Sine, cosine, tangent

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

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

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

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 triangular areas are enclosed. What is the area of this convex hexagon?

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

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 sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.