👉 To accurately explain the math behind calculating the area of a circle, we start with the formula \( A = \pi r^2 \), where \( A \) represents the area and \( r \) is the radius of the circle. This formula is derived from the concept of limiting the sum of the areas of an infinite number of infinitesimally thin rings, known as the method of exhaustion, a precursor to integration. The constant \( \pi \) (pi), approximately 3.14159, is the ratio of the circumference of a circle to its diameter and is fundamental in this calculation. By squaring the radius, we account for the circle's curvature, ensuring the formula accurately reflects its two-dimensional area. This precise mathematical relationship allows us to compute the area of any circle given its radius, illustrating the power of geometric principles in solving real-world problems.
