Regular polygons and circles

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?
problem

Roll On

Age

14 to 16

Challenge level

Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?
problem

Gold yet Again

Age

16 to 18

Challenge level

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."
problem

Semicircle in a Semicircle

Age

14 to 16

Challenge level

The diagram shows two semicircular arcs... What is the diameter of the shaded region?
problem

Oh so Circular

Age

14 to 16

Challenge level

The diagram shows two circles and four equal semi-circular arcs. The area of the inner shaded circle is 1. What is the area of the outer circle?
problem

Circle Time

Age

14 to 16

Challenge level

Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?
problem

Gaudi's Design

Age

11 to 16

Challenge level

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

What Shape and Colour?

Age

5 to 7

Challenge level

Can you fill in the empty boxes in the grid with the right shape and colour?
problem
Favourite

2 Rings

Age

5 to 7

Challenge level

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?