Regular polygons and circles

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

Are these statements always true, sometimes true or never true?

How could you find out the area of a circle? Take a look at these ways.

Eight lines are drawn in a regular octagon to form a pattern. What fraction of the octagon is shaded?

Investigate constructible images which contain rational areas.

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.