Question

O is the centre of the circle. OR is the diameter of the smaller circle. RO = 7 cm. Find the area of the shaded region.

Answer 1

Given:

i. O is the centre of the larger circle.

ii. RO = 7 cm is the radius of the smaller circle.   ...(Since OR is its diameter)

iii. The shaded region lies between the larger semicircle and two parts:

• the smaller semicircle above inside,
• And triangle ΔPSQ below the diameter.

We are to find the area of the shaded region, which is:

Shaded Area = Area of large semicircle – (Area of small semicircle + Area of ΔPSQ)

Step 1: Radius of the smaller circle

RO = 7 cm

⇒ Diameter = 2r = 14 cm

So the diameter of the large circle

RS = RO + OS

= 7 + 7

= 14 cm

⇒ Radius = 7 cm

Step 2: Area of the large semicircle

`"Area" = 1/2 πr^2`

= `1/2 xx 22/7 xx 7^2`

= `1/2 xx 22/7 xx 49`

= 77 cm²

Step 3: Area of the small semicircle

Area = `1/2 πr^2`

= `1/2 xx 22/7 xx 7^2`

= 77 cm² 

But this is not correct – the radius is 7 cm, so diameter is 14 cm, which would be larger, not smaller.

RO = 7 cm and it is the diameter of the smaller circle, so:

Radius of small circle = `7/2` = 3.5 cm

Corrected Step 3: Area of the small semicircle

Area = `1/2 πr^2`

= `1/2 xx 22/7 xx 3.5^2`

= `1/2 xx 22/7 xx 12.25`

= `(22 xx 12.25)/14`

= `269.5/14 ≈ 19.25  cm^2`

Step 4: Area of triangle ΔPSQ

This is an equilateral triangle within the semicircle.

Since diameter = 14 cm:

PQ = 14 cm   ...(Straight line across)

Height = Radius = 7 cm

So:

`"Area" = 1/2 xx  "base" xx "height"`

= `1/2 xx 14 xx 7`

= 49 cm²

Step 5: Final calculation

Shaded Area = Large semicircle – (Small semicircle + Triangle)

= 77 – (19.25 + 49)

= 77 – 68.25

= 8.75 cm²

Possibility: The shaded region = triangle + lower semicircle minus upper semicircle.

That is:

Shaded = Area of lower semicircle + Area of triangle – Upper semicircle

= 77 + 49 – 59.5

= 66.5 cm²