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

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

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

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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,. . . .

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

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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.

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

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The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

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

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

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Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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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?

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

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

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

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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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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.

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What shape and size of drinks mat is best for flipping and catching?

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

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

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