How do you inscribe a circle into a sector?
- 1 decade agoFavorite Answer
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.