Sine, cosine, tangent

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

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Trigonometry, circles and triangles combine in this short challenge.

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

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?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees before being reflected across to the opposite wall and so on until it hits the screen.

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.