Question
Question: In the given figure a semicircle is drawn with the centre O as centre and AB as diameter. Semicircle...
In the given figure a semicircle is drawn with the centre O as centre and AB as diameter. Semicircles are drawn with AO and BO as diameter. If AB = 28 cm, find the perimeter of the shaded region.
A.98 cm
B.48 cm
C.80 cm
D.88 cm
Solution
Hint: Perimeter of circle i.e. circumference of a circle is given as 2πr, where r is the radius of the circle, and value of π is 722. Perimeter is defined as the sum of the path of that curve. Diameter is given as twice the radius of any circle. Use these concepts to solve the given problem.
Complete step by step answer:
So, we need to find the perimeter of the shaded region. And as we know the perimeter of any curve is the length of the path of that curve. It means we need to add the path length of the shaded region i.e. length of the Arc ATB + length of Arc AT1O + length of Arc OT2B.
As we know the diameter of the larger semicircle is 28 cm and hence the diameter of smaller semicircle will be 228=14cm
Now, we know the circumference/ perimeter of any full circle is 2πr , where r is the radius of that circle.
It means the perimeter of the curve part of the semi-circle (leaving diameter) is given as 22πr=πr .
Now, as we have perimeter of shaded region as
Perimeter of shaded region = Arc ATB + Arc AT1O + Arc OT2B
Hence, radius of larger semicircle =2diameter
=228=14cm
And radius of smaller semi-circle =2diameter
=214=7cm
Hence, we can write the perimeter of shaded region as
=π× radius of larger semi-circle +2π× radius of smaller semi-circle
=π×14+2π×7
=14π+14π
=28π
So, perimeter of shaded region =28π
We know the value of π is 722
So, we get perimeter of shaded region =28×722
=4×22
=88cm
Hence, option (D) is the correct answer.
Note: Don’t consider the diameter of the larger semi-circle in the calculation of the perimeter of it or with the shaded region as well. So, be clear with the definition of perimeter and involve only the path of the shaded region. Don’t use any other length to get the perimeter.Be clear with the formula of circumference and area of circle. Area is given as πr2 and circumference is given as 2πr. Don’t use the formulae in a reverse manner, it is the general confusion among the students.