Area of Circle

We can define the area of circle as the region occupied by the circle in a two-dimensional plane. It can be easily calculated using the formula, $\mathrm{A}=\pi \mathrm{r}^2$, (Pi r-squared), $r$ being the term used for radius. We generally take the unit of area as the square unit, such as $\mathrm{m}^2, \mathrm{~cm}^2$, etc.

Area of Circle $=\pi {r}^ 2$ and also $\pi {d}^2 / 4$, sq units where $\pi=22 / 7$ or 3.14

This important formula is useful for calculating the space occupied by a circular field or plot. We have various real life examples depicting the usefulness of practical applications of area of a circle. Like if some guests are arriving at our home so how much area of the table cloth we need in case our table is in the form of a circle. If we imagine ourselves as a farmer who has to grow wheat or paddy in his circular field so how much net sown area and the yield of the crop will be obtained. These are just a few examples. We know that a circle doesn't have a volume because of the fact that it is a 2 dimensional figure. Now we will learn about these crucial concepts in detail.

What is a Circle

We can define a circle as a closed plain geometric shape in the most simple terms.While technically, a circle is a locus of a point moving around a fixed point at a fixed distance away from the point. So in a collective sense it can be said that a circle is a closed curve with its outer line equidistant from the center. The fixed distance from the point is known as the radius of the circle. We define few terms :

Radius: It can be simply defined as the line that joins the center of the circle to the outer boundary or circumference. It is generally represented by ' $r$ ' or ' $R$ '.

Diameter: It is defined as the line that divides the circle into two equal parts or halves and is represented by '$d$' or '$D$'. Hence,

$
d=2 r \text { or } D=2 R
$

And so,

$
r=d / 2 \text { or } R=D / 2
$

Circumference of Circle: It is defined as the length of the boundary of the circle. We define the circumference of the circle with the knowledge of a term known as 'pi'.

The perimeter of the circle is equal to the length of its boundary or circumference. If we try wrapping a string around a circle and then unfold it to calculate its length, then upon unfolding it and measuring it we get the circumference of the circle. The formula used is:

Circumference or Perimeter $=2 \pi r$ units
where $r$ is the radius of the circle.
$\pi$, can be pronounced as 'pi', the ratio of the circumference of a circle to its diameter. This ratio is the same for every circle.

• $\pi=$ Circumference/Diameter
• $\pi=\mathrm{C} / 2 \mathrm{r}$ ($r$ = radius), ($C$ = circumference)
• $\mathrm{C}=2 \pi r$

What is the Area of Circle

Any geometrical shape always has its own area. The area of a circle is simply defined as the region covered or enclosed by its boundary and is calculated using the formula $A=\pi r^2$ measured in square units.

Area of Circle Formula

The area of the circle can be calculated from the radius which can further be taken out from the diameter. But these formulae provide the shortest method to find the area of a circle. Suppose a circle has a radius 'r' then the area of circle $=\pi r^2$ or $\pi d^2 / 4$ in square units, where $\pi=22 / 7$ or 3.14, and $d$ is the diameter.

Area of a circle formula, $A=\pi r^2$ square units
Circumference or Perimeter $=2 \pi r$ units

Formula of Area of Circle with Radius ($r$)
The area of a circle having radius $r$ can be calculated by using the formulas:

• Area of circle $=\pi \times r^2$, where ' $r$ ' is the radius.

Area of Circle with Diameter ($d$)

• Area $=(\pi / 4) \times d^2$, where ' $d$ ' is the diameter.
• Area $=C^2 / 4 \pi$, where ' $C$ ' is the circumference.

The table below summarises all the formulas discussed.

How to Find Area of Circle

To find the area of circle we should know the radius or diameter of the circle.

For example, if the radius of the circle is 10 cm, then its area will be:

Area of circle with 7 cm radius $=\pi r^2=\pi(10)^2=22 / 7 \times 10 \times 10=314.28 \mathrm{sq} . \mathrm{cm}$.
We can find the area with the help of the following relations:

$
\begin{aligned}
& \mathrm{C}=2 \pi r \\
& \mathrm{r}=\mathrm{C} / 2 \pi
\end{aligned}
$

Surface Area of Circle

A circle is 2-D representation of a sphere. The total area that is taken inside the boundary of the circle is only the surface area of the circle. Hence, we may conclude by saying that both the terms provide us with the same result only. Sometimes, even the volume of a circle is also used to direct towards the area of a circle.

Therefore, the surface area of circle $=A=\pi x r^2$

Difference Between Area and Circumference of Circle

Solved Examples based on the Area of Circle

Example 1: What is the radius of the circle whose surface area is 100 sq.cm?
Solution:
By formula of area of circle, we know that:

$
A=\pi \times r^2
$

Now, substituting the value in the formula for area of circle:

$
\begin{aligned}
& 100=\pi \times r^2 \\
& 100=3.14 \times r^2 \\
& r^2=100 / 3.14 \\
& r^2=31.84 \\
& r=\sqrt{31.34} \\
& r=5.64 \mathrm{~cm}
\end{aligned}
$

Example 2: What is the circumference and the area of circle if the radius is 9 cm .

Solution:
Given: Radius, $r=9 \mathrm{~cm}$.
We know that the circumference/ perimeter of the circle is $2 \pi r \mathrm{~cm}$.
Now, substitute the radius value in the formula for circumference of circle,

$
\begin{aligned}
& \mathrm{C}=2 \times(22 / 7) \times 9 \\
& \mathrm{C}=56.57 \mathrm{~cm}
\end{aligned}
$

Thus, the circumference of the circle is 56.57 cm .
Now, the area of circle is $\pi \mathrm{r}^2 \mathrm{~cm}^2$

$
\begin{aligned}
& A=(22 / 7) \times 9 \times 9 \\
& A=254.57 \mathrm{~cm}^2
\end{aligned}
$

Example 3: Find the area of circle whose longest chord is 11cm.

Solution:
Given that the longest chord of a circle is 11 cm .
We know that the longest chord of a circle is nothing but the diameter of circle.
Hence, $d=11 \mathrm{~cm}$.
So, $r=d / 2=11 / 2=5.5 \mathrm{~cm}$.
The formula to calculate the area of circle is given by,
$A=\pi r^2$ square units.
Substitute $\mathrm{r}=5.5 \mathrm{~cm}$ in the formula of area of circle, we get
$A=(22 / 7) \times 5.5 \times 5.5 \mathrm{~cm}^2$
$A=(22 / 7) \times 30.25 \mathrm{~cm}^2$
$\mathrm{A}=665.5 / 7 \mathrm{~cm}^2$
$A=95.07 \mathrm{~cm}^2$
Therefore the area of circle is $113.14 \mathrm{~cm}^2$.

Example 4: Rohan and his friends ordered a pizza on Saturday night. Each slice was 5 cm in length.
Calculate the area of the pizza that was ordered by Rohan. You can assume that the length of the pizza slice is equal to the pizza's radius.

Solution:
A pizza is circular in shape. So we can use the area of circle formula to calculate the area of the pizza. Radius is 5 cm .

Area of Circle formula $=\pi r^2=3.14 \times 5 \times 5=78.5$
Area of the Pizza $=78.5 \mathrm{sq} . \mathrm{cm}$.

Example 5: What is the area of circle with a radius 4 m ?

Solution:
The measure of the radius is given to us which is 4 m . The only thing we have to do is substitute the value of the radius into the formula then simplify.

Therefore $A=3.14 \times 4 \times 4=50.24 \mathrm{~m} \mathrm{sq}$.

Summary

The area of circle can be used infinitely in various diverse and wide ranging sectors. For example, this can be used in different bakeries to make pizza or cake with a definite radius and area covered on the pan. Also, the area covered by the football stadium or cricket stadium can be find out using the area of a circle. In various celebrated fairs, different Ferris wheels or the wheel of cars can be used to find the area of the circle. The list does not end here since it has numerous applications in the field of physics, astronomy, etc.2. What is the formula of area of circle