Snail's Pace
A snail is at the corner of a cube. What proportion of the cube's surface can the snail reach within an hour?
Age
14 to 16
Challenge level
Problem
A snail is at one corner of the top face of a cube with side length 1m.
The snail can crawl at a speed of 1m per hour.
What proportion of the cube's surface is made up of points which the snail could reach within one hour?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Image
On each of the three reachable faces, the points which the snail can reach form a quarter of a circle of radius 1m.
So the required fraction is $\frac{3 \times \frac{1}{4}\pi \times 1 \times 1}{6 \times 1 \times 1} = \frac{\pi}{8}$.