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Today, we will take a look at one of the most common shapes in geometrical problems. A circle! You can observe many circular objects around you, such as a coin, a plate, or a disc. Even the Earth or the full moon we see on some nights may appear circular in shape when observed from a distance.

What is a circle?

What is the geometric definition of a circle? When we try to describe a circular thing, we say it is round. To be more exact, a circle is a round figure which is symmetrical in every direction. A circle always has a center, which is the point that is equally distant from any and every point on the edge of the circle. 

If we measure a circle’s angle from the center with a protractor, the angle is always found to be 360˚ or 360 degrees, which makes the circle a perfect shape. Let’s make a circle

To make a circle, we will need a compass. We can also use a coin instead and trace its boundary path with a pencil. Now let’s take a thread, and wrap it around that coin or the circle we have just created. This is the perimeter of the circle, or also known as the circumference. We will further use the thread from this example.

But first, let’s see some of the properties of a circle.

Features of a circle

1. Circumference:

 The boundary or edge of the circle, which we just drew is called a circumference, or ‘C.’ If the endpoints of a straight line are joined, it makes a circle. The line’s length will give us the length of the circumference, or the perimeter of that circle — if you are familiar with the term.

2. Diameter:

The line that goes from one end of the edge of the circle to the other and passes through the center is called diameter. It divides the circle into two equal parts or arcs. We will denote it as ‘d’. The diameter can also be defined as the longest line that can be drawn in a  circle. The diameter is also the longest chord, but that is a term we will talk about later. .

3. Radius:

A radius, simply, is half of the diameter. It is defined as the line drawn from the center of the circle to any point on the circumference. We will denote it as ‘r.’

4. Area of a circle

We have learned that a line is a one-dimensional figure. It only has length. When a straight line is curved until one endpoint meets another, we get a circle. Now, this circle becomes a two-dimensional geometric figure. Circle has an area as well. Any figure with more than one dimension will have an area.

The area is defined as the amount of space enclosed within the boundary of a figure. Now, we shall see how to find the area of a circle. We have everything needed, except for a pi!

5. Pi or ∏

Recall how we used a thread to trace the edge of the coin and find its circumference! Let us make a diameter for this circle. Now, assume that the diameter is one unit. Let us say that the unit is a centimeter. If possible, make a circle with a compass now with a diameter of 1 cm or a radius of 0.5 cm. Now, let’s wrap the thread around the edge till the circumference has been traced. Cut out any extra portion of the thread.

Now, when this thread is opened up into a straight line, measure its length. It will approximately or almost be 3.14 units or cm.

If we divide the circumference (C) by the diameter (d), we get

C/d = 3.14/1 = 3.14

And, if we take any value of the diameter and find its circumference using the above method, the ratio of C/d will always be approximately 3.14 units.

This ratio is called pi. It is denoted by the symbol ∏. The pi is a constant, which means its value always remains the same.

∏ = 3.14, but this is not its exact value.

The value of pi can stretch to infinite digits after the decimal point. For the sake of calculation, we only take its value as two to five digits after the decimal.

It is chosen arbitrarily, which means that it is a number decided purposely to calculate  the area of a circle. It is also given by dividing 22 by 7. You can try it yourself: 22/7 will give you ~ 3.14 units.

As a fun fact, its use was invented more than four thousand years ago, and was also used by an ancient scientist called Archimedes. You have heard his name, most likely in science class.

Finding the circumference of a circle

First, let us examine how the circumference of a circle can be found. We saw how the value of pi is derived or calculated.

∏ = C/d

Thus, C = ∏ x d

Since the radius is half the diameter,

r = d/2. Or, d = 2 x r

So, the formula for circumference can also be written as

C = ∏ x 2r or 2∏r

We will see why this formula is useful later on!

Finding  area of a circle

There are a few methods through which the area of a circle can be calculated without its formula. Some are simple, while some are a bit complicated. Here, we will see a simple method.

Say that the circle is a large pizza. It is divided into slices or ‘sectors’ that are each equal to the other. Let’s say that each slice or sector is one degree in angle from the center. So, we will have 360 equal sectors, as a circle is always 360 degrees. Now, let’s take the sectors out and assemble them in a line. We get a figure that is very similar looking to a rectangle.

We can divide the interior of the circle into an infinite number of such identical sectors. We will get an even more exact rectangular figure.

Upon examining this rectangle, the breadth of the rectangle will be the same as the radius of the circle. And the length of this rectangle will be found to be the same as half the circumference of the circle – a semi-circle!

Remember:

Area (A) of a Rectangle = length (l) x breadth (b)

You can see how the area of a rectangle is derived from our post on measurement tips and tricks.

The area of this rectangle will give us the area of the circle.

A = l x b = ½ C x r

C is circumference, r is the radius.

Now, we can derive the Formula from the information so far.

The Formula for the area of a circle

We looked into the Formula for measuring the circumference of the circle. Now, we will see its usefulness in finding the area of the circle.

C = ∏ x 2r

From the Formula for the area of the rectangle:

A = ½ C x r

A = ½ ∏ x 2r x r

From the above equation, 

½ multiplied by 2 will be equal to 1. 

r multiplied by r will give us the square of r, that is, r2

Thus, A = ∏r2

The area of a circle is given by finding the value of ∏r2

Example

Given that

A circle has a radius of 3 cm. Find its area.

Using the Formula,

A = ∏r^2

A = 3.14 x (3)^2 cm = 3.14 x 3 cm x 3 cm

A = 3.14 x 9 cm2

A = 28.26 cm2

To be noted

Don’t get confused between 2r and r2. In the first case, the radius is doubled or multiplied by 2. In the case of the area, it is squared, which means that the number is multiplied by itself.

Since a diameter or a radius are lines, they are one dimensional. They can only have length. So, when the circumference is calculated for the above example.

C = 2∏r = 2 x 3.14 x 3 cm

C = 18.84 cm

The circumference is a one-dimensional quantity, and it is measured in ‘cm,’ whereas area is a two-dimensional quantity. It will always be measured in square cm or cm2. 

Summary

A circle is a round and symmetrical two-dimensional figure. It has a center that is equidistant from each point on the circumference (C). The circumference is the edge or boundary of the circle. The circle’s central angle is 360 degrees. It can be created by joining the endpoints of a line or using a compass.

The diameter (d) is the straight line that connects two points on the circumference and passes through the center. The radius (r) is half the diameter.

The length of the line that makes the circle is the measure of the circumference.

Pi or ∏ is the constant ratio given by C/d. It is always 22/7 or approximately 3.14 units.

The formulae

Circumference C = 2∏r

Area A = ∏r2

Learn about how to measure the area and other quantities for various kinds of shapes and geometric figures. Check out our posts on measurement tips and tricks on Cuemath.