The diagram shows a rectangular metal plate \(20\) cm by \(12\) cm. A quarter circle of radius \(4\) cm is removed from each of the four corners. (a) Find t...

(a) Each corner loses a quarter of a circle of radius 4 cm. Four quarters make one complete circle [M1], so the total area removed is the area of a single circle of radius 4 cm.

Area removed \(=\pi r^{2}=\pi\times 4^{2}=16\pi\) cm\(^{2}\) [A1].

(b) The whole plate has area \(20\times 12=240\) cm\(^{2}\) [M1], so the metal remaining is

\((240-16\pi)\) cm\(^{2}\) [A1].

The two terms cannot be combined, because one is a plain number and the other is a multiple of \(\pi\). Leaving the answer in this form keeps it exact.

(c) Taking \(\pi=3.14\): \(16\times 3.14=50.24\), so the remaining area is \(240-50.24=189.76\) cm\(^{2}\), which to the nearest square centimetre is 190 cm\(^{2}\) [B1] cao.

The idea worth carrying forward is that quarter circles cut from the four corners of a rectangle always combine into exactly one circle, provided they all have the same radius. That turns four separate calculations into one.