Question

If a circular sheet of perimeter 2πr touching each side of a given quadrilateral sheet of perimeter 2p is cut off from the quadrilateral, then what is the area of the remaining sheet?

Answer 1

Let ABCD be the quadrilateral from which a circular sheet is cut off touching each side of the quadrilateral. 

Also, given AB + BC + CD + DA = 2p             ...(i) 

Circumference of the circle = 2πr ⇒ Radius of circle = r 

∴ Area of quadrilateral = Area of (ΔOAB + ΔOBC + ΔOCD + ΔODA)

= \(\frac{1}{2}\) r (AB) + \(\frac{1}{2}\) r (BC) + \(\frac{1}{2}\) r (CD) + \(\frac{1}{2}\) r (DA)

= \(\frac{1}{2}\) r (AB + BC + CD + DA) = \(\frac{1}{2}\) r 2p = pr.

∴ Required remaining area = Area of quadrilateral – Area of circle = pr – πr² = r(p – πr).