F'arc'tion
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.
Problem
At the corner of a cube, circular arcs are drawn as shown.
What fraction of the surface area of the cube is shaded?
Student Solutions
Here is the solution from Christiane Eaves, Alicia Maltby, Kathy Lam, Rachael Evans andFiona Conroy (Y10) The Mount School,York:
The total surface area of the cube is $ 6r^2. $ The area shaded on one face is $ \frac{\pi r^2}{4}$ so the total shaded area is $ \frac{3\pi r^2}{4}.$
The fraction of the total surface area shaded is thus $\frac{3\pi r^2}{4}$ divided by $6r^2$ which is: $$\frac{3\pi r^2}{4} \times\frac{1}{6r^2}= \frac{\pi}{8}$$