Which formula gives the cross-sectional area of a circle based on its radius?

Which formula gives the cross-sectional area of a circle based on its radius?

• A. A = 2πr
• B. A = π r^2
• C. A = π D^2
• D. A = 4πr^2

Answer: (B)

The area of a circle is determined by how quickly it grows when you increase the radius, and the precise relationship is A = π r^2. This comes from how π connects a circle’s radius to its geometry: π is the constant that relates a circle’s circumference to its diameter, and when you square the radius you capture the way area scales with size. So for any circle, cross-sectional area equals π times the radius squared. For the other formulas: A = 2πr gives the circumference, not area. A = π D^2 uses the diameter inside, which, since D = 2r, becomes A = π(2r)^2 = 4πr^2—four times the true area. A = 4πr^2 is the surface area of a sphere of radius r, not a circle’s cross-sectional area. So the correct expression for the circle’s cross-sectional area is A = π r^2. For example, if the radius is 3 units, the area is π × 9 ≈ 28.27 square units.

The area of a circle is determined by how quickly it grows when you increase the radius, and the precise relationship is A = π r^2. This comes from how π connects a circle’s radius to its geometry: π is the constant that relates a circle’s circumference to its diameter, and when you square the radius you capture the way area scales with size. So for any circle, cross-sectional area equals π times the radius squared.

For the other formulas: A = 2πr gives the circumference, not area. A = π D^2 uses the diameter inside, which, since D = 2r, becomes A = π(2r)^2 = 4πr^2—four times the true area. A = 4πr^2 is the surface area of a sphere of radius r, not a circle’s cross-sectional area. So the correct expression for the circle’s cross-sectional area is A = π r^2. For example, if the radius is 3 units, the area is π × 9 ≈ 28.27 square units.