Two Circles, Apart For A Triangle

Geometry Level 2

If r = 1 r=1 , then what is the area coloured in purple?

Details:

  • The blue sector is a perfect circle tangent to all three sides of the red equilateral triangle.
  • The red equilateral triangle is inscribed to the purple circle.
2 π − 3 3 2\pi-3\sqrt{3} π − 3 3 \pi-3\sqrt{3} 4 π − 3 3 4\pi-3\sqrt{3} 4 π − 6 3 4\pi-6\sqrt{3} 2 π − 3 2\pi-\sqrt{3} 4 π − 3 4\pi-3

Andrei Li by Andrei Li , 16, Canada

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1 solution

Andrei Li
Aug 14, 2018

From this diagram, we know that the radius of the large circle equals 2 2 ; that the base of the equilateral triangle equals 2 3 2\sqrt{3} ; and that the height of the equilateral triangle equals 3 3 . It follows that the area of the purple sector, A p u r p l e A_{purple} , is

A p u r p l e = π R 2 − 1 2 b h = 4 π − 3 3 A_{purple}=\pi R^2-\frac{1}{2}bh=4\pi-3\sqrt{3}

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