Question

Calculate the area of the shaded region in the fig6.18, where ⎕ABCD is a square withside 8 cm each. ($\mathrm{\pi }=3.14$)

Answer 1

We will join diagonal AC of the square.
The figure is symmetric along diagonal AC.
Thus, we have:
Area of segment APC = Area of segment AQC
Area of the shaded region = 2 × Area of segment AQC
We know:
Area of a segment = r² $\left[\frac{\mathrm{\pi \theta }}{360}-\frac{\mathrm{sin\theta }}{2}\right]$
∴ Area of segment AQC = $r^2\left[\frac{3.14\times 90}{360}-\frac{\sin 90°}{2}\right]$
(Side of the square = Radius of the sector = 8 cm)
= $64\left[0.785-\frac{1}{2}\right]$

= $64\left[0.785-0.5\right]$
= 18.24 cm²
Thus, we have:
Area of the shaded region = 2 × 18.24 = 36.48 cm²