Regular polygons and circles

problem
The pillar of Chios

Age

14 to 16

Challenge level

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
problem
Encircling

Age

14 to 16

Challenge level

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
problem
Approximating pi

Age

14 to 18

Challenge level

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?
problem
Crescents and triangles

Age

14 to 16

Challenge level

Can you find a relationship between the area of the crescents and the area of the triangle?
problem
Semi-square

Age

14 to 16

Challenge level

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?
problem
Squaring the circle and circling the square

Age

14 to 16

Challenge level

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

Age

11 to 18

Challenge level

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?
problem
Circumnavigation

Age

14 to 16

Challenge level

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
problem
The dodecahedron

Age

16 to 18

Challenge level

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?
problem
Hexagon cut out

Age

11 to 14

Challenge level

Weekly Problem 52 - 2012
An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?