Area of an Ellipse

Area of an Ellipse

The formula is given below.

Ellipse with semi-axes a (horizontal) and b (vertical), equation (x−h)²/a²+(y−k)²/b²=1, and area formula Area=πab.

See also

Key Formula

A = \pi a b

Where:

$A$ = Area of the ellipse
$a$ = Length of the semi-major axis (half the longest diameter)
$b$ = Length of the semi-minor axis (half the shortest diameter)

Worked Example

Problem: Find the area of an ellipse with a semi-major axis of 5 cm and a semi-minor axis of 3 cm.

Step 1: Identify the semi-axes:

a = 5

and

b = 3

Step 2: Substitute into the formula.

A = \pi(5)(3) = 15\pi

Step 3: Approximate the result.

A \approx 47.12 \text{ cm}^2

Answer: The area of the ellipse is

15\pi \approx 47.12

cm².

Why It Matters

This formula generalizes the area of a circle. When

a = b = r

, the ellipse becomes a circle and

\pi ab

reduces to

\pi r^2

. Ellipse area calculations appear in engineering, orbital mechanics, and architecture wherever oval shapes are used.

Common Mistakes

Mistake: Using the full axis lengths instead of the semi-axis lengths.

Correction: The formula uses semi-axes (half each diameter). If you are given the full major axis of 10 and minor axis of 6, divide each by 2 first to get

a = 5

and

b = 3

before applying

A = \pi ab

Related Terms

Ellipse — The shape whose area this formula computes
Formula — General term for mathematical equations like this
Area of a Circle — Special case when both semi-axes are equal
Semi-Major Axis — The longer semi-axis used in the formula
Pi — The constant approximately 3.14159 used here