Eyelids
Can you work out the shaded area surrounded by these arcs?
$AB$ is a diameter of a circle of radius 1 cm. Two circular arcs of equal radius are drawn with centres $A$ and $B$. These arcs meet on the circle, as shown. What is the shaded area?
Image
![Eyelids Eyelids](/sites/default/files/styles/large/public/thumbnails/content-id-4987-circle_q.gif?itok=2mnyYos-)
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
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$^2$.
Image
![Eyelids Eyelids](/sites/default/files/styles/large/public/thumbnails/content-id-4987-circle_a.gif?itok=CRUux4sJ)