**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 = π*r*^{2}

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**

**Radius of a Circle**

**Radius,**

*r*=*d*/2**Diameter of a Circle**

**Diameter,**

*d*= 2*r*

**Circumference**

*r*, then the circumference of the circle is given by:

**Circumference = 2**

**π**

*r***Area of a Circle Formula**

**Area of a circle = π r^{2}**

**Derivation of Area of a Circle Formula**

**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 π*r *=π*r *

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

Area of the circle = Area of the
rectangle formed

= length × breadth

= π*r *× *r *=
π*r*^{2} sq. units.

**Hence,
the area of a circle = π r^{2}**

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**

*r*

^{2}

*r*=

*d*/2

*r*in above formula, we get

**Area of a circle = π(**

*d*/2)^{2}= π*d*^{2}/4*d*is the diameter of the circle.

**Area of a Circle Using Circumference**

*r*

^{2}

*C*= 2π

*r*

*r*= C/2π

*r*in above formula, we get

**Area of a circle = π(**

*C*/2π)^{2}=*C*^{2}/4π*C*is the circumference of the circle.

**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 = π*r*^{2},
where *r *is the radius of the circle.

Here, radius (r) = 7 cm

Area of the circle = π*r*^{2}
= 22/7 × (7)^{2} cm^{2}

= 154
cm^{2}

**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 = π*r*^{2}
= 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 = π*r*^{2}

1386 = 22/7 × *r*^{2}

*r*^{2} = 1386 ×
7/22 = 441

r = 21 cm

Diameter = 2 × radius

= 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 = 2π*r*

66
= 2 × 22/7 × *r*

*r* = 66 × 7/44 = 10.5

r = 10.5 cm

Area of the circle = π*r*^{2}

= 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
*r*_{1} and *r*_{2} be the radii of two circles and
let their areas be denoted by A_{1}

and A_{2}, respectively. Then,
A_{1} = π*r*_{1}^{2} and A_{2} = π*r*_{2}^{2}

Here, A_{1}/A_{2 }= π*r*_{1}^{2}/π*r*_{2}^{2}
= 9/25

⇒ *r*_{1}^{2}/*r*_{2}^{2 }= 9/25

⇒ (*r*_{1}/*r*_{2})^{2} = (3/5)^{2}

⇒ *r*_{1}*/ r*_{2} =
3/5

⇒ *r*_{1}* *:* r*_{2}
= 3 : 5

**Terms Related to a Circle**