H1. The diagram shows four equal arcs placed on the sides of a square. Each arc is a major arc of a circle with radius $1 \mathrm{~cm}$, and each side of the square has length $\sqrt{2} \mathrm{~cm}$.
What is the area of the shaded region?
Solution
Join the centres of the circles, as shown in the diagram alongside, to form a quadrilateral whose sides have length $1 \mathrm{~cm}+1 \mathrm{~cm}$. Each angle of this quadrilateral is equal to $90^{\circ}$ from the converse of Pythagoras' Theorem, since we are given the 'inner' square has sides of length $\sqrt{2} \mathrm{~cm}$. Because it has equal sides. it follows that the quadrilateral is a square.
The shaded region comprises this square and four sectors of circles, each of radius $1 \mathrm{~cm}$ and angle $270^{\circ}$, thus its area is equal to $2 \times 2+4 \times \frac{3}{4} \times \pi \times 1^{2}$, in $\mathrm{cm}^{2}$.
Therefore, in $\mathrm{cm}^{2}$, the shaded area is equal to $4+3 \pi$.