Regular polygons and circles

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?

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

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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