What is the area of a circle formula?

The area of a circle is calculated using the formula A = πr², where A represents the area and r represents the radius of the circle. This formula is derived from the concept of how the area relates to the radius: when you square the radius (r²) and multiply it by π (approximately 3.14159), you obtain the total space contained within the circular boundary.

This relationship comes from integral calculus when deriving the area of a circle based on the accumulation of infinitesimally small circular rings. Each ring's area accumulates to give the total area, resulting in the formula A = πr².

Additionally, it's important to understand that radius squared (r²) indicates that the area grows with the square of the radius. Thus, if the radius doubles, the area increases by a factor of four, which is a unique property of two-dimensional shapes like circles.

In contrast, other options either represent different geometric properties of circles or contain incorrect relationships that do not pertain to calculating area. For example, A = 2πr represents the circumference of a circle, while A = πr suggests a misunderstanding of the area relationship. A = 2πr² does not represent a known formula related to circles