Two equilateral triangles and a quarter circle that goes through both apexes. If the smaller triangle has area 1, what is the total shaded area?

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The shaded area is 5.

Solution by Jane Miller.

First extend the quarter to a full circle. Then reflect the large triangle in the diameter as shown. The angles on the left side of B add up to 180. Therefore, DE is a straight line.

From Thales we know that angle DEC is right. Now consider triangle BEC. It is a 30-60-90 triangle, and it is well known that the short side is exactly half the hypotenuse. So |CB|=2|EB|. The large triangle is therefore four times larger than the small one.

Nice illustration of proportions by Jonathan Crownshaw.

Visual solution

Solution without words by David Andriana.


Gizeh and it’s pyramids
Keops, Kephren, Mykerinos
So beautiful
And aesthetic
So powerfull
And dynamic
A treasure for generations
A gift for civilisations
With 1000 angles of beauty
Climbing the steps, heavenly
To the egyptian dynasty
A glorious hymn to trinity
Gizeh is eternity


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