Eye of Providence

A regular pentagon and an equilateral triangle. What is the angle α?

Solution

The angle α is 24°.

Replace the yellow pentagon with the orange, whose vertices are the midpoints of the edges of the yellow one. Encircle the orange pentagon. Draw a blue circle, with center at one of the common vertices of the blue and orange figures. Then the central angle of a piece of the pentagon, 72 degrees, divides in 2 to make 36, using the orange circle, then divides in 2 again, to make 18, using the blue circle. The angle alpha is 60 – 2*18 = 24 degrees.

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