Question

....

Find the area of the shaded design in Fig. 12.17 , where ABCD is a square of side 10cm and and semicircles are drawn with each side of the square has diameter ?
( use π = 3.14 ) .


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Answer 1

Method of Solution :


In this Question, It is given that ABCD is a square of side 10cm and and semicircles are drawn with each side of the square has diameter.


▶ Given :


Side of Square = 10 Centimeters


▶ To Find : Area of Square = ?


✒ Area of Square = (Side)² cm

✒ Area of Square = (10)² cm

✒ Area of Square = 100 cm²


Now, Considering on Circles which have in Square!


Note : In a Square, There are four (4) Semicircles!

So Lets Find out What is the value of 1 Semicircle:


{ Area of Semicircle = πr²/2 }


Here, Side of Square is Equal to Diameter of Semicircle is 10 cm , So Radius is to be 5 cm.

➡ { Area of Semicircle = πr²/2 }

➡ Area of Semicircle = 3.14 × (5)² / 2

➡ Area Of Semicircle = 3.14 × 25/2

➡ Area of Semicircle = 78.5/2

➡ Area of Semicircle = 39.25 Cm²


[Hence, Area of 1 Semicircle = 39.25 cm² ]

Now, According to the Question's Statement!


Statement : Find the Area of 4 Semicircles ?


✏Area of four Semicircles = 4 × Semicircle✏


✒ Area Of four Semicircles = 4× 39.25 cm²

✒ Area Of four Semicircles = 157 cm²


Finally, We have need to Find the Area of Shaded Portion ?


Area of Shaded Portion = Area of four Semicircles - Area of Square


✒ Area of Shaded Portion = 157 - 100 Cm²


✒ Area of Shaded Portion = 57 cm²



✍ Therefore, Required Area of Shaded Portion is 57 cm². ✍

Answer 2

Solution :

Draw  the problem

So , from Figure  :

We have a Square ABCD

It has  4 Semicircles

We also have Shaded region

To Find :

Area of Shaded region

Imagination

Area of 4 semicircles - Area of Shaded region = Area of Square ABCD

Area of Shaded Region = Area of 4 Semicircles - Area of Square  ABCD

Find the Area of the Square ABCD

Area of Square = Side^2

⇒ 10^2

⇒ 100 cm^2

Find the Area of the semicircles

Radius = D/2

⇒ 10/2

⇒ 5 cm

Area of Semicircle = AD = 1/2 * area of the circle

⇒ 1/2 * pie  r^2

⇒ 1/2 *pie* 5^2

⇒ ( 3.14 * 25) /2

Radius is same for semicircle . Thus , AD = BC = CD = AB

Find the area of Shaded region

Area of Shaded region = 4 *  ( 3.14 * 25) /2  - 100

⇒ 2 × 3.14 × 25 - 100

⇒ 157 - 100

⇒ 57 cm^2

Answer :

 57 cm^2 is  the area of the shaded design in Fig. , where ABCD is a square of side 10cm and and semicircles are drawn with each side of the square has diameter .