Temple geometry

This task is designed to assess how well students understand applying Pythagoras’ theorem in an unfamiliar situation and calculating areas of circles.

During the Edo period (1603-1897) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. The task presented here is an example of such a puzzle.

The puzzle is set up as follows:

• Inside a large circle with radius 2r, two circles of radius r are drawn
• Four smaller circles, of radius p, are drawn to touch the large circle and circles of radius r

The task is to find the relationship between r and p. The solution involves using Pythagoras’ theorem and isosceles triangles. This task is scaffolded into several steps. The final task is to calculate the shaded area.

This task is from the Mathematics Assessment Resource Service, a collaboration between UC Berkeley and the Shell Centre team at the University of Nottingham. A full rubric is given for the task, together with examples of student work and suggestions for helping students overcome misconceptions. Further information on the collection is given here.