Hi guzzlers.

Today’s puzzles are about the shade, by which I mean the shaded areas in the geometrical diagrams below. The images are to be studied and contemplated, until the pleasurable moment of insight arrives…

(Background help: the area of a triangle is half the base times the height. And Pythagoras’s Theorem states that the square of the hypotenuse is equal to the squares of the other two sides.)

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

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?

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

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

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

All five puzzles are taken from More Geometry Snacks by Ed Southall and Vincent Pantaloni, the follow-up to last year’s mouthwatering Geometry Snacks. I love the concept and design of both books, which are pocket-sized and with almost no text at all. It is deeply pleasurable to revisit the basics of Euclidean geometry in such well-presented, bite-sized morsels. The authors’ mission is to stimulate thinking about how to solve geometry problems and they always present two ways to solve each problem. I’ll include both solutions. You, however, are only required to find one of them…

PLEASE NO SPOILERS

I keep the comments open because they are often funny and sometimes contribute interesting ideas about the mathematics behind the puzzles. But since I can’t stop smart Alecs writing the answer, if you want to solve the puzzle yourself DO NOT SCROLL DOWN!

*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.*

*I’m the author of several books of popular maths and puzzles, as well as the series Football School, for 7 to 13-year-olds, which explains the world though football. Note to parents thinking about Christmas gifts: the box set of Football School Seasons 1 to 3 is out now!*