Regular polygons and circles

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Can you find a relationship between the area of the crescents and the area of the triangle?

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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.

What fractions of the largest circle are the two shaded regions?

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.