The area of a Reuleaux triangle

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Let r be the radius of the arcs (also the diameter of the triangle), the area of a Reuleaux triangle is the sum of the area of equilateral triangle and three arc sections that are shaded in the figure above. (NOTE: here each arc section is one sixth of a circle (a pie) minus the triangle)

Then we have:

r * (√3/2) * r / 2 + 3 * (1/6 * PI * r * r – r * (√3/2) * r / 2)

=  r * r * ( PI/2 –  √3/2)

 

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