Area of a Circle Formula

# Area of a Circle Formula

## Area of a Circle

The measure of the region bounded by a circle is called the area of the circle. A circle is a plane 2-demensional shape which is round in shape.

The locus of all those points which are equidistant from a fixed point is called a circle. The area of a circle is the space occupied by the boundary of the circle. Or, the number of unit squares covered the circle is called the area of the circle. The area of a circle is measured in square units, i.e., sq. cm, sq. m, sq. km, sq. mm, etc.

The area of a circle is calculated using the formula, A = πr2

Where r is the radius of the circle and π is a mathematical constant. It is equal to the ratio of the circumference to the diameter. Thus, π = Circumference/Diameter

The value of π is equal to approximately 22/7 or 3.14.

### Circle and Parts of a Circle

A line segment joining the centre of a circle to any point on the circle is called the radius of the circle. In the following figure, OA is the radius of the circle. The radius of a circle is half of the diameter of the circle.
Diameter of a Circle

A line segment joining any two points on the circle and passing through the centre is called the diameter of the circle. The diameter of a circle is twice the radius of the circle. In the above figure, AB is the diameter of the circle.

Diameter, d = 2r

### Circumference

The length of the boundary of the circle is called the circumference. If the radius of a circle is r, then the circumference of the circle is given by:
Circumference = 2πr

### Area of a Circle Formula

Area of a circle = πr2

### Derivation of Area of a Circle Formula

To obtain the formula for the area of a circle, let us perform the following activity.

Activity: Draw a circle of any convenient radius (let r cm) on a cardboard. Take out the cut-out of the circle. Divide the circular region into 16 equal parts as shown in the following figure.

Cut out these equal parts in the form of sectors of a circle and arrange them on a sheet of paper as shown below.

When the number of sectors is very large, the figure takes the shape of a rectangle.

Now, find the length and the breadth of the rectangle formed.

We can find the area of the circle by finding the area of the rectangle formed.

The length of the rectangle formed = Half the circumference of the circle = ½ × 2 πr r

The breadth of the rectangle formed = Radius of the circle = r

Area of the circle = Area of the rectangle formed

= πr × r = πr2 sq. units.

Hence, the area of a circle = πr2

Where r is the radius of the circle and π is the ratio of the circumference and the diameter, whose value is 22/7 or 3.14.

### Area of a Circle Using Diameter

We know that the area of a circle = πr2
Again, the radius of the circle, r = d/2
Putting this value of r in above formula, we get
Area of a circle = π(d/2)2 = πd2/4
Where d is the diameter of the circle.
If the diameter of a circle is given, then we can find the area of the circle by using this formula.

### Area of a Circle Using Circumference

We know that the area of a circle = πr2
Again, the circumference of the circle, C = 2πr
Or r = C/2π
Putting this value of r in above formula, we get
Area of a circle = π(C/2π)2 = C2/4π
Where C is the circumference of the circle.
If the circumference of a circle is given, then we can find the area of the circle by using this formula.

### Area of a Circle Examples

Example 1: Find the area of a circle whose radius is 7 cm.

Solution: We know that,

Area of a circle = πr2, where r is the radius of the circle.

Here, radius (r) = 7 cm

Area of the circle = πr2 = 22/7 × (7)2 cm2

= 154 cm2

Example 2: Find the cost of levelling a circular portion in a garden at the rate of Rs 50 per sq. m, if the diameter of the circle is 28 m.

Solution: Given: Diameter of the circle = 28 m, so radius = 28/2 = 14

Area of the circle = πr2 = 22/7 × 14 × 14 = 616 sq. m

The cost of levelling the ground = 616 × 50 = Rs 30,800

Example 3: The area of a circle is 1386 sq. cm. Find the diameter of the circle.

Solution: Given: Area of the circle = 1386 sq. cm

We know that, area of a circle = πr2

1386 = 22/7 × r2

r2 = 1386 × 7/22 = 441

r = 21 cm

= 2 × 21 = 42 cm

Hence, the diameter is 42 cm.

Example 4: The circumference of a circle is 66 cm. Find the area of the circle.

Solution: Given: Circumference of the circle = 66 cm

We know that, circumference of a circle = 2πr

66 = 2 × 22/7 × r

r = 66 × 7/44 = 10.5

r = 10.5 cm

Area of the circle = πr2

= 22/7 × 10.5 × 10.5 = 346.5 sq. cm

Example 5: The areas of two circles are in the ratio 9 : 25. Find the ratio of their radii.

Solution: Let r1 and r2 be the radii of two circles and let their areas be denoted by A1

and A2, respectively. Then, A1 = πr12 and A2 = πr22

Here, A1/A2 = πr12/πr22 = 9/25

r12/r22 = 9/25

(r1/r2)2 = (3/5)2

r1/ r2 = 3/5

r1 : r2 = 3 : 5