Asymmetric bunny ears?

Geometry Level 3

There are three semi-circles. The largest radius is 5 units. What is the combined area in orange?


The answer is 24.

Mahdi Raza by Mahdi Raza , 16, India

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1 solution

Pop Wong
Apr 18, 2021

The whole figure area = two smaller semi-circles area + the triangle area

The bunny ears area

= two smaller semi-circles area + the triangle area - largest semi-circle area

= π 2 3 2 + π 2 4 2 + 1 2 × 6 × 8 − π 2 5 2 = π 2 ( 3 2 + 4 2 − 5 2 ) + 24 = 24 = \cfrac{\pi}{2} 3^2 + \cfrac{\pi}{2} 4^2 + \cfrac{1}{2} \times 6 \times 8 - \cfrac{\pi}{2} 5^2 \\ = \cfrac{\pi}{2} (3^2+4^2-5^2) + 24 \\ = \boxed{24}

3 Helpful 1 Interesting 2 Brilliant 0 Confused

This is the problem known as the Quadrature of the lunes of Alhazen, which is a generalization of the famous Quadrature of the lune of Hippocrates .

Thanos Petropoulos - 1 month, 3 weeks ago

lunes of Alhazen

Mahdi Raza - 1 month, 3 weeks ago

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