### Arclets

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

### How Far Does it Move?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.

### Speeding Up, Slowing Down

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.

# Floored

##### Stage: 3 Challenge Level:

Congratulations to Nisha Doshi, Year 9, The Mount School, York for this beautifully explained solution.

If these triangles are made into a tessellation, they will form regular hexagons with circles overlaid. Each triangle consists of 3 x 1/6 = 1/2 of a circle, plus the shaded area, so to find the area of the shaded section, you can do : (area of triangle - area of 1/2 of a circle)

Using Pythagoras' Theorem the height of the triangle is $\sqrt((2r)^2 - r^2 ) = r\sqrt3$. So the area of the triangle is $r^2\sqrt3$. The area of ${1\over 2}$ a circle is ${1\over 2}\pi r^2$. So the area shaded is $r^2\sqrt3 - {1\over 2}\pi r^2$ and the proportion of the tessellation that is shaded is $$\frac{r^2\sqrt3 - {1\over 2}\pi r^2}{r^2\sqrt3} = 1 - \frac{\pi}{2\sqrt3}$$

Well done Arwa Jamil, Year 8, the International School Brunei who calculated correctly, to 3 significant figures, that 9.31% of the floor is shaded.