How can you find the area of a circle geometrically?

You can do an upper and lower bound by inscribing and circumscribing polygons. The more sides the polygon has, the more precise your will be. You inscribe a polygon by having the corners touch the circle's interior, and you circumscribe a polygon by having the midpoint of the sides touch the circle's exterior.

Note that the polygon must by equilateral and equiangular for this method to be reasonably simple. Then simply find the area of the inscribed polygon - you know the circle is bigger than it, because the circle contains the polygon and has more space as well. Thus that number is your lower bound.

Then find the area of the circumscribed polygon- same logic for the polygon being bigger than the circle. Area of circumscribed is your upper bound. Then typically average your upper and lower bound to get a reasonable estimate of the area of the circle.

Of course, solving the problem algebraically is both simpler and more precise, but since you wanted a geometric answer, you got one. More.

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