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Area of Ellipse — Formula, Derivation, and Examples

Geometry

TL;DR

The area of an ellipse is A=πab, where a is the semi-major axis and b is the semi-minor axis (the half-lengths of the longest and shortest diameters). This article covers the area of ellipse formula, two ways to derive it — by stretching a circle and by integration — a variable glossary, and worked examples. When a=b, the ellipse becomes a circle and the formula collapses to πr².

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Bhanzu Team Last updated on June 25, 2026 7 min read

What Is The Area Of An Ellipse?

The area of an ellipse is the amount of flat space enclosed by the oval curve, given by:

A=πab

Here a is the semi-major axis — half the length of the longest diameter — and b is the semi-minor axis, half the length of the shortest diameter. Both are measured from the center to the edge along the two axes, just as a radius is measured in a circle. An ellipse has two such radii instead of one; multiply them, scale by π, and you have the area.

A point worth flagging: a and b are semi-axes, the half-lengths. If a problem hands you the full major and minor axis lengths, halve each before multiplying. This is the single most common source of wrong answers on ellipse-area problems.

How Is The Area Of An Ellipse Formula Derived?

There are two clean ways to see where A=πab comes from. The first needs no calculus.

By stretching a circle. Start with a circle of radius a. Its area is πa². Now squash the circle vertically by the factor b/a, so that every vertical distance shrinks while horizontal distances stay the same — the circle becomes an ellipse with semi-axes a and b. Scaling one direction by a factor scales the area by the same factor:

A=πa²×(b/a)=πab

That is the formula, derived from the circle the ellipse came from.

By integration. The top half of an ellipse x²/a² + y²/b² = 1 is the curve y=b√(1 - x²/a²). Integrating across the width and doubling gives the area:

A=2∫(−a to a)b√(1 - x²/a²)dx

The integral of √(1 − x²/a²) across [-a, a] evaluates to πa², so:

A=2b×(πa/2)=πab

Both routes land on the same place. The stretch argument is the one to keep in mind because it shows why the formula multiplies the two semi-axes.

Examples of Area of Ellipse

Example 1

Find the area of an ellipse with semi-major axis a=6 cm and semi-minor axis b=4 cm. Use π≈3.14.

A=πab=A=3.14×6×4=75.36 cm².

Final answer: 75.36 cm².

Example 2

An ellipse has a major axis of 14 cm and a minor axis of 10 cm. A student computes the area as π×14×10=439.6 cm². What went wrong, and what is the correct area? Use π≈3.14.

The first instinct is to multiply the full axis lengths straight into the formula. But A=πab uses the semi-axes. Halve each axis: a=14/2=7 cm, b=10/2=5 cm

Then apply the formula: A=πab=A=3.14×7×5=109.9 cm².

Final answer: 109.9 cm².

Example 3

An ellipse has semi-axes a=9 m and b=5 m. Find its area in terms of π.

A=πab=A=π×9×5=45π m².

Final answer: 45π m² (about 141.4 m²).

Example 4

The area of an ellipse is 60π cm² and its semi-major axis is a=12 cm. Find the semi-minor axis b.

Start from the area formula and solve for b: A=πab 60π=π×12×b b=5 cm.

Final answer: b=5 cm.

Example 5

Show that the ellipse area formula gives the circle area when a=b=r.

A=πab=A=π×r×r=πr².

Final answer: the formula reduces to πr², the area of a circle.

Example 6

An elliptical garden bed is 8 m long and 6 m wide. A gardener needs 0.5 kg of seed per square meter. How much seed is needed? Use π≈3.14.

Halve the full axes: a=8/2=4 m, b=6/2=3 m.

Find the area: A=πab=A=3.14×4×3=37.68 m².

Multiply by the seed rate: Seed=37.68×0.5=18.84 kg.

Final answer: about 18.84 kg.

Why The Ellipse Formula Matters Beyond The Classroom

The ellipse is the shape of orbits. Planets trace ellipses around the Sun. Computing those swept areas starts with knowing the area of the whole ellipse. The area formula is vital in engineering designs where elliptical shapes appear.

Common Mistakes With The Area Of An Ellipse

Mistake 1: Using the full axes instead of the semi-axes

Don't do this: Substitute the full axis lengths directly as a and b.

The correct way: Halve each full axis.

Mistake 2: Squaring a semi-axis as if it were a circle

Don't do this: Write A=πa² or πb².

The correct way: An ellipse needs the product of two semi-axes.

Mistake 3: Forgetting the area units

Don't do this: Leave the answer as a bare number.

The correct way: Report the answer in square units.

Conclusion

• The area of an ellipse is A=πab.
• a and b are the semi-major and semi-minor axes.
• The formula comes from scaling a circle's area.
• Always halve full axis lengths to semi-axes before substituting.

Practice And A Next Step

Practice these problems to solidify your understanding. Want a live Bhanzu trainer to walk through more area-of-ellipse problems? Book a free demo class.