Triangles and petals

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Problem

Triangles and Petals printable sheet

Look at the equilateral triangle rotating around the equilateral triangle. It produces a flower with three petals:

Placing two identical equilateral triangles next to each other and rotating one around the other's vertex, tracing out the shape of 3 circles with overlap.

If each equilateral triangle has side length $r$, can you work out the perimeter of the flower's petals?

Now consider a flower made by the triangle rotating about a square with side length $r$ - what is the perimeter of the petals now?

Image
Moving an equilateral triangle around a square, both of which have the same side length, and tracing out a shape with 4 circles overlapping.

What is the perimeter when the centre of the flower is a regular pentagon, hexagon, heptagon...?

What can you say about the increase in perimeter as the number of sides of the centre shape increases?

Can you explain this increase?

 

What would be the perimeter of a flower whose centre is a regular $100$-sided polygon with side length $r$?

It may help to work in terms of $\pi$ throughout this problem.