Area of an Ellipse
Reference for ellipse area A = π × a × b from semi-major and semi-minor axes, plus perimeter approximation using the Ramanujan formula and eccentricity.
The Formula
A = π × a × b
This formula calculates the area of an ellipse (an oval shape).
A circle is a special case of an ellipse where a = b = r.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the ellipse |
| π | Pi, approximately 3.14159 |
| a | Semi-major axis (half the longest diameter) |
| b | Semi-minor axis (half the shortest diameter) |
Example 1
Find the area of an ellipse with semi-major axis 8 cm and semi-minor axis 5 cm
a = 8, b = 5
A = π × 8 × 5 = π × 40
A ≈ 125.66 cm²
Example 2
An oval-shaped pool is 10 m long and 6 m wide. What is its surface area?
The full length is 10 m, so a = 10 / 2 = 5 m
The full width is 6 m, so b = 6 / 2 = 3 m
A = π × 5 × 3 = π × 15
A ≈ 47.12 m²
When to Use It
Use the area of an ellipse formula when:
- Calculating the area of oval-shaped objects (pools, tracks, mirrors)
- Working with planetary orbits (which are elliptical)
- Designing elliptical features in architecture or engineering
- Any time you have an oval where you know the two axis lengths
Key Notes
- Formula: A = π × a × b: a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). When a = b = r, the ellipse becomes a circle with A = πr².
- Area is simple; perimeter is not: The area of an ellipse has an exact closed-form formula, but the perimeter does not. The perimeter requires an elliptic integral, often approximated by Ramanujan's formula: P ≈ π(3(a+b) − √((3a+b)(a+3b))).
- Eccentricity describes the shape: Eccentricity e = √(1 − b²/a²). A circle has e = 0. As e approaches 1, the ellipse becomes more elongated. Earth's orbit around the Sun has e ≈ 0.0167 (nearly circular).
- Foci of an ellipse: An ellipse has two foci at distance c = √(a² − b²) from the center. For any point on the ellipse, the sum of its distances to the two foci equals 2a — the defining property of an ellipse.
- Applications in astronomy: Planetary orbits are ellipses (Kepler's first law). The Sun sits at one focus, not the center. This means a planet is closer to the Sun at perihelion and farther at aphelion.