Perimeter of Ellipse

The perimeter of an ellipse is the length of its boundary. It is also known as the circumference of the ellipse (recall that perimeter is also known as the circumference for the curved shapes). Some examples of an ellipse in real life can be orbits of planets in the solar system, the 2D drawings of watermelon, egg, etc. Mathematically, an ellipse is defined as the set of all points where the distance of each point from two fixed points (which are known as foci) is always constant. Let us explore the perimeter of ellipse (or circumference of ellipse) in detail along with solved examples and practice questions.

The perimeter of ellipse is the length of the continuous line forming the boundary of the ellipse. Unfortunately, unlike other shapes, there is no formula to calculate the exact (or) accurate value of the perimeter of an ellipse, or any other figure of the conic section. But there are many approximation formulas to calculate the approximate value of perimeter such as:

• Approximation formulas
• Ramanujan Formulas
• Formulas using Infinite series
• Formulas using integration

Let us learn each of these formulas in the upcoming sections. For all these formulas, consider an ellipse of the semi-major axis of length 'a' and the semi-minor axis of length 'b' (i.e., a > b). i.e., we are going to find the perimeter of ellipse (x²/a²) + (y²/b²) = 1. Assume that its perimeter is P.

We have three approximation formulas to calculate the circumference of the ellipse. The first one is used when the shape of the ellipse is almost a circle (i.e., the values of 'a' and 'b' are approximately equal). The second (or) third formulas is used when one of the values of 'a' and 'b' is considerably large compared to the other value.

1. P ≈ π (a + b)
2. P ≈ π √[ 2 (a² + b²) ]
3. P ≈ π [ (3/2)(a+b) - √(ab) ]

When the values of 'a' and 'b' are not approximately equal, we can use either of the formulas 2 or 3 to find the perimeter. But formula 2 gives a larger value whereas formula 3 gives a smaller value than the actual value. So we can get a relatively close answer by taking the average of the values obtained from formula 2 and formula 3.

Ramanujan, one of the famous mathematicians, came up with some formulas that would give better approximations of the perimeter of ellipse. Ramanujan's formulas for finding the perimeter of ellipse became famous as they are simple and easy to use. Though these formulas do not give the exact perimeter, they can give reasonably a very close answer. The formulas are:

1. P ≈ π [ 3 (a + b) - √[(3a + b) (a + 3b) ]]
2. P ≈ π (a + b) [ 1 + (3h) / (10 + √(4 - 3h) ) ], where h = (a - b)²/(a + b)²

We have some formulas involving infinite series to calculate the perimeter of an ellipse. One of them includes 'e' which is called the eccentricity of ellipse and its value is e = √(a² - b²) / a. Famous formulas involving infinite series to find the perimeter of ellipse are:

1. \( p \approx 2a \pi\left[1-\left(\dfrac{1}{2}\right)^{2} e^{2} \right.\)
   \(\left.-\left(\dfrac{1 \cdot 3}{2 \cdot 4}\right)^{2} \dfrac{e^{4}}{3}-\left(\dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^{2} \dfrac{e^{6}}{5}-\ldots\right] \).
2. \(p \approx \pi(a+b)\left(1+\dfrac{1}{4} h+\dfrac{1}{64} h^{2}+\dfrac{1}{256} h^{3}+\ldots\right)\)
   Here, \(h=\dfrac{(a-b)^{2}}{(a+b)^{2}}\).

We have two formulas to find the circumference of an ellipse using integration. Both formulas are obtained by applying the two arc length formulas of integration (where one uses the direct function and the other uses the parametric form of ellipse).

1. Perimeter of ellipse using arc length is,
   P = 4 \(\int_{0}^{a} \sqrt{1+\dfrac{b^{2} x^{2}}{a^{2}\left(a^{2}-x^{2}\right)}} \, dx\)
2. Perimeter of ellipse using parametric equations is,
   P = \(4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \sin ^{2} \theta}\, \mathrm{d} \theta\)
   Here, e = eccentricity of the ellipse = [√(a² - b²) ] / a.

Let us derive each of these formulas.

Perimeter of Ellipse Using Arc Length

We know that the arc length of a function y = f(x) over an interval [a, b] is, \(\int_a^b \sqrt{1+\left(y'\right)^{2}} d x\). Solving (x²/a²) + (y²/b²) = 1 for y, we get y = (b/a) (√(a² - x²). Then its derivative is,

y' = (-bx)/(a√(a² - x²))

By applying the above arc length formula over the interval [0, a], we get the perimeter of the ellipse that is present in the first quadrant only. Thus,

Perimeter of the ellipse in the first quadrant = \(\int_{0}^{a} \sqrt{1+\dfrac{b^{2} x^{2}}{a^{2}\left(a^{2}-x^{2}\right)}} \, dx\)

To find the entire perimeter of ellipse (that is present in all four quadrants), we just have to multiply the above integral by 4. Thus,

Perimeter of ellipse = 4 \(\int_{0}^{a} \sqrt{1+\dfrac{b^{2} x^{2}}{a^{2}\left(a^{2}-x^{2}\right)}}\,dx\)

Perimeter of Ellipse Using Parametric Equations

We have the semi-major axis length of the ellipse to be 'a' and the semi-minor axis length of the ellipse to be 'b'. Thus, the parametric equations of the ellipse are,

x = a cos θ and y = b sin θ.

Using the arc length formula of parametric equations, we have the arc length of a function (x(θ), y(θ)) over the interval [a, b] is given by \(\int_a^b (x'(\theta))^2+(y'(\theta))^2 \, dt\). Applying this formula for the ellipse over the interval [0, π/2], we get its perimeter that is present only in the first quadrant. Thus, as explained in the previous section, the total perimeter is obtained by multiplying the resultant integral by 4.

\(\begin{align}
P &=4\int_{0}^{\pi / 2} \sqrt{(-a \sin \theta)^{2}+(b \cos \theta)^{2}} \,\mathrm{~d} \theta \\
&=4\int_{0}^{\pi / 2} \sqrt{a^{2}\left(1-\cos ^{2} \theta\right)+b^{2} \cos ^{2} \theta}\, \mathrm{d} \theta \\
&=4\int_{0}^{\pi / 2} \sqrt{a^{2}-\left(a^{2}-b^{2}\right) \cos ^{2} \theta} \mathrm{d}\, \theta \\
&=4a \int_{0}^{\pi / 2} \sqrt{1-\left(1-\dfrac{b^{2}}{a^{2}}\right) \cos ^{2} \theta}\, \mathrm{d} \theta \\
&=4a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \cos ^{2} \theta} \, \mathrm{d} \theta
\end{align}\)

Here, 'e' is the eccentricity of the ellipse and e = [√(a² - b²) ] / a.

We know that cos θ = sin (π/2 - θ). Now we use the substitution π/2 - θ = t. Then dθ = -dt. Also, the limits of the above integral would change to π/2 to 0. Then the above integral becomes,

\(P = 4a \int_{\pi / 2}^{0} \sqrt{1-e^{2} \sin^{2} t} \, (-dt)\)

Interchanging the limits would change the sign of the integral. Thus,

\(P = 4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \sin ^{2} t} \, dt\) (or)

\(P = 4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \sin ^{2} \theta} \mathrm{d} \theta\)

What Is Perimeter of Ellipse?

The perimeter of ellipse is the length of its boundary line. Unfortunately, there is no simple formula that can give the perimeter of ellipse right away but there are some formulas for approximation.

What Are Some Approximation Formulas of Perimeter of Ellipse?

Consider an ellipse of semi-major axis 'a' and semi-minor axis 'b'. Here are some formulas used to find the approximate value of perimeter of the ellipse.

• P ≈ π (a + b)
• P ≈ π √[ 2 (a² + b²) ]
• P ≈ π [ (3/2)(a+b) - √(ab) ]
• P ≈ π [ 3 (a + b) - √[(3a + b) (a + 3b) ]]
• P ≈ π (a + b) [ 1 + (3h) / (10 + √(4 - 3h) ) ], where h = (a - b)²/(a + b)²

What Are Ramanujan Formulas of Circumference of Ellipse?

There are two popular formulas by Ramanujan which are simple and which give a very close perimeter of an ellipse. They are:

• P ≈ π [ 3 (a + b) - √[(3a + b) (a + 3b) ]]
• P ≈ π (a + b) [ 1 + (3h) / (10 + √(4 - 3h) ) ], where h = (a - b)²/(a + b)²

Here, 'a' is the length of the semi-major axis and 'b' is the length of the semi-minor axis.

What Are Infinite Series Formulas of Perimeter of Ellipse?

There some formulas to find the perimeter of an ellipse using the infinite series. We may need to calculate more terms of the series to get a relatively close answer. Anyhow, these formulas would give just approximation but not the exact value of the perimeter.

• \( p \approx 2a \pi\left[1-\left(\dfrac{1}{2}\right)^{2} e^{2} \right.\)
  \(\left.-\left(\dfrac{1 \cdot 3}{2 \cdot 4}\right)^{2} \dfrac{e^{4}}{3}-\left(\dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^{2} \dfrac{e^{6}}{5}-\ldots\right] \).
• \(p=\pi(a+b)\left(1+\dfrac{1}{4} h+\dfrac{1}{64} h^{2}+\dfrac{1}{256} h^{3}+\ldots\right)\)
  Here, \(h=\dfrac{(a-b)^{2}}{(a+b)^{2}}\).

In both of these formulas, a = the length of the semi-major axis, b = the length of the semi-minor axis, and 'e' is called the eccentricity of ellipse and its value is e = √(a² - b²) / a

What Are the Formulas of the Perimeter of Ellipse Using Integration?

The perimeter of an ellipse can be found by applying the arc length formula to its equation in the first quadrant and then multiplying the resultant integral by 4. The perimeter of an ellipse x²/a² + y²/b² = 1 (where a > b) formulas using the integration are as follows:

• Perimeter of ellipse using arc length is,
  P = 4 \(\int_{0}^{a} \sqrt{1+\dfrac{b^{2} x^{2}}{a^{2}\left(a^{2}-x^{2}\right)}} \, dx\)
• Perimeter of ellipse using parametric equations is,
  P = \(4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \sin ^{2} \theta} \,\mathrm{d} \theta\)
  Here, e = eccentricity of the ellipse = [√(a² - b²) ] / a.

When Can We Use the Circumference of a Circle Formula To Find the Circumference of an Ellipse?

When the lengths of the semi-major axis and the semi-minor axis of an ellipse are equal, then the ellipse becomes a circle. In that case, the circumference of a circle formula can be used to find the circumference of an ellipse as well.