Perimeter of Ellipse Calculator
Enter the semi-major and semi-minor axes of an ellipse to find its perimeter.
Updated
Perimeter of Ellipse Calculator
Our Perimeter of Ellipse Calculator finds the distance around any ellipse or oval from its two semi-axes. It works out the exact value to full precision, shows Ramanujan's approximations beside it, and adds the axes, eccentricity, area and the perimeter in other units.
Quick answer
An ellipse with semi-axes a = 5 cm and b = 3 cm has a perimeter of P ≈ 25.53 cm.
- No simple exact formula: the true value is P = 4a × E(e), an elliptic integral
- Best quick formula: Ramanujan's P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))), with h = (a − b)² ÷ (a + b)²
- From the full axes: halve them first, so a 10 cm by 6 cm oval uses a = 5 and b = 3
- Circle check: when a = b = r, the perimeter is 2πr
What Is the Perimeter of an Ellipse?
The perimeter, also called the circumference, is the length of the curved edge of an ellipse: the distance an ant would walk to go once around it. It depends on two lengths measured from the centre:
- Semi-major axis (a): centre to edge along the longest width.
- Semi-minor axis (b): centre to edge along the shortest width.
Perimeter of an Ellipse Formula
The exact perimeter uses the eccentricity e and the complete elliptic integral of the second kind, E(e). The calculator evaluates it with the arithmetic-geometric mean, so the answer is exact to every digit shown.
P = 4a × E(e)
For hand work, Ramanujan's second formula is the one to use. It is almost exact for everyday shapes:
P ≈ π(a + b) ×
(1 + 3h ÷ (10 + √(4 - 3h)))
- Find h. With a = 5 and b = 3: (5 − 3)² ÷ (5 + 3)² = 4 ÷ 64 = 0.0625.
- Work out the bracket. 1 + 0.1875 ÷ (10 + √3.8125) ≈ 1.015687.
- Multiply by π(a + b). 8π × 1.015687 ≈ 25.527 cm.
Ramanujan's first formula, P ≈ π[3(a + b) − √((3a + b)(a + 3b))], is a little simpler and gives 25.52699 here.
Why There Is No Simple Formula
A circle bends by the same amount everywhere, so its perimeter is simply 2πr. An ellipse bends sharply at the ends of the long axis and gently along the sides, so adding up its length needs an integral that has no answer in terms of π, roots and powers alone.
What is easy is trapping it. The diamond through the four ends of the axes is shorter than the ellipse, and the box around it is longer:
So 4√(a² + b²) < P < 4(a + b). For a = 5 and b = 3 that is 23.32 < 25.53 < 32.
Ellipse Perimeter Examples
| a | b | Perimeter |
|---|---|---|
| 1 m | 1 m | ≈ 6.28 m (a circle) |
| 2 m | 1 m | ≈ 9.69 m |
| 4 cm | 2 cm | ≈ 19.38 cm |
| 5 cm | 3 cm | ≈ 25.53 cm |
| 6 m | 4 m | ≈ 31.73 m |
| 10 ft | 1 ft | ≈ 40.64 ft |
Doubling both semi-axes doubles the perimeter. Doubling only one does not: 2 by 1 gives 9.69, not twice the 6.28 of the 1 by 1 circle.
Approximations Compared
Short formulas are close for round ellipses and drift apart as the shape gets flatter:
| Method | 5 by 3 | 10 by 1 |
|---|---|---|
| Exact | 25.52700 | 40.63974 |
| Ramanujan II | 25.52700 | 40.63927 |
| Ramanujan I | 25.52699 | 40.60553 |
| 2π√((a² + b²) ÷ 2) | 25.90624 | 44.65042 |
| π(a + b) | 25.13274 | 34.55752 |
As b shrinks towards 0, the ellipse flattens into a line of length 2a, and the perimeter heads towards 4a (there and back).
Perimeter From the Full Axes
A tape measure usually gives the full length and width of an oval. Halve both, then use the formula:
b = 5 ÷ 2 = 2.5 ft
P ≈ 20.69 ft
That is the edging needed around an 8 ft by 5 ft oval rug. Buy a little extra for joins and waste.
Common Mistakes
- Using the full axes as a and b. The answer comes out 2 times too big.
- Using π(a + b). It is always too short, by about 15% for the 10 by 1 shape above.
- Using 2π√((a² + b²) ÷ 2) as exact. It is always too long.
- Mixing units. Convert both semi-axes to the same unit first (the calculator does this for you).
Key Terms
- Semi-major (a)
- Centre to edge along the long axis.
- Semi-minor (b)
- Centre to edge along the short axis.
- Eccentricity
- How stretched it is, 0 for a circle.
- E(e)
- Elliptic integral, π/2 for a circle.
How to Use the Perimeter of Ellipse Calculator
- Enter the semi-major axis. Type a, the distance from the centre to the edge along the long axis, such as 5. If you measured the full length, halve it first.
- Enter the semi-minor axis. Type b, the distance from the centre to the edge along the short axis, such as 3. Pick a unit for each if you want one (mm, cm, m, km, in, ft, yd or mi); mixed units are converted.
- Read and copy the result. The perimeter appears as you type, with a drawing, the working, Ramanujan's approximations, the full axes, the eccentricity, the area and the perimeter in other units. Press Copy to copy it, or Clear or Esc to start again.
Example: an oval running track area is 60 m long and 40 m wide. Enter a = 30 and b = 20 and pick m. The perimeter is about 158.654396 m and the area is about 1,884.96 m². For the area alone, use the Area of Ellipse Calculator.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you find the perimeter of an ellipse?
Measure the semi-axes a and b, then use P = 4a × E(e), or Ramanujan's close approximation P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))) with h = (a − b)² ÷ (a + b)². An ellipse with a = 5 cm and b = 3 cm has a perimeter of about 25.53 cm.
Is there an exact formula for the perimeter of an ellipse?
Not in terms of π, roots and powers alone. The exact perimeter is an elliptic integral, P = 4a × E(e), where e = √(1 − b² ÷ a²). It can be worked out to any accuracy, which is what this calculator does.
What is Ramanujan's formula for the perimeter of an ellipse?
His first formula is P ≈ π[3(a + b) − √((3a + b)(a + 3b))]. His second, more accurate one is P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))). For a = 5 and b = 3 both give about 25.527.
Is the circumference of an ellipse the same as its perimeter?
Yes. Both words mean the length of the curved edge, the distance once around the ellipse.
What is the perimeter of an ellipse with a = 10 and b = 6?
About 51.054. It is exactly twice the perimeter of the a = 5, b = 3 ellipse, because both semi-axes doubled.
Why is π(a + b) not the perimeter of an ellipse?
It only works for a circle. For any other ellipse it is too short: for a = 10 and b = 1 it gives 34.56 instead of 40.64.
Is the Perimeter of Ellipse Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Perimeter of Ellipse Calculator to find the distance around any ellipse or oval from its two semi-axes, in any units.