Height and area of circular segment?

Update:

#1. Before somebody asks, this is NOT homework. This problem was inspired by working up ballistics data for 45 ACP ammo. Quickload Software indicated that powder ABC will only fill 67% of the straight-walled 45 ACP case ... which prompted this generalized question. Thanks.

Relevance
• iceman
Lv 7
3 years ago

The formula to find the area of the segment is given by:

½ R² (π/180 * θ − sinθ)

Where:

θ = the central angle in degrees

R = the radius of the circle

General formula for circle center to chord midpoint distance d:

d = √[(R² - (¼ C²)]

Where:

C = chord length

Note: let O be the center , OA & OB be the two radius R in the graph above, then the area in the green can also be found by calculating the area of the whole sector(pie shaped) and subtracting the area of the isosceles triangle △OAB.

Segment height h:

h = R - d

Area of the segment can also be calculated if h is known as follow:

Area = R² * arccos[(R - h)/R] - (R - h) * √(2Rh - h²)

I hope this helps.