How do you find an area of a segment of a circle?

Answer 1

The solution depends on the information supplied.

Basically, you find the area of the sector containing the segment and then deduct the area of the triangle formed by the chord and the two radii enclosing the sector.

If you are given the radius(r) of the circle and the height(h) then construct a radius that is perpendicular to and bisects the chord. This will create two congruent triangles which together form the main triangle. Using Pythagoras enables the half-chord length to be calculated as the hypotenuse is r and the height (also the length of the third side) is r-h. With this information the full chord length can be established and thus the area of the main triangle. Using sine or cosine methods enables the sector angle at the centre to be calculated and thus the sector area. Simple subtraction produces the area of the segment.

If you are given the radius and the chord(c) length then the construction referred to above enables the height of the main triangle to be calculated and a similar process will generate the area of that triangle and the sector area. This, in turn, will enable the segment area to be determined.