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
Circumference
Area of a Circle Formula
Area of a circle = πr2
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 πr =πr
The breadth of the rectangle formed =
Radius of the circle = r
Area of the circle = Area of the
rectangle formed
= length × breadth
= 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
Area of a Circle Using Circumference
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
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 = 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