What is the approximate area of a segment with a radius 12m if the length of the chord is 20m?

Answer 1

Draw the sector OAB, and suppose that radius OA = OB = 12 m, and arc AB = 20m.

The area of the segment AB equals to the difference of the area of the sector OAB and the area of the triangle OAB.

1. (the sector area) As = (n/360) pi r2 = (144pi*n)/360 = n(71pi)/180, where n is the number of degrees of the angle AOB

2. (the area of the triangle) At = (1/2)(r*r)(sin O) = 144/2 sin O = 71 sin O (since the angle O is between the two radii)

Let's find the measure in degrees of the angle O.

C = 2 pi r = 2(12) pi = 24 pi m (the circumference of the circle)

20/24pi = 5/6pi (the ratio of the arc length to the circumference)

So we have a proportion:

5/6pi = n/360 (n/360, the ratio of the measures in degrees of the angle AOB to the circle)

5/pi = n/60

n = 300/pi (by the the rule of 3)

So that At = 71 sin 300/pi m2, and As = n(71pi)/180 = [(300/pi)(71pi)]/180 = 375/3 m2

the segment area = As - At = 375/3 - 71 sin 300/pi = 54.32... ≈ 54 m2

Note: Try to avoid as much as you can the approximations during the work, to have an answer too close to the real one.