How do you inscribe a circle into a sector?

Answer 1

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Imagine the sector in the x-y plane with radius R and let alpha be its angle. Suppose its origin is at (0,0) and it is symmetrical with respect to the x-axis. Say r is the radius of the inscribed circle.

Then the center is along the x-axis at the point where x = R - r. The line from the center perpendicular to a side of the sector creates a right triangle, showing that

x sin(alpha/2) = r.

Using this equation and x = R - r, one can find that

r = R sin(alpha/2) / ( 1 + sin(alpha/2)).

Knowing r and so also x, one can construct the circle.