Question

If a wire is bent into the shape of a square, then the area of the square is 81 cm². When wire is bent into a semi-circular shape, then the  area of the semi-circle will be ______.

Options

• 22 cm²

• 44 cm²

• 77 cm²

•  154 cm²

Answer 1

If a wire is bent into the shape of a square, then the area of the square is 81 cm². When wire is bent into a semi-circular shape, then the  area of the semi-circle will be 77 cm².

Explanation:

We have given that a wire is bent in the form of square of side a cm such that the area of the square is `81 cm^2`. If we bent the same wire in the form of a semicircle with radius r cm, the perimeter of the wire will not change.

∴ perimeter of the square = perimeter of semi circle 

`4a=1/2 (2pir)+2r`             ...(1)

We know that area of the square = 81 cm²

∴ a² = 81

∴ a = 9

Now we will substitute the value of a in the equation (1),

`4xx9=1/2(2pi r)+2r`

`∴ 36=1/2(2pir)+2r`

`∴36=(pir)+2r`

`∴36=r(pi+2)`

Now we will substitute `pi=22/7`

`∴36=r(22/7+2)`

`∴36=r((22+14)/7)`

`∴36=r(36/7)`

Multiplying both sides of the equation by 7 we get,  36 × 7 = r × 36

Now we will divide both sides of the equation by 36 we get, r = 7

Therefore, radius of the semi circle is 7cm.

Now we will find the area of the semicircle.

Area of the semicircle = `1/2xxpir^2`

= `1/2xxpixx7^2`

= `1/2xx22/7xx7^2`

= 11 × 7

= 77

Therefore, the area of the semicircle is 77 cm².