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Let θ = the angle between any radius r and the x-axis. If we divide A into many equal sectors, each with arc-length Δθ , each sector has area ΔA = r² Δθ. As we keep dividing A into more and more sections of smaller arc-length, then, as Δθ → 0, the "area" of the sector ΔA → r.

So, we integrate r dθ from 0 to 2π radians:

A = ∫ r dθ

= r² θ / 2 | θ = 0 to 2π

= (2π r² / 2) - 0

= π r²

Radius square =area of circle

or

diameter divided by 2 = area of circle

or

diameter divided by 2 = area of circle