Five circles in a square. Total red area is 24. What is the total orange area?

Catriona Shearer's Twitter feed in May 2020.

First, let's explore a tedious, algebraic approach to computing Θ by measuring the sides of the square in two ways: horizontally and vertically. Let the radius of the orange circle be 1. Horizontal length of the square is 2 + 4 sin Θ. Vertical length of square is 2 cot (Θ / 2) + 4 cos Θ. These must be equal, so

2 + 4 sin Θ = 2 cot (Θ / 2) + 4 cos Θ

We may simplify the equation by applying formulas for half-angles to get a degree-4 equation in sin Θ. Turns out that Θ equals 60 degrees (!) Wow. Can we derive this in a more elegant way?

Here is a solution borrowed from Catriona Shearer's twitter thread:

Catriona Shearer: She is a math teacher in Cambridge. Check out her fascinating puzzles! She has a 13-page file with 150+ geometric puzzles. She also wrote a book with 85 puzzles: Geometry Puzles in Felt Tip (88 pages, 2019).

This article explains that Catriona Shearer is getting high praise from some big shot mathematicians for her puzzles — they are puzzled as to how she conjures her puzzles :) Mathematicians would like to find a general theory that explains how her brain functions.