Circular horn puzzle

The area of the blue circular horn triangle can be calculated using either GCSE or A-Level approaches...

circular horn triangle
Solution

Connect the three centres of the red circles to produce an equilateral triangle, with sides of length $2R$

The height of this equilateral triangle will be $\sqrt{3}R$. The area of the triangle is $$ A_\Delta = \frac{1}{2}bh = \frac{1}{2}\cdot 2R \cdot \sqrt{3}R = \sqrt{3}R^2 $$ The three circular segments intersecting the triangle each have a central angle of $60^\circ$. Each has an area of $\frac{1}{6}\pi R^2$. The desired region has an area of $$A = A_\Delta - 3A_S = \left(\sqrt{3} - \frac{\pi}{3}\right)R^2$$ However, this only gives us an expression in terms of the radius of the small circles, a currently unknown quantity.

Consider the distance from the centre of the circle to the top of the circle. This will be two thirds of the height of the triangle plus an additional small radius $$ 1 = \frac{2}{3}\sqrt{3}R + R $$ Solving this for $R$ gives $$ R = \frac{3}{3 + 2\sqrt{3}} = \frac{3(3 - 2\sqrt{3})}{9 - 12} = 2\sqrt{3} - 3 $$ The final area of the blue region is therefore $$ A_B = \left(\sqrt{3} - \frac{\pi}{3}\right)\left(2\sqrt{3} - 3\right)^2 = (3\sqrt{3} - \pi)(7 - 4\sqrt{3}) $$

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