Regular polygons and circles

problem

Spirostars

Age

16 to 18

Challenge level

2 out of 3

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?
problem

Circumspection

Age

14 to 16

Challenge level

3 out of 3

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.
problem

Semi-Square

Age

14 to 16

Challenge level

3 out of 3

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

Circumnavigation

Age

14 to 16

Challenge level

3 out of 3

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

Gold yet Again

Age

16 to 18

Challenge level

3 out of 3

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

Gaudi's Design

Age

11 to 16

Challenge level

3 out of 3

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

What Shape and Colour?

Age

5 to 7

Challenge level

1 out of 3

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

Shaping It

Age

5 to 11

Challenge level

1 out of 3

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.
problem

Part the Polygons

Age

7 to 11

Challenge level

1 out of 3

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.
problem

Overlapping Circles

Age

7 to 11

Challenge level

1 out of 3

What shaped overlaps can you make with two circles which are the same size?