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Area of a Circle – Formula, Derivation & Examples

area of a circle

The area of a plane 2D shape is the region occupied by it (region bounded by the perimeter) in a 2D plane. The area is measured in square units of length such as square centimetre $\left(cm^{2}\right)$ and square metre$\left(m^{2} \right)$. The area of a circle is the region bounded by its circular boundary, i.e., the perimeter commonly known as the circumference.  

Let’s learn what is meant by the area of a circle and how to find the area of a triangle and what are the different methods of finding it. 

Circle – A 2D Plane Figure

A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance from a fixed point on the plane. The fixed point is called the centre of the circle and the fixed distance of the points from the centre is called the radius.

area of a circle

Some of the important terms related to the circle are

  • Semicircle: A semicircle is one of the two halves when a circle is cut along the diameter. The two halves (semicircles) are of equal measure.
  • Arc of a Circle: An arc of a circle is referred to as a curve, that is a part or portion of its circumference.
  • Segment of a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments – minor segment, and major segment.
  • Sector of a Circle: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors – minor sector, and major sector.

What is the Area of a Circle?

The area of a circle is the space occupied by the circle in a two-dimensional plane. In other words, the space occupied within the boundary/circumference of a circle is called the area of the circle. The formula for the area of a circle is $A = \pi r^{2}$, where $r$ is the radius of the circle. The unit of area is the square unit, for example, $m^{2}$, $cm^{2}$, $in^{2}$, etc. 

Area of a circle is $\pi r^{2}$ or $\pi \frac {d^{2}}{4}$ in square units, where $\pi = \frac {22}{7}$ or $3.14$. $\pi$ which is a mathematical constant is the ratio of circumference to diameter of any circle.

area of a circle

Derivation of Area of a Circle Formula

Let’s consider a circle of radius $r$ and divide it into small sectors as shown in the figure below:

area of a circle

The perimeter (circumference) or the length of the boundary of a circle is $2 \pi r$.

The circle can now be cut up to form a rectangle. When the central angle of each sector becomes very small or nearly diminishes, the curves on the bottom and top of the rectangle straighten out to form a straight line with length $\pi r$ units while the width of the rectangle is $r$ units.

area of a circle

When you consider a very small portion of the circumference of a circle, it is approximately a straight line. For that reason only, you see the surface of the Earth as flat (although it is curved),  because you are seeing a very small portion of the Earth’s surface while on the ground.

Thus the area of the circle approximates the area of the rectangle.

And now you can find the area of the rectangle obtained as Area = $\pi r \times r = \pi r^{2}$.

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Examples

Ex 1: Find the area of a circle whose radius is $10.5 m$.

Radius of circle $r = 10.5 m$

Area of circle $A = \pi r^{2} = \frac {22}{7} \times 10.5^{2} =  \frac {22}{7} \times 110.25 = 346.5 m^{2}$

Ex 2: Find the radius of a circle whose area is $38.5 cm^{2}.

Area of circle $A = 38.5 cm^{2}$

$A = \pi r^{2} => 38.5 = \frac {22}{7} \times r^{2} => r^{2} = \frac {38.5 \times 7}{22}$

$ => r^{2} = 12.25 => r = \sqrt{12.25} => r = 3.5 cm$

Ex 3: Find the area of a circle whose diameter is $21 in$.

Diameter of circle $d = 21 in$

Area of circle = $\pi \frac {d^{2}}{4} = \frac {22}{7} \times \frac {21^{2}}{4} = 346.5 in^{2}$

Ex 4: Find the area of a circle whose circumference is $132 in$.

The circumference of a circle is $2 \pi r$

$=> 2 \pi r = 132 => 2 \times \frac {22}{7} \times r = 132 => r = \frac {132 \times 7}{22 \times 2} => r = 21 cm$

Area of a circle $A = \pi r^{2} = \frac {22}{7} \times 21^{2} = 1386 in^{2}$.

Therefore, the area of a circle whose circumference is $132 in$ is $1386 in^{2}$.

Area of a Sector of a Circle

A sector of a circle is a pie-shaped part of a circle made of the arc along with its two radii. A portion of the circumference (also known as an arc) of the circle and $2$ radii of the circle meet at both endpoints of the arc forming a sector. The shape of a sector of a circle looks like a pizza slice or a pie.

For example, in the circle shown below, $AB$ is the arc of the circle with centre $O$. Then the area covered by this arc $AOB$ is the area of a sector of a circle.

area of a circle

The area of a sector of a circle of radius $r$ subtending an angle $\theta$ at the centre of the circle is $A = \frac {\theta}{360} \pi r^{2}$.

Derivation of Area of a Sector of a Circle

Area of a circle of radius $r$ is given by $A = \pi r^{2}$

Angle contained in a circle is $360^{\circ}$. Therefore, the area of a portion of a circle subtending an angle of $360^{\circ}$ is $ \pi r^{2}$.

It means, that the area of a part of a circle subtending an angle of $1^{\circ}$ will be $\frac { \pi r^{2}}{360^{\circ}}$ (Unitary Method).

Therefore, the area of a sector subtending an angle $\theta$ at the centre of the circle is given by $A = \frac {\theta}{360} \pi r^{2}$.

Greatest Math Discoveries

Examples

Ex 1: Find the area of a sector of a circle of radius $7 cm$ subtending an angle $30^{\circ}$ at its centre.

Radius of a circle $r = 7 cm$

The angle subtended by an arc at the centre $\theta = 30^{\circ}$

Area of a sector = $ \frac {\theta}{360} \pi r^{2} =  \frac {30}{360} \times \frac {22}{7} \times 7^{2} = 12.83 cm$

Ex 2: The area of a quadrant of a circle is $32 mm^{2). Find the area of a complete circle.

The angle subtended by a quadrant of a circle is $90^{\circ}$.

Let the radius of the circle be $r$

Therefore, $ \frac {90}{360} \pi r^{2} = 32 => \frac {1}{4} \pi r^{2} = 32 => \pi r^{2} = 32 \times 4 = 128 mm^{2}$.

The area of a circle is $128 mm^{2}$.

Conclusion

The area of a circle is the region bounded by the curved boundary around a circle. Numerically the area of a circle is equal to the product of the square of a radius of a circle and a constant $\pi$.

Practice Problems

  1. Find the area of a circle whose radius is
    • $17.5 in$
    • $28 mm$
  2. Find the diameter of a circle whose area is
    • $616 mm^{2}$
    • $13866 cm^{2}$
  3. Find the area of a sector of a circle whose radius is $14 cm$ and the angle subtended at the centre of a circle is $45^{\circ}$.
  4. Find the area of a sector of a circle whose diameter is $21 cm$ and the angle subtended at the centre of a circle is $270^{\circ}$.

Recommended Reading

FAQs

What is the area formula of a circle?

The formula to find the area of a circle of radius $r$ is given by $A = \pi r^{2}$, where $\pi = \frac {22}{7}$ is a mathematical constant.

The formula to find the area of a circle of diameter $d$ is given by $\pi \frac {d^{2}}{4}$.

How do you find the area with diameter?

The formula to find the area of a circle of diameter $d$ is given by $\pi \frac {d^{2}}{4}$.

What is the value of $\pi$?

$\pi$ is a mathematical constant and is an irrational number. For convenience in calculation, its approximate value $\frac {22}{7}$ is considered.

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