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The formula A = bh can be used to find a good estimate for the area of the cut-out diagram. What would happen to this estimate if the circle were cut into 100 sections? 1,000 sections?
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GPT-5 mini
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When you cut the circle into many equal wedge-shaped sectors and rearrange them alternating up and down, the figure approaches a rectangle whose base is half the circumference and whose height is the radius. For a circle of radius r that gives
base ≈ πr, height ≈ r → A ≈ (πr)(r) = πr^2,
which is the exact area of the circle. As you increase the number of sectors the jagged top and bottom become finer and the approximation gets better; in the limit (infinitely many, infinitesimally thin sectors) it is exact.
So with 100 sections the estimate is already very close to πr^2, and with 1,000 sections it is even closer (the error goes to zero as the number of sections increases). For example, for a unit circle (r = 1) the area estimate approaches 3.14159…; 100 slices give a visually indistinguishable approximation and 1,000 slices are essentially exact for ordinary purposes.
base ≈ πr, height ≈ r → A ≈ (πr)(r) = πr^2,
which is the exact area of the circle. As you increase the number of sectors the jagged top and bottom become finer and the approximation gets better; in the limit (infinitely many, infinitesimally thin sectors) it is exact.
So with 100 sections the estimate is already very close to πr^2, and with 1,000 sections it is even closer (the error goes to zero as the number of sections increases). For example, for a unit circle (r = 1) the area estimate approaches 3.14159…; 100 slices give a visually indistinguishable approximation and 1,000 slices are essentially exact for ordinary purposes.
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