Circle properties and circle theorems

problem
Dodecagon angles

Age

11 to 14

Challenge level

Weekly Problem 50 - 2012
The diagram shows a regular dodecagon. What is the size of the marked angle?
problem
Salinon

Age

14 to 16

Challenge level

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
problem
Circle scaling

Age

14 to 16

Challenge level

Describe how to construct three circles which have areas in the ratio 1:2:3.
problem
Circle box

Age

14 to 16

Challenge level

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
problem
Flower

Age

16 to 18

Challenge level

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.
problem
Lunar angles

Age

16 to 18

Challenge level

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?
problem
Circles in circles

Age

16 to 18

Challenge level

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.
problem
ArRh!

Age

14 to 16

Challenge level

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?
problem
Belt

Age

16 to 18

Challenge level

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
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.