Area of Circle

Circle

A circle is a closed, round geometric shape. Technically, a circle is a point moving around a fixed point at a fixed distance away from the point. It can be said that a circle is a closed curve where its outer line is equal in distance from the center. The fixed distance from the point is known as the radius of the circle.

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

Circumference of a Circle

Circumference of Circle 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 of circumference 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 $\frac{22}{7}$ or $3.14$.

What is the Area of Circle?

Any 2D geometrical shape always has its own area. The area of a circle is simply defined as the region covered by its boundary and is calculated using the formula $A=\pi r^2$ measured in square units. It means the space covered by it in xy plane which is also known as 2D plane.

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 $=\pi r^2$. The area of circle with diameter is $\frac{\pi d^2}{4}$ in square units, where $\pi=\frac{22}{7}$ or $3.14$, and $d$ is the diameter.

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

Formula of Area of Circle with Diameter ($d$)

- Area $=(\frac{\pi}{4}) \times d^2$, where ' $d$ ' is the diameter.
- Area $=\frac{C^2}{ 4} \pi$, where ' $C$ ' is the circumference.

The table below summarises all the formulas discussed.

Area of Circle Using other Shapes

Area of a circle can be visualized and proved using two methods, as follows :

• Determining the circle's area using rectangles

• Determining the circle's area using triangles

Let us understand both methods one-by-one.

Using Areas of Rectangles

The circle is divided into $16$ equal sectors, and the sectors are arranged as shown in figure. The area of the circle will be equal to that of the parallelogram-shaped figure formed by the sectors cut out from the circle. Since the sectors have equal area, each sector will have an equal arc length. The orange coloured sectors will contribute to half of the circumference, and green coloured sectors will contribute to the other half. If the number of sectors cut from the circle is increased, the parallelogram will finally look like a rectangle with length equal to $\mathrm{\pi r}$ and breadth equal to $r$.

The area of a rectangle (A) will also be the area of a circle. So, we have
- $\mathrm{A}=\pi \times \mathrm{r} \times \mathrm{r}$
- $A=\pi r^2$

Using Area of Triangles

We fill the circle having radius r with concentric circles. After cutting the circle along the indicated line in figure and spreading the lines, the result will be a triangle. The base of the triangle will be equal to the circumference, and its height will be equal to the radius of the circle.

So, the area of the triangle (A) will be equal to the area of the circle. We have

$
\begin{aligned}
& A=\frac{1}{2} \times \text { base } \times \text { height } \\
& A=\frac{1}{2} \times(2 \pi r) \times r \\
& A=\pi r^2
\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 $=A=\pi x r^2$

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

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

Difference Between Area and Circumference of Circle

Relation Between Area of Square and Area of Circle

The area of circle is estimated to be $80 \%$ of area of square, provided that the diameter of the circle and length of side of the square is the same.

For example, if area of square is 100 sq. unit, then the area of circle will be approximately 80 sq.unit of it.

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=\frac{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 or 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(\frac{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=(\frac{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, $r= \frac{d}{2}=\frac{11}{2}=5.5 \mathrm{~cm}$.
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
$\begin{gathered}A=\left(\frac{22}{7}\right) \times 5.5 \times 5.5 \mathrm{~cm}^2 \\ A=\left(\frac{22}{7}\right) \times 30.25 \mathrm{~cm}^2 \\ A=\frac{665.5}{7} \mathrm{~cm}^2 \\ A=95.07 \mathrm{~cm}^2\end{gathered}$
Therefore the area of circle is $95.07 \mathrm{~cm}^2$.

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

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