Question

The given figure shows the cross-section of the railway tunnel. The radius OA of the circular path is 2 m. If $\angle AOB={90}^{\circ }$, calculate the area of the cross-section.

Answer 1

Given Radius$=2m$ and $\angle AOB\left(\theta \right)={90}^{\circ }$

Area of cross-section = ar. circle - ar minor segment APB

ar. minor segment APB = ar sector AOB - ar $△$AOB

= $\frac{\theta }{{360}^0}\times \pi r^2-\frac{1}{2}\times OA\times OB$

= $\frac{{90}^0}{{360}^0}\times \pi \times 2\times 2-\frac{1}{2}\times 2\times 2=(\pi -2)m^2$

Area of circle = $\pi r^2=\pi \times 2^2=4\pi m^2$

$\therefore$ Area of cross-section = $4\pi -(\pi -2)=(3\pi +2)m^2$