The area of a circle with radius r is

• A. pi r
• B. 2 pi r
• C. pi r^2
• D. r^2

Answer: (C)

Area grows with the square of the radius, because when you scale a circle, the amount of space it covers increases with the square of that linear size. The constant that links that squared radius to area is π, which comes from how circles are defined and relates to the ratio of a circle’s circumference to its diameter. Put together, the area is A = π r^2. You can see why this makes sense by imagining doubling the radius: the area becomes four times as large, not eight times, illustrating the r^2 relationship. Other options don’t fit: πr would be a length, not an area; 2πr is the circumference; r^2 omits the π factor.