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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

Triangles and Petals

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.