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Ellipse Perimeter Calculator

Introduction

Welcome to our Ellipse Perimeter Calculator, which will help you approximate with high level of accuracy the perimeter of the ellipse using multiple methods. An ellipse is a closed two-dimensional figure which looks like a squished circle. The ellipse has two axes, major and minor axes. The longest diameter of an ellipse is called the major axis. The shortest diameter of an ellipse is called the minor axis.

Two focus points, F1 and F2, lie on the major axis and the locus of the points such that the sum of the distances from F1 and F2 to those points will form the ellipse. The sum of the distances from the points on the ellipse will be equal to the length of the ellipse’s major axis.

The perimeter of the ellipse is the total distance around the shape. We generally measure it in units like centimeters, inches, meters, feet, etc.

How to use the Ellipse Perimeter Calculator?

Using the Ellipse Perimeter calculator, you can calculate the perimeter of the ellipse by using one of the following methods.

Perimeter of Ellipse Approximation 1

We can use the following formula to approximate the perimeter of the ellipse.

The variables in the Ellipse Perimeter Calculator include

Semi-Major Axis (a) The distance from the center to the farthest point on the ellipse.

Semi-Minor Axis Length (b) The distance from the center to the shortest point on the ellipse.

Perimeter of the Ellipse (P) The total distance around the ellipse is calculated using the following formula

P≈π×(a+b)P \approx \pi \times (a +b)

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Approximation 2

We can use the following formula to approximate the perimeter of the ellipse.

The variables in the Ellipse Perimeter Calculator include

Semi-Major Axis (a) The distance from the center to the farthest point on the ellipse.

Semi-Minor Axis Length (b) The distance from the center to the shortest point on the ellipse.

Perimeter of the Ellipse (P) The total distance around the ellipse is calculated using the following formula

P≈π2×(a2+b2)P \approx \pi \sqrt{2 \times (a^2 +b^2)}

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Approximation 3

We can use the following formula to approximate the perimeter of the ellipse.

The variables in the Ellipse Perimeter Calculator include

Semi-Major Axis (a) The distance from the center to the farthest point on the ellipse.

Semi-Minor Axis Length (b) The distance from the center to the shortest point on the ellipse.

Perimeter of the Ellipse (P) The total distance around the ellipse is calculated using the following formula

P≈π[32×(a+b)−a×b]P \approx \pi [\normalsize \dfrac{3}{2} \times (a + b) - \sqrt{a \times b}]

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Ramanujan Formula 1

We can use the following formula that was developed by Indian Mathematician Ramanujan. This is a better approximation to calculate the perimeter of the ellipse.

The variables in the Ellipse Perimeter Calculator include

Semi-Major Axis (a) The distance from the center to the farthest point on the ellipse.

Semi-Minor Axis Length (b) The distance from the center to the shortest point on the ellipse.

Perimeter of the Ellipse (P) The total distance around the ellipse is calculated using the following formula

P≈π[3(a+b)−(3a+b)×(a+3b)]P \approx \pi [3(a+b) - \sqrt{(3a +b)\times(a + 3b)}]

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Ramanujan Formula 2

This is another formula developed by Indian Mathematician Ramanujan to approximate the Perimeter of the Ellipse.

The variables in the Ellipse Perimeter Calculator include

Semi-Major Axis (a) The distance from the center to the farthest point on the ellipse.

Semi-Minor Axis Length (b) The distance from the center to the shortest point on the ellipse.

Perimeter of the Ellipse (P) We can calculate the total distance around the ellipse using the following formula

First, we have to calculate h using the following formula

h=(a−b)2(a+b)2h = \normalsize \dfrac{(a - b)^2}{(a + b)^2}

Then, we substitute the value of h to approximate the value of the perimeter of the ellipse

P≈π(a+b)×(1+3×h10+(4−3h))P \approx \pi (a + b) \times \bigg(1 + \normalsize \dfrac{3 \times h}{10 + \sqrt{(4 - 3h)}} \bigg)

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

h = The value calculated using the formula /frac(a−b)2(a+b)2/frac{(a - b)^2}{(a + b)^2}

What is an Ellipse?

An ellipse is a closed two-dimensional shape formed by connecting the points, where the sum of the distances from the focus points F1 and F2 is constant, and both the focus points (together called the foci) lie on the major axis. The longest diameter of the ellipse is called the major axis. The shorter diameter of the ellipse is called the minor axis.

The ellipse’s Eccentricity is defined as the ratio of the distance from the center to a focal point and the distance from that focus point to the co-vertex (also known as the length of the semi-major axis).

Eccentricity=ca\text{Eccentricity} = \normalsize \dfrac{c}{a}

Where,

c = The distance from the center to one of the foci

a = The distance from that focus point to the co-vertex.

The Eccentricity of an ellipse will always be less than one. If the eccentricity is 1, the ellipse will be squished entirely into a single line. If the eccentricity is 0, then the ellipse will become a circle.

Properties of an Ellipse

  1. An ellipse will have two focal points, F1 and F2, called foci.

  2. The sum of distances from the foci to any point on the ellipse will be a constant.

  3. The ellipse has two axes, major and minor axes.

  4. The ellipse will have an eccentricity of less than 1.

How to calculate the Perimeter of the Ellipse?

There are many ways to calculate the perimeter of the ellipse. There are approximations and exact formulas. Let’s look at some of these methods here.

Perimeter of Ellipse Approximation 1

P≈π×(a+b)P \approx \pi \times (a +b)

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Approximation 2

P≈π2×(a2+b2)P \approx \pi \sqrt{2 \times (a^2 +b^2)}

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Approximation 3

P≈π[32×(a+b)−a×b]P \approx \pi [\normalsize \dfrac{3}{2} \times (a + b) - \sqrt{a \times b}]

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Ramanujan Formula 1

P≈π[3(a+b)−(3a+b)×(a+3b)]P \approx \pi [3(a+b) - \sqrt{(3a +b)\times(a + 3b)}]

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Perimeter of Ellipse Ramanujan Formula 2

First, we have to calculate h using the following formula.

h=(a−b)2(a+b)2h = \normalsize \dfrac{(a - b)^2}{(a + b)^2}

Then we substitute the value of h to approximate the value of the perimeter of the ellipse

P≈π(a+b)×(1+3×h10+(4−3h))P \approx \pi (a + b) \times \bigg(1 + \normalsize \dfrac{3 \times h}{10 + \sqrt{(4 - 3h)}} \bigg)

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

h = The value calculated using the formula (a – b)^2/(a + b)^2

Perimeter of Ellipse using Infinite Series

There are also formulas to calculate the exact value of the perimeter of the ellipse. These formulas use an infinite series to calculate the perimeter of the ellipse, and this calculator does not support those functions yet. But still, let’s look at these formulae to understand how to calculate the perimeter of the ellipse in a better way.

Perimeter of Ellipse using Infinite Series 1

We can calculate the perimeter of the ellipse using the following formula, which consists of an infinite series.

P=2aπ(1−∑i=1∞(2i)!2(2i⋅i!)4⋅e2i2i−1)P = 2a\pi \bigg(1 - \sum_{i=1}^\infin \normalsize \dfrac{(2i)!^2}{(2^i\cdot i!)^4}\cdot\dfrac{e^{2i}}{2i-1} \bigg)
P=2aπ[1−(12)2e2−(1×32×4)2e43−(1×3×52×4×6)2e65  −...]P = 2 a \pi \normalsize \bigg [1 - \bigg(\dfrac{1}{2}\bigg)^2 e^2 - \bigg(\dfrac{1 \times 3}{2 \times 4}\bigg)^2 \dfrac{e^4}{3} - \bigg(\dfrac{1 \times 3 \times 5}{2 \times 4 \times 6}\bigg)^2 \dfrac{e^6}{5} \; - ... \bigg]

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

e = Eccentricity of the Ellipse

e=a2−b2ae = \dfrac{\sqrt{a^2 - b^2}}{a}

Perimeter of Ellipse using Infinite Series 2

First, we have to calculate h using the following formula.

h=(a−b)2(a+b)2h = \normalsize \dfrac{(a - b)^2}{(a + b)^2}

We can calculate the perimeter of the ellipse using the following formula, which consists of an infinite series.

P=π(a+b)∑n=0∞(0.5n)2hnP = \pi (a+b) \sum_{n=0}^\infin \binom{0.5}n^2 h^n
(0.5n)\binom{0.5}{n}

is the Binomial Coefficient with half-integer factorials.

P=π(a+b)(1+14h+164h2+1256h3+...)P = \pi (a+b) \bigg(1 + \dfrac{1}{4}h + \dfrac{1}{64}h^2 + \dfrac{1}{256}h^3 + ... \bigg)

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

h = The value calculated using the formula (a – b)^2/(a + b)^2

Perimeter of Ellipse Using Integrals

Arc Length Equation

We can find the perimeter of the ellipse using an integration of the equations specified below.

P=4∫0a1+b2x2a2(a2−x2dxP = 4 \int_0^a \sqrt{1 + \dfrac{b^2x^2}{a^2(a^2 - x^2}} \text{dx}

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

b = Half the length of the minor axis or the distance from the center to the shortest point on the ellipse

Parametric Equation

P=4a∫0π21−e2  sin2θ  dθP = 4a \int_0^{\small \frac{\pi}{2}} \sqrt{1 - e^2 \;\text{sin}^2\theta} \; \text{d}\theta

Where,

a = Half the length of the major axis or the distance from the center to the farthest point on the ellipse

e = Eccentricity of the Ellipse

e=a2−b2ae = \dfrac{\sqrt{a^2 - b^2}}{a}
Author

hexacalculator design team

Our team blends expertise in mathematics, finance, engineering, physics, and statistics to create advanced, user-friendly calculators. We ensure accuracy, robustness, and simplicity, catering to professionals, students, and enthusiasts. Our diverse skills make complex calculations accessible and reliable for all users.

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