# What is the area of a reuleaux triangle, but with square angles as corners?

Supposing we have a triangle where each of its edges comes from a section of a circle's edge, and that every corner is a right angle, I need to calculate the area and perimeter of it.

If the three circles that create this shape have radius one, one can show that their centers must be at $$\sqrt2$$ distance appart. But what is its area and perimeter? Thanks.

• I suppose you have drawn a diagram. Divide the area into an ordinary triangle and three circular segments, and algebra the shit out of it. – hmakholm left over Monica Nov 7 '18 at 12:58

Assuming that the sides of the inner equilateral triangle have unit length, the blue arcs are arcs of circles with radius $$\sqrt{3}$$, associated to chords with unit length. It follows that the length of each arc is $$\frac{\pi}{6}\sqrt{3}$$. The enclosed area is the area of the equilateral triangle, $$\frac{1}{4}\sqrt{3}$$, plus three times the area of a circle segment, which can be seen as the difference between a circle sector with area $$\frac{\pi}{4}$$ and a triangle with area $$\frac{abc}{4R}=\frac{3}{4}$$.