Circle properties and circle theorems

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

Similarly So

Age

14 to 16

Challenge level

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
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

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

Circle Scaling

Age

14 to 16

Challenge level

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

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

Three Balls

Age

14 to 16

Challenge level

Do points P and Q lie inside, on, or outside this circle?
problem

Tied Up

Age

14 to 16

Challenge level

How much of the field can the animals graze?