Challenge Level

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

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?

Challenge Level

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Challenge Level

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

Challenge Level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Challenge Level

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Challenge Level

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Challenge Level

Explain how to construct a regular pentagon accurately using a straight edge and compass.

Challenge Level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Challenge Level

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

Challenge Level

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

Challenge Level

Investigate constructible images which contain rational areas.

Challenge Level

What fractions can you divide the diagonal of a square into by simple folding?

Challenge Level

Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B. Use a geometric construction to locate a point, P, on the line,. . . .

Challenge Level

Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

Challenge Level

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Challenge Level

The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

Challenge Level

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

Challenge Level

Drawing a triangle is not always as easy as you might think!

Challenge Level

How can you represent the curvature of a cylinder on a flat piece of paper?

Challenge Level

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.