Area of an Ellipse Calculator
Calculate the area and approximate circumference of an ellipse from its semi-major and semi-minor axes. See also Area of Circle Calculator and Area of Sector Calculator.
How to Calculate the Area of an Ellipse
An ellipse is an oval shape defined by two axes: the semi-major axis (a) — the longer radius — and the semi-minor axis (b) — the shorter radius. To find the area, multiply π by both axes: A = π × a × b. When a = b, the ellipse becomes a circle and the formula reduces to A = πr². The circumference of an ellipse has no simple exact formula, so this calculator uses Ramanujan's approximation, which is accurate to within 0.01% for most ellipses.
Ellipse Area Formula
A = π × a × b
Circumference ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h)))
where h = ((a − b)/(a + b))²
Example
Find the area of an ellipse with a = 8 and b = 5:
A = π × a × b
A = π × 8 × 5
A = 40π
A ≈ 125.6637 square units
Ellipse Area Reference Table
| a | b | Area | Circumference |
|---|---|---|---|
| 2 | 1 | 6.2832 | 9.6884 |
| 3 | 2 | 18.8496 | 15.8654 |
| 4 | 3 | 37.6991 | 22.1035 |
| 5 | 3 | 47.1239 | 25.5270 |
| 6 | 4 | 75.3982 | 31.7309 |
| 8 | 5 | 125.6637 | 41.3863 |
| 10 | 6 | 188.4956 | 51.0540 |
| 10 | 8 | 251.3274 | 56.7233 |
| 12 | 8 | 301.5929 | 63.4618 |
| 15 | 10 | 471.2389 | 79.3272 |
| 20 | 12 | 753.9822 | 102.1080 |
| 25 | 15 | 1178.0972 | 127.6350 |
When to Use the Ellipse Area Formula
Use the ellipse area formula whenever you encounter an oval shape — any shape that looks like a stretched or compressed circle. Common situations include:
- Oval swimming pools — to calculate water surface area, pool covers, or chemical treatment requirements
- Racetrack inner areas — the infield of an oval track is approximately elliptical for landscaping or drainage planning
- Elliptical garden beds — for calculating soil, mulch, or planting area within an oval flower bed
- Oval tabletops and mirrors — for ordering custom glass or material
- Planetary orbits — in astronomy, orbits are ellipses and the area relates to orbital period via Kepler's laws
If the shape is perfectly round (both axes equal), it's a circle — use A = πr² instead. An ellipse has two different radii: the semi-major axis (longer) and semi-minor axis (shorter).
Solved Examples
Example 1: Oval Swimming Pool Cover
An oval swimming pool is 10 meters long and 6 meters wide. How much material is needed for a pool cover?
Semi-major axis a = 10/2 = 5 m
Semi-minor axis b = 6/2 = 3 m
A = π × a × b
A = 3.14159 × 5 × 3
A ≈ 47.12 square meters of cover material
Example 2: Racetrack Infield Area
A running track's inner boundary is approximately an ellipse with semi-major axis 45 m and semi-minor axis 30 m. What is the infield area for turf installation?
a = 45 m, b = 30 m
A = π × a × b
A = 3.14159 × 45 × 30
A ≈ 4,241.15 square meters of turf
Example 3: Elliptical Flower Bed
A garden has an elliptical flower bed measuring 3.6 m long and 2.4 m wide. How much mulch (in m²) is needed to cover it?
a = 3.6/2 = 1.8 m, b = 2.4/2 = 1.2 m
A = π × 1.8 × 1.2
A = 3.14159 × 2.16
A ≈ 6.79 square meters of mulch
Example 4: Oval Mirror Frame
An oval bathroom mirror has a semi-major axis of 40 cm and semi-minor axis of 30 cm. What is the mirror area?
a = 40 cm, b = 30 cm
A = π × 40 × 30
A = 3.14159 × 1,200
A ≈ 3,769.91 cm² (about 0.377 m²)
Common Mistakes When Calculating Ellipse Area
Using the full length/width instead of semi-axes
The formula uses semi-axes (half the total length and width). If a pool is 10 m × 6 m, the semi-axes are 5 and 3, not 10 and 6. Using full axes gives 4× the correct answer.
Confusing with circle formula
A circle uses A = πr² (one radius squared). An ellipse uses A = πab (two different radii multiplied). Squaring one axis and ignoring the other gives the wrong result.
Assuming the circumference is 2π × average radius
Unlike area, the circumference of an ellipse has no simple formula. Using C = 2π × (a+b)/2 can have significant error. Use Ramanujan's approximation for accuracy.
Key Takeaways
- Ellipse area: A = π × a × b, where a = semi-major axis and b = semi-minor axis.
- If given total length and width, divide each by 2 to get the semi-axes first.
- When a = b, the ellipse is a circle and the formula simplifies to A = πr².
- An ellipse always has less area than a circle with radius equal to the semi-major axis (because b < a).
- The circumference requires Ramanujan's approximation: C ≈ π(a+b)(1 + 3h/(10+√(4-3h))) where h = ((a-b)/(a+b))².
- Eccentricity e = √(1 - b²/a²) measures how elongated the ellipse is (0 = circle, close to 1 = very elongated).
Frequently Asked Questions
What is the difference between an ellipse and a circle?
A circle is a special case of an ellipse where both axes are equal (a = b = r). An ellipse has two different axis lengths, making it oval-shaped.
What is the eccentricity of an ellipse?
Eccentricity (e) measures how elongated an ellipse is: e = √(1 − b²/a²). A circle has e = 0, and as the ellipse gets more elongated, e approaches 1.
Why is the circumference approximate?
Unlike a circle, there is no simple closed-form formula for the exact circumference of an ellipse. It requires an elliptic integral. Ramanujan's approximation used here is extremely accurate for practical purposes.
What are the foci of an ellipse?
The foci are two special points inside the ellipse. The sum of distances from any point on the ellipse to both foci is constant and equals 2a. The distance from center to each focus is c = √(a² − b²).