Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

Can you explain what is happening and account for the values being displayed?

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

There are many different methods to solve this geometrical problem - how many can you find?

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Prove Pythagoras' Theorem for right-angled spherical triangles.

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?