Regular polygons and circles

problem
LOGOsquares

Age

16 to 18

Challenge level

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.
problem
Kissing

Age

16 to 18

Challenge level

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
problem
Circles ad infinitum

Age

16 to 18

Challenge level

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?
problem
Baby circle

Age

16 to 18

Challenge level

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
problem
Some(?) of the parts

Age

14 to 16

Challenge level

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle
problem
Part the polygons

Age

7 to 11

Challenge level

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

Age

7 to 11

Challenge level

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
problem
Sweets in a box

Age

7 to 11

Challenge level

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
problem
Bracelets

Age

7 to 11

Challenge level

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
problem
Clock hands

Age

7 to 11

Challenge level

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.