Area of an ellipse

What the symbols mean

$A$

the area of the ellipse

$a$

the semi-major axis: half the longest width

$b$

the semi-minor axis: half the shortest width

When to use it

Use it for oval shapes: running tracks laid out as ellipses, oval tables and mirrors, elliptical orbits, the cross-section of a squashed pipe. Measure the full length and width and halve both. There is no equally simple formula for an ellipse's perimeter.

Why it works

An ellipse is a circle of radius a squashed by the factor $b/a$ in one direction. Squashing scales area by the same factor, so $\pi a^2\times b/a=\pi ab$. When $a=b$ it is a circle and the formula becomes $\pi r^2$.

Common mistakes

• Using the full length and width instead of half of each, which gives four times the area.
• Applying the same trick to the perimeter: the circumference is not $\pi (a+b)$ exactly, only approximately.