Two circles

Two circles with the same radius r = 1 are given. The center of the second circle lies on the circumference of the first. What is the area of a square inscribed in the intersection of given circles?

Result

S =  0.5

Solution:

r=1 r2=r/2=1/2=12=0.5 u=2⋅ r2=2⋅ 0.5=1  a=u/2=1/2≐0.7071  S=a2=0.70712=12=0.5r = 1 \ \\ r_{ 2 } = r/2 = 1/2 = \dfrac{ 1 }{ 2 } = 0.5 \ \\ u = 2 \cdot \ r_{ 2 } = 2 \cdot \ 0.5 = 1 \ \\ \ \\ a = u / \sqrt{ 2 } = 1 / \sqrt{ 2 } \doteq 0.7071 \ \\ \ \\ S = a^2 = 0.7071^2 = \dfrac{ 1 }{ 2 } = 0.5

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Following knowledge from mathematics are needed to solve this word math problem:

Pythagorean theorem is the base for the right triangle calculator.

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