Copyright © University of Cambridge. All rights reserved.

## 'Triangles and Petals' printed from http://nrich.maths.org/

### Why do this problem?

This problem offers the opportunity to practise calculating arc
lengths, working in terms of $\pi$, and calculating interior angles
of regular polygons. It also provides an opportunity to generalise
from simple examples, and to explain patterns in terms of the
underlying structure of the problem.

### Possible approach

Start by showing the
animation of the triangle rotating around another triangle. Then
hide the animation and ask learners to sketch what they saw. Show
the animation once more so they can confirm what they saw.

Explain that we are
interested in calculating the perimeter of the flower that is
traced out. Give them time in pairs to calculate the perimeter,
then share answers and methods.

Now show the animation of
the triangle rotating around a square. Again, ask learners to
sketch what they saw, and show them the animation once more.
Challenge them to work out the perimeter of this flower, and to
then sketch and work out the perimeter of flowers whose centres are
regular pentagons, hexagons and so on. This is a good opportunity
to discuss the benefits of working in terms of $\pi$ rather than
calculating a decimal answer on the calculator.

After the class have had
some time to engage with the task, collect some results together on
the board and encourage them to look for patterns. Can they now
work out the perimeter of a flower whose centre is a regular
$100$-sided polygon?

### Key questions

What angles do you know?
What angles can you work out?

How can we work out the
angle for each arc on a 3, 4, 5... petalled flower?

### Possible extension

Construct a proof for the
formula for finding the perimeter of a flower with a regular
$n$-sided polygon at its centre.

What will the flower look
like if we continue this process indefinitely?

### Possible support

An Unusual Shape
offers opportunities to visualise sectors of circles and to
consider simple loci.