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Use a compass to inscribe polygons in a circle.

Q: How do you inscribe a polygon in a circle?

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You can get as close to 100*pi as you like. As you increase the number of sides, the limiting value of the area of the polygon is 100*pi.

You can do an upper and lower bound by inscribing and circumscribing polygons. The more sides the polygon has, the more precise your answer 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.

A circle can be a polygon. Sometimes a circle can be a polygon that has infinite number of sides.

A polygon is a plane area bounded by straight lines. A circle consists of a curved line, not a straight line. Therefore a circle is not a polygon and conversely, no polygon can be a circle.

It means drawing a circle around a polygon in such that each vertex of the polygon is on the circumference of the circle.

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Then you would draw the polygon inside of the circle, or in other words, "inscribe" the polygon.

An inscribed polygon

You can get as close to 100*pi as you like. As you increase the number of sides, the limiting value of the area of the polygon is 100*pi.

5*pi

You can do an upper and lower bound by inscribing and circumscribing polygons. The more sides the polygon has, the more precise your answer 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.

A circle is not a polygon.

No, a circle is not a polygon

That is correct

A circle can be a polygon. Sometimes a circle can be a polygon that has infinite number of sides.

the circle is inscribed in the polygon

A circumscribed polygon is a polygon all of whose vertices are on the circumference of a circle. The circle is called the circumscribing circle and the radius of the circle is the circumradius of the polygon.

False