Sine, cosine, tangent

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

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?

Can you work out the dimensions of the three cubes?

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.

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?

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?

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

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?