commonly used formulas of a circle:

Diameter \( =2 \times radius\)

Circumference \( = 2\pi r \)

Area \( = \pi {r^2} \)

  
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