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Out of context: Reply #1372

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  • imbecile2

    re: https://qbn.com/reply/4103219/

    Let's say we have a circle with a radius r centered at the origin. The equation of the circle is


    from which we get

    The area of the circle is

    Thus, one third of the area is

    To find the x-value where we place the first vertical line (starting from the center of the circle and moving outward), we want to find x such that the area from the y-axis to x is one third of the circle's area.

    Using the formula for the area of a segment:


    We set this equal to


    From this equation, we can solve for x to determine the point at which to place the first vertical line to trisect the circle by area. The second vertical line would be its reflection across the y-axis since the circle is symmetric.

    This is a transcendental equation. A closed-form solution in terms of elementary functions is not possible, so a numerical method like the Newton-Raphson method is typically used to approximate x.

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