Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

Weekly Problem 42 - 2006

Stage: 3 Challenge Level:

Let $O$ be the centre of the circle and let the points where the arcs meet be $C$ and $D$ respectively. $ABCD$ is a square since its sides are all equal to the radius of the arc $CD$ and $\angle ACB=90^{\circ}$ (angle in a semicircle).

In triangle $OCB$, $CB^2 = OC^2 + OB^2$; hence $CB=\sqrt{2}$ cm. The area of the segment bounded by arc $CD$ and diameter $CD$ is equal to the area of section $BCD -$ the area of the triangle $BCD$, i.e.

$$\left(\frac{1}{4}\pi\left(\sqrt{2}\right)^2-\frac{1}{2}\times\sqrt{2}\times \sqrt{2}\right)\textrm{cm}^2$$,i.e. $(\frac{1}{2}\pi - 1)$ cm$^2$.

The unshaded area in the original figure is, therefore, $(\pi-2)$ cm$^2$. Now the area of the circle is $\pi$ cm$^2$, and hence the shaded area is 2 cm$^2$.

This problem is taken from the UKMT Mathematical Challenges.

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