The area of the circle formula can be visually proven by dividing the area into concentric rings, unwrapping the rings and placing them next to each to form a triangle. This is shown in the following illustrations that divide the area of the circle into 3, 5 and 10 rings respectively.
As the area of the circle is divided into more and more rings, the shape on the right-hand side approaches the shape of a triangle.
Since the triangle is formed by the rings of the circle, the area of this triangle is equal to the area of the circle. We know the area of a triangle is given by the formula[1]:
So we can substitute the length of the base and height into the formula to calculate the area.
After simplifying, this gives us the area of the circle formula.
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Derive Area of Triangle Formula