Question

PQ is the diameter of the semicircle with centre O. OR is the diameter of the small circle inscribed in it. Find the shaded area if PQ = 21 cm.

Answer 1

We are given:

• A semicircle with diameter PQ = 21 cm
• A small circle inscribed in the semicircle, with diameter OR
• Need to find the shaded area, i.e.:
  Area of semicircle – Area of inscribed small circle

Step-by-step:

1. Radius of semicircle:

`r = (PQ)/2`

= `21/2`

= 10.5 cm

2. Area of semicircle:

`A_("semi") = 1/2 πr^2`

= `1/2 xx 22/7 xx (10.5)^2`

= `11/7 xx 110.25`

= 173.25 cm²

3. Radius of inscribed circle:

In a semicircle, a circle inscribed inside it touching the curved boundary and diameter has radius exactly half that of the semicircle.

`r_("small") = 10.5/2`

= 5.25 cm

4. Area of small circle:

`A_("small") = πr^2`

= `22/7 xx (5.25)^2`

= `22/7 xx 27.5625`

= 86.625 cm²

Final Step: Shaded Area

Shaded Area = `A_("semi") - A_("small")`

= 173.25 – 86.625

= 86.625 cm²