Question

If the area of rectangle PQRS (green) is 21 $c{m}^2$, AB = 7 cm and width of circular path is 1 cm, then find area of circular path(grey). (Take $\pi$ = 3.14)

Answer 1

Circular path has width 1 cm, hence AP=BQ=CR=DS = 1 cm
Area of rectangle PQRS
21 = Length$\times$Breadth
21 = PQ$\times$QR
21 = AB$\times$QR = 7$\times$QR
QR = $\frac{21}{7}$ = 3 cm

BC = QR + BQ + CR
BC = 3+1+1 = 5 cm

Area of circular path
= Area of rectangle ABQP
+ Area of rectangle SRCD
+ Area of semicircular rings AEDSP
+ Area of semicircular rings BFCRQ

= (AB$\times$BQ) + (CD$\times$CR)
+ $\frac{\pi }{2}$$\times$($R^2-r^2$) + $\frac{\pi }{2}$$\times$($R^2-r^2$)
= (7$\times$1) + (7$\times$1) +$\pi$($R^2-r^2$)
= 7+7+ 3.14$\times$($5^2-3^2$)
.....( where R =$\frac{BC}{2}$ and r =$\frac{QR}{2}$)
= 14 + 3.14$\times$(25-9)
= 14 + 3.14$\times$16
= 14 + 50.24 = 64.24 $c{m}^2$