Ellipse perimeter calculator

What is the perimeter of an ellipse?

The perimeter of an ellipse is the length of its boundary. An ellipse is a geometric figure that generalizes a circle and is defined by two axes: the major axis (a) and the minor axis (b). Because of its shape, finding the perimeter of an ellipse is a more complex task than calculating the circumference of a circle. There is no single formula for exactly calculating the perimeter of an ellipse using elementary means, and for this reason, various approximate formulas are used.

One of the most well-known approximate formulas for calculating the perimeter of an ellipse is Ramanujan’s formula. The Indian mathematician Srinivasa Ramanujan proposed it in the early 20th century, and it has since gained wide application due to its accuracy in approximation. This formula illustrates how the ellipse can be considered in the context of geometric problems and everyday calculations.

History of Ramanujan’s formula

Ramanujan’s formula for the approximate calculation of the perimeter of an ellipse was proposed in the early 1900s. Srinivasa Ramanujan, a renowned Indian mathematician, developed this formula after numerous experiments and analysis of various approximation methods. His approach significantly simplified the calculation of the ellipse’s length with high accuracy without the need for complex mathematical tools.

The formula was published in one of his letters to G.H. Hardy, with whom Ramanujan had a professional collaboration. Even though the formula itself is approximate, it has proven its effectiveness in many practical applications, providing results with high precision.

Application of the formula and its accuracy

Although Ramanujan’s formula is not the only one available, its value lies in its simplicity and accessibility for calculations. It is used in various engineering and scientific tasks where knowledge of the perimeter of an ellipse is required, such as in architecture, mechanical engineering, and astronomy.

Ramanujan’s formula avoids the use of complex integrals and differential equations that would be required for exact computation of the ellipse’s curve length. However, for the most accurate calculations, more complex computational methods, such as numerical integration, may be used.

Formula

Ramanujan’s formula for approximately calculating the perimeter of an ellipse is as follows:

$$P \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]$$

where $a$ is the major semi-axis of the ellipse, and $b$ is the minor semi-axis of the ellipse.

This formula allows the perimeter to be calculated based on elementary arithmetic operations and the square root function.

Examples

Example 1
For an ellipse with a major semi-axis $a = 5$ and a minor semi-axis $b = 3$, the perimeter is approximately calculated as:

$$P \approx \pi \left[3(5+3) - \sqrt{(3 \times 5 + 3)(5 + 3 \times 3)}\right]$$

Calculating gives:

$$P \approx \pi \left[24 - \sqrt{(15+3)(5+9)}\right]$$ $$P \approx \pi \left[24 - \sqrt{18 \times 14}\right]$$ $$P \approx \pi \left[24 - 15.3\right] \approx 27.28$$

Example 2
Suppose $a = 10$ and $b = 7$, calculate the perimeter of the ellipse:

$$P \approx \pi \left[3(10+7) - \sqrt{(3 \times 10 + 7)(10 + 3 \times 7)}\right]$$ $$P \approx \pi \left[51 - \sqrt{(30+7)(10+21)}\right]$$ $$P \approx \pi \left[51 - \sqrt{37 \times 31}\right]$$ $$P \approx \pi \left[51 - 34.06\right] \approx 53.42$$

Notes

Ramanujan’s formula is sufficient for most practical needs, but its accuracy may decrease for very elongated ellipses, where the ratio between the major and minor axes differs significantly.

For greater flexibility and accuracy, especially for professional applications, more complex methods, such as numerical integration, may be used to account for the specifics of the mathematical model of the ellipse.

FAQ

Why is this formula approximate?

Ramanujan’s formula approximates the perimeter because the geometry of the ellipse does not have an exact elemental solution for the length of its periphery.

How to find the perimeter of an ellipse, given the semi-axis lengths are 2.5 and 3.5 cm?

Using Ramanujan’s formula:

$$P \approx \pi \left[3(2.5+3.5) - \sqrt{(3 \times 2.5 + 3.5)(2.5 + 3 \times 3.5)}\right]$$ $$P \approx \pi \left[18 - \sqrt{11 \times 13.5}\right]$$ $$P \approx \pi \left[18 - \sqrt{148.5}\right]$$ $$P \approx \pi \left[18 - 12.19\right] \approx 18.30$$

Are the values of the semi-axes of an ellipse sufficient to calculate its area?

Yes, the values of the semi-axes $a$ and $b$ are sufficient to compute the area of an ellipse. The formula for the area of an ellipse is: $A = \pi \cdot a \cdot b$. For convenience, you can use the ellipse area calculator.

What is the correct term: the circumference of an ellipse or the perimeter of an ellipse?

The correct term is “the perimeter of an ellipse.” The term “circumference” is traditionally used for concepts associated with circles, while an ellipse is not generally a circle.