Example 1

Review question R6838 at the Circles station

Two coplanar circles, each of radius \(\quantity{5}{cm}\), have their centres \(\quantity{6}{cm}\) apart. Calculate the area of the region common to the interiors of both the circles, giving your answer in \(\mathrm{cm}^2\) correct to two significant figures.

Using this Review question

This problem requires students to work out what is happening (ideally by drawing a sketch of the situation) and then to devise a strategy for solving the question asked. This could be used either before or after introducing radians.

It could be used to introduce the idea of calculating areas of sectors and segments; by asking students to generalise their results, they could derive these formulae for themselves. It could also be used at some point after learning this skill as a review of these skills.

A related problem could be to work out the perimeter of the common region.

The question could also be generalised to ask how the answer would change if the numbers in the question were changed.

Aspects of mathematical thinking

  • Mathematical behaviours to look out for

    • Devising a strategy to enter the problem

    • Drawing appropriate sketches: marking appropriate lengths and angles on the diagram

    • Recognising which triangle techniques are needed (Pythagoras, appropriate trigonometry)

    • Making use of proportion to calculate sector areas

  • Overarching ideas

    • Pervasive idea: Symmetry (offering a way into the problem)

    • Proportional reasoning

Some related resources