Here’s a circle, of radius r:

What is its area? Because the radius is perpendicular to the circumference, we can measure the area in approximately square units like so:

If the circumference stayed the same no matter what the radius, this method would work, and the area of the circle would be r * circumference, the same as the blue rectangle here:

In fact, however, the circumference shrinks linearly with the radius, so we must adjust the width of the rectangle continuously to match the radius, like so:

This makes it a triangle. You can easily see that the triangle has exactly half the area of the rectangle, so this makes our area formula r * circumference / 2. Since the circumference is r * 2pi we can substitute that in, then simplify:

r * (r * 2pi) / 2

r * r * 2pi / 2

pi * r²

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