How can the area of an ellipse be calculated?

• A. a = 2πxy
• B. a = πxy
• C. a = πr²
• D. a = 4πr²

Answer: (B)

The area of an ellipse can be calculated using the formula (A = \pi \cdot a \cdot b), where (a) and (b) represent the lengths of the semi-major and semi-minor axes, respectively. When this formula is simplified, it is often presented as (a = \pi xy), with (x) and (y) standing in for the semi-axis lengths.

This formula works because the area is a product of the dimensions of the ellipse, effectively paralleling the calculation for the area of a circle, where the radius is squared and multiplied by (\pi). However, since an ellipse is stretched in two dimensions, we adjust the formula to include both semi-axis lengths.

The other choice indicating (a = 2\pi xy) incorrectly suggests a factor of 2, which does not reflect the correct mathematical relationship governing the area of an ellipse. The choice of (a = \pi r^2) applies specifically to circles, where (r) is the radius, not suitable for ellipses at all. Lastly, the formula (a = 4\pi r^2) is related to the area of a sphere,