Circle properties and circle theorems

problem
Sports equipment

Age

7 to 11

Challenge level

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?
interactivity
Favourite

Virtual geoboard

Age

7 to 16

This virtual geoboard allows you to create shapes by stretching rubber bands between pegs on the board.
problem
Triangles in circles

Age

11 to 14

Challenge level

Can you find triangles on a 9-point circle? Can you work out their angles?
problem
Subtended angles

Age

11 to 14

Challenge level

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
problem
Right angles

Age

11 to 14

Challenge level

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
problem
Cyclic quadrilaterals

Age

11 to 16

Challenge level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
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
Circles in quadrilaterals

Age

14 to 16

Challenge level

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
problem
Long short

Age

14 to 16

Challenge level

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
problem
Sitting pretty

Age

14 to 16

Challenge level

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?