What is the area of a circle with radius r?

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Multiple Choice

What is the area of a circle with radius r?

The area of a circle depends on the square of its radius. If you think of filling the circle with many thin rings from the center out to radius r, each ring at radius x has area roughly its circumference 2πx times a small thickness dx, so dA = 2πx dx. Adding up all those rings from 0 to r gives ∫0^r 2πx dx = πr^2. Therefore the area is πr^2. Why the other forms don’t fit: 2πr is the circumference, not area. πr is missing a factor of r, so it’s not area. 2r^2 has the incorrect scaling and lacks π.

The area of a circle depends on the square of its radius. If you think of filling the circle with many thin rings from the center out to radius r, each ring at radius x has area roughly its circumference 2πx times a small thickness dx, so dA = 2πx dx. Adding up all those rings from 0 to r gives ∫0^r 2πx dx = πr^2. Therefore the area is πr^2.

Why the other forms don’t fit: 2πr is the circumference, not area. πr is missing a factor of r, so it’s not area. 2r^2 has the incorrect scaling and lacks π.

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