Here's this week's collection. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.
I don’t know if you are fond of puzzles, or not. If you are, try this. … A gentleman (a nobleman let us say, to make it more interesting) had a sitting-room with only one window in it–a square window, 3 feet high and 3 feet wide. Now he had weak eyes, and the window gave too much light, so (don’t you like ‘so’ in a story?) he sent for the builder, and told him to alter it, so as only to give half the light. Only, he was to keep it square–he was to keep it 3 feet high–and he was to keep it 3 feet wide. How did he do it? Remember, he wasn’t allowed to use curtains, or shutters, or coloured glass, or anything of that sort.
Semi Circle in a Triangle
This right-angled triangle above has sides of lengths 12cm, 5cm and 13cm. The diameter of the semicircle lies on the 12cm side and the 13cm side is a tangent to the circle. What is the radius of the semi circle?