Area of any region is the space occupied by the figure on a two dimensional plane surface. We are familiar with the area of a square, rectangle which are obtained by multiplying the length by the width. Circle is a closed figure where a point moves such that it has constant distance from the fixed point. The fixed point is called the center of the circle and the constant distance is called the radius of the circle. We have different terms in circle called, diameter, chord, sector, segment and circumference of a circle. In this section let us discuss with the area of a circle and a semicircle.

Diameter and Radius of a circle: Diameter of a circle is the line segment ( chord ) joining any two points on the circumference of the circle, which passes through the center.
Radius of a circle is the line segment joining the center of a circle to any point on the circumference of the circle.
The following diagram describes the above terms.

Area of a circle is calculated using the formula $\pi$ $r^{2}$
Area of a Circle
where ' r ' is the radius of the circle and $\pi$ an irrational constant = 3.141592653589793238462643383279.
So approximately we take $\pi$ = 3.14 correct to 2 decimal places.

Surface Areas: Surface areas are calculated for three dimensional solid figures like, cube, cuboid, sphere, cylinder, cone etc. Since the circle is a two dimensional figure which is drawn on a plane surface, the surface area of a circle is the same as the area of the circle.
We follow the following steps to find the area of a circle :
1. Write the given quantity radius or diameter as ' r ' or ' d ' respectively.
2. If the diameter is given, find the radius by dividing the diameter by 2.
3. Write the formula $\pi$ $r^{2}$ to find the area of the circle.
4. Plug in the value of $\pi$ = 3.14 and the radius from step ( 1 ) or ( 2 ).
5. Multiply the values accordingly using calculator or manually.
6. Express the final answer in square units.

Solved Examples

Question 1: Find the area of a circle of of radius 10 cm.
Solution:
 
We have radius of the circle = 10 cm
                               Area of the circle = $\pi$ $r^{2}$
                                                        = 3.14 x 102  [ by plugging in r = 10 cm ]
                                                        = 3.14 x 100
                                                        = 314 sq. cm
If we take $\pi$ = 3.1416 correct to 4 decimal places, we have,
                         the Area of the circle = $\pi$ $r^{2}$
                                                       = 3.1416 x 102
                                                       = 3.1416 x 100
                                                       = 314.16 cm2

 

Question 2: Find the area of a circle whose diameter is 15 cm.
Solution:
 
We have the diameter of the circle = 15 cm
                                                       Radius = $\frac{15}{2}$
                                                                  = 7.5 cm
                                        Area of the circle = $\pi$ $r^{2}$
                                                                 = 3.14 x 7.5 2  [ by plugging in r = 7.5 ]
                                                                 = 3.14 x 56.25
                                                                 = 176.625 cm 2
                                                                 = 176.63 cm 2 ( correct to 2 decimal places )
 

Semi circle is the one which is the half of a circle.
Area of a semi circle = $\frac{1}{2}$ of area of a circle

= $\frac{1}{2}$ x $\pi$ $r^{2}$
Area of a semi circle = $\frac{1}{2}$ $\pi$ $r^{2}$

Solved Example

Question: Find the area of a semi circle whose radius = 12 cm.
Solution:
 
Radius of the circle = 12 cm
                 Area of the semi circle = $\frac{1}{2}$ $\pi$ $r^{2}$

                                                  = $\frac{1}{2}$ $\pi$ $12^{2}$ [by plugging in r = 12 cm]

                                                  = $\frac{1}{2}$ x $\pi$ x 144
 
                                                  = $\frac{144}{2}$ x $\pi$

                                                  = 72 $\pi$
                                                  = 72 x 3.14
     Area of the semi circle         = 226.08 cm 2

 


Solved Examples

Question 1: A square is inscribed in a circle of radius 5 cm. Calculate the area of the shaded region shown in the figure.
Calculating Area of Circle

Solution:
 
We are given that the radius of the circle = 5 cm
                                         Therefore, the diameter = 2 r
                                                                           = 2 ( 5 )
                                                                           = 10 cm
                                     Area of the shaded region = Area of the circle - Area of the square
                                                 Area of the Circle = $\pi$ $r^{2}$
                                                                           = $\pi$ 52
                                                                           = 25 $\pi$
              Let us assume that the side of the square = a cm
and the diagonal ( diameter of the circle ) of the square = d cm
Since each angle of a square measure 90o ,
According to Pythagoras Theorem,
                                                                   a2 + a2 = d2 = 102
                                                           =>         2 a2 = 100
                                                           =>            a2 = $\frac{100}{2}$

                                                                              = 50
                             Therefore, the area  of the square = 50 cm2
                                        Area of the shaded region = 25 $\pi$ - 50
                                                                              = 25 x 3.14 - 50
                                                                              = 78.5 - 50
                                                                              = 28.5 cm2
 

Question 2: A wire is bent in the form of a square encloses an area of 484 cm2. The same wire is bent in the form of a circle. Find the area enclosed by the circle.
Finding Area of Circle

Solution:
 
We are given that the area of the square = 484 cm2
                                Therefore, side of the square = $\sqrt{484}$
                                                                          = 22 cm
                                       Perimeter of the square = 4 a
                                                                          = 4 x 22
                                                                          = 88 cm
The length of 88 cm is bent to form a circle of radius r cm
                   Therefore, Circumference of the circle = Perimeter of the square
                                                  =>   2 $\pi$ r     = 88
                                                  =>     $\pi$ r      = $\frac{88}{2}$
                                                                          = 44
                                                 =>          3.14 r  = 44
                                                 =>                 r  = $\frac{44}{\pi}$

                                               Area of the circle = $\pi$ r2
                                                                        = $\pi$ $\left (\frac{44}{\pi }  \right )^{2}$
                                                                      
                                                                       = $\pi$ $\frac{44\times 44}{\pi \times \pi }$

                                                                       = $\frac{44\times 44}{\pi}$

                                                                       = $\frac{1936}{\pi}$

                                                                       = 616.25 cm2
                   Therefore, the Area of the circle = 616.25 cm2
 

Question 3: The diameter of a circular park is 60 m. A 3m wide road runs on the outside around it. Calculate the cost of constructing the road at $\$$ 7 per square meter.
Calculate Area of Circle

Solution:
 
We are given that the Diameter of the park = 60 m
                                Therefore, the radius of the park = $\frac{60}{2}$
                                                                               = 30 m
    Therefore, inner
                                                    Width of the road = 3 m
              Therefore, the outer radius of the circle ( R ) = 30 + 3 = 33 m
                                                      Area of the road = Area of the outer circle - Area of inner circle
                                                                              = $\pi$ R2 - $\pi$ r2
                                                                              = $\pi$ ( R2 - r2 ) = $\pi$ ( 332 - 302 )
                                                                              = $\pi$ ( 33 + 30 ) ( 33 - 30 )
                                                                              = $\pi$ ( 63 ) ( 3 ) = 593.76 m2
                     Cost of constructing the road per sq. m = $\$$ 7
         Cost of constructing the road of area 593.76 m2 = 593.76 x 7
                                                                              = $\$$ 4156.32
                 Therefore, cost of constructing the road = $\$$ 4156.32
 

Question 4: TRUE is a square of side 20 cm with centers T, R, U and E. Four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.
Area of Circle Example

Solution:
 
We are given that the side of the square = 20 cm
Let us observe the diagram closely to find the area of the shaded region.
The shaded region is inside the square TRUE and the square has $\frac{1}{4}$ circle( quadrants ) at each corner.
                      Therefore area of the shaded region = Area of the square - Area of the four quadrants
                                              Area of the square = 202
                                                                         = 20 x 20
                                                                         = 400 cm2
                                          Radius of each circle = $\frac{20}{2}$
                                                                         = 10 cm
                                      Area of the 4 quadrants = 4 x ( $\frac{1}{4}$ $\pi$ r2 )
                                                                         = $\pi$ r2
                                                                         = $\pi$ 102
                                                                         = 3.14 x 100
                                                                         = 314 cm2
                               Area of the shaded region = 400 - 314
                                                                         = 86 cm2