What is the formula for the area of a circle?

The formula for the area of a circle is A = πr², where A represents the area, r is the radius of the circle, and π (pi) is a constant approximately equal to 3.14159. This formula derives from the concept that the area of a circle is proportional to the square of its radius. By squaring the radius (r) and then multiplying by π, you account for the way circles expand in two dimensions.

Understanding why this formula works involves realizing that the area of a circle can be thought of as the number of square units that fit inside it. Since the radius represents the distance from the center to the edge of the circle, squaring the radius gives a measure of how that distance multiplies out in both dimensions, effectively covering the entire circular region. This is a fundamental concept in geometry and underpins many applications in mathematics, physics, and engineering.

The other options represent different mathematical principles: one relates to the circumference of a circle (which is a perimeter measurement), while others do not accurately describe the area in a way that incorporates necessary dimensionality. Understanding the distinction between perimeter and area is crucial in geometry, and this formula is a cornerstone for calculating areas of circular shapes.