Height and area of circular segment?


#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.

Attachment image

1 Answer

  • iceman
    Lv 7
    3 years ago
    Favorite Answer

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

    ½ R² (π/180 * θ − sinθ)


    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)]


    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.

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