Ellipse Perimeter Calculator

Created by Purnima Singh, PhD
Reviewed by Madhu Raman
Last updated: Apr 12, 2022

Omni's ellipse perimeter calculator allows you to calculate the perimeter of an ellipse using the Ramanujan approximation.

Continue reading to learn about the formula for the perimeter of an ellipse. You will also find an example of using this tool to determine an ellipse's area, perimeter ( or circumference), and eccentricity.

Let us first start by reviewing some of the basic concepts related to the ellipse.

The ellipse

As you know that an ellipse is a two-dimensional shape that looks like a squeezed circle (see figure 1). We can define an ellipse as all points in a plane where the sum of the distances from two fixed points, called foci of the ellipse (points F1 and F2 in figure 1), is constant.

Ellipse with semiaxes, foci, center and vertices marked.
Fig 1: Ellipse with semiaxes, foci, center and vertices marked

Every ellipse has two radii. The longer radius is called the semi-major axis (a in figure 1), and the shorter one is called the semi-minor axis (b in figure 1).

The center of the ellipse (point C in figure 1) lies at the point of intersection of the line joining its vertices (V1, V2, V3, V4).

We can express the standard form of the equation of an ellipse with center (c1, c2) as:

(xc1)2a2+(yc2)2b2=1\small \frac{(x - c_1)^2}{a^2} + \frac{(y - c_2)^2}{b^2} = 1

Perimeter of an ellipse

Before going any further, let us first try to understand what we mean by perimeter? The perimeter is a distance around the outlines or edge of any shape.

A practical example of measuring the perimeter of an ellipse would be the distance you cover when you walk along the edges of an elliptical-shaped field. Or the length of fence you need to surround it.

In the next section, we will see how to calculate the perimeter of an ellipse.

How to calculate the perimeter of an ellipse? - Ellipse perimeter formula

It may come as a surprise to you but calculating the exact perimeter of an ellipse is not that straightforward. In fact, a lot of effort has gone into determining the accurate approximation for the perimeter of an ellipse.

In this calculator, we will use the formula for ellipse perimeter proposed by the great mathematician Ramanujan:

pπ(a+b)(1+3h10+43h)\small p \approx \pi (a+b) \left ( 1 + \frac {3h}{10 + \sqrt{4-3h}} \right )

where:
aa - Semi-major axis of the ellipse; and
bb - Semi-minor axis of the ellipse.

To calculate the value of hh in the above equation, we will use the formula:

h=(ab)2(a+b)2\small h = \frac{(a-b)^2}{(a+b)^2}

If you want to calculate the area AA or the eccentricity ϵ\epsilon of the ellipse, you can use the following formulas:

A=πabϵ=a2b2a\small \begin{align*} A &= \pi a b\\ \epsilon & = \frac{\sqrt{a^2 - b^2}}{a} \end{align*}

How to use the ellipse perimeter calculator

Now let us see how you can use our ellipse perimeter calculator to determine the perimeter of an ellipse with a few clicks.

We will calculate the circumference of an ellipse whose semi-major axis (aa) is of length 5 units and semi-minor axis (bb) is of length 3 units.

  1. Enter the values a=5a =5 and b=3b=3 in the respective fields.

  2. The tool will calculate the ellipse's perimeter/circumference (25.527), area (47.12), and eccentricity (0.8).

  3. You can also use this ellipse perimeter calculator to find out one of the axes if the area and other axis are known.

Other ellipse calculators

We recommend checking out our range of calculators that deal with other metric properties of ellipse:

FAQ

How do I find the perimeter of an ellipse?

To find the perimeter of an ellipse, follow the given instructions:

  1. Determine the values of the semi-major axis a and semi-minor axis b.

  2. Calculate the value h = (a - b)²/(a + b)².

  3. Find out the perimeter by using the formula, perimeter = π × (a + b)[1 + (3 × h/(10 + √(4 - 3h)))].

Is circumference the same as perimeter?

Yes, the circumference, and perimeter are the same. Both circumference and perimeter refer to the distance or path surrounding any circular or elliptical shapes.

Purnima Singh, PhD
Ellipse with semiaxes, foci, center and vertices marked
(x - c₁)² / a² + (y - c₂)² / b² = 1
a
b
Perimeter
Area
Eccentricity
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