Earlier today I set you five puzzles from More Geometry Snacks. Here are the questions each followed by two methods of solution. Sometimes the simple action of drawing in a few extra lines reveals the solution clearly.

1. A point inside a square is connected to its four vertices. What fraction of the square is shaded?

Solution: 1/2

2. A shaded region is constructed within an equilateral triangle using the midpoint A of a side, and the midpoint between AB. What fraction is shaded?

Solution: 1/4

3. In this regular hexagon, three diagonals have been drawn to form this parallelogram. What fraction of the hexagon is shaded?

Solution: 1/3

4. A square is shown with its inscribed and circumscribed circle. What fraction of the larger circle is shaded?

Solution: 1/2

r^{2} = R^{2}/2

5. This last puzzle is about angles. Four identically-sized squares are arranged in an L. What’s the angle marked in blue?

Solution: 45 degrees

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

Thanks to Ed Southall and Vincent Pantaloni for today’s puzzles.