In the given figure, AOB is a sector of angle 60° of a circle with centre O and radius 17 cm . If AP ⟂ OB and AP = 15 cm, find the area of the shaded region. 

 

AOB is a sector of angle 60° of a circle with centre 𝑂 and radius 17 𝑐𝑚

If 𝐴𝑃 ⟂ 𝑂𝐵 𝑎𝑛𝑑 𝐴𝑃 = 15 𝑐𝑚

Now we have to find the area of the shaded region.

 In ∆𝑂𝑃𝐴, ∠𝑂 = 90⁰ 

By using Pythagorus theorem 

𝐴𝑂² = 𝐴𝑃² + 𝑂𝑃² 

17² = 15² + 𝑂𝑃² 

𝑂𝑃² = 289 − 225 

𝑂𝑃² = 64

𝑂𝑃 = 8 𝑐𝑚 

Area of shaded region = Area of sector 𝐴𝑂𝐵𝐴 − Area of ∆𝑂𝑃𝐴

$\begin{array}{r}=\frac{x}{360}\times \pi r^2-\frac{1}{2}\times b\times h \\ =\frac{60}{360}\times \frac{22}{7}\times 17\times 17-\frac{1}{2}\times 8\times 15\\ =\frac{1}{6}\times \frac{22}{7}\times 289-60 \\ =151.38-60 \\ =91.38{\mathrm{cm}}^2 \end{array}$

Hence the area of the shaded region is 91. 38 𝑐𝑚²