Regular polygons and circles

problem
Retracircles

Age

16 to 18

Challenge level

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.
problem
Escriptions

Age

16 to 18

Challenge level

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
problem
Star gazing

Age

14 to 16

Challenge level

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
problem
Coins on a plate

Age

11 to 14

Challenge level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
problem
Not so little x

Age

11 to 14

Challenge level

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?
problem
Bull's eye

Age

11 to 14

Challenge level

What fractions of the largest circle are the two shaded regions?
problem
Hex

Age

11 to 14

Challenge level

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
problem
LOGO challenge 1 - star square

Age

7 to 16

Challenge level

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.
problem
LOGO challenge 10 - circles

Age

11 to 16

Challenge level

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.
problem
LOGO challenge 6 - triangles and stars

Age

11 to 16

Challenge level

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.