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Find the area of a segment formed by a chord 8" long in a circle with radius of 8".

  • Geometry -

    as we all know, the side of an inscribed hexagon is equal to the radius.

    So, the angle subtending the side is π/3

    a = 1/2 r^2 (θ - sinθ)
    = 1/2 * 64 (π/3 - √3/2)
    = 32/6 (2π - 3√3)

    More generally, the angle θ which subtends a chord of length s in a circle of radius r is

    sin θ/2 = (s/2)/r = s/2r

    In this problem, s=r=8, so

    sin θ/2 = 1/2
    θ = π/3

    and away we go.

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