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Q) PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal. Semi-circles are drawn such that PQ = QR = RS. Semicircles are drawn on PQ and QS as diameters as shown in figure. Find the perimeter and area of the shaded region.

PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal. Semi-circles are drawn such that PQ = QR = RS. Semicircles are drawn on PQ and QS as diameters as shown in figure. Find the perimeter and area of the shaded region.

Ans: 

Given that the radius = 6 cm

∴  Diameter PQRS = 2 x 6 = 12 cm

Since PQRS = PQ + QR + RS and its given that PQ = QR = RS

∴  12 = 3 PQ

∴ PQ = \frac{12}{3} = 4 cm

∴  PQ = QR = RS = 4 cm

Let’s start with calculating Perimeter of Shaded region:

∵ We know that the perimeter of a circle is given by =  2 π r or  π d

∴ the perimeter of a half circle is given by:  π r or π \frac{d}{2}

Now the Perimeter of shaded region =  Perimeter of Semi-circle of dia. PS + Perimeter of Semi-circle of dia. PQ + Perimeter of Semi-circle of dia. QS

=  π \frac{PS}{2} + π \frac{PQ}{2} + π \frac{QS}{2}

Here PS = 12 cm, PQ = 4 cm, QS = QR + RS = 4 + 4 = 8 cm

∴ Perimeter of shaded region =  π \frac{12}{2} + π \frac{4}{2} + π \frac{8}{2}

=  6 π + 2 π + 4 π = 12 π

= 12 x \frac{22}{7} = \frac{264}{7} = 37.714

∴ Perimeter of shaded region = 37.714 cm

Next, we will calculate Area of the Shaded region:

∵ We know that the area of a circle is given by =  π r2 or  π \frac{d^2}{4}

∴ the area of a half circle is given by:  π \frac{r^2}{2} or π \frac{d^2}{8}

Area of shaded region =  Area of Semi-circle of dia. PS + Area of Semi-circle of dia. PQ – Area of Semi-circle of dia. QS

∴ Area of shaded region =  π \frac{PS^2}{8} +  π \frac{PQ^2}{8} – π \frac{QS^2}{8}

Here PS = 12 cm, PQ = 4 cm, QS = 8 cm

∴ Area of shaded region =  π \frac{12^2}{8} +  \frac{4^2}{8}  – π \frac{8^2}{8}

=  π \frac{(12^2 + 4^2 - 8^2)}{8}

=  π \frac{(144 + 16 - 64)}{8}

=  12 π = 12 x \frac{22}{7} = \frac{264}{7} = 37.714

∴ Area of shaded region = 37.714 cm2

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