Perimeter of Ellipse - Definition, Formula, and Examples

Have you ever thought about how to measure the outer edge of a flattened circle or the journey the Earth follows around the Sun? This form is called an ellipse in geometry. Finding the perimeter of ellipse is a little harder than finding the circumference of a circle because the shape's "stretch" changes. This article breaks down the idea and gives you simple formulas and examples to help you understand this mental maths topic.

Perimeter of Ellipse Definition

An ellipse is a closed circle that goes around two fixed points, which are called foci. The total distance from any point on the curve to these two foci stays the same. The perimeter of an ellipse is the total length of the line that goes around the object. You could also see this called the circumference of the ellipse in a lot of literature.

A circle can help you comprehend this better. The distance from the center of a circle to any point on the periphery is always the same. But an ellipse is longer than that. One side is long and the other is short. Because of this changing curvature, there is no one "simple" formula like $2\pi r$ that works for all ellipses. But mathematicians have come up with quite accurate estimates that we use in our research.

Parts for Ellipse Perimeter

You need to know the two primary dimensions of an ellipse before you can figure out its perimeter. These dimensions are essential for any formula you apply.

• Major Axis: This is the longest diameter of the ellipse, running through the centre and the foci.

• Semi-major Axis (a): This is half of the major axis (the distance from the centre to the furthest edge).

• Minor Axis: This is the ellipse's shortest diameter, and it is at a right angle to the major axis.

• The semi-minor axis (b): Half of the minor axis, which is the distance from the centre to the edge that is closest to it.

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Perimeter of Ellipse Formula

Finding the correct number for the perimeter is difficult. But we use common approximation formulas for fundamental mental maths.  The most popular and easiest way to make an estimate in class is:

P ≈ 2π √[(a² + b²) / 2]

In this formula:

• P is the perimeter in this formula.

• Pi (π) is around 3.14.

• a is the semi-major axis.

• B is the semi-minor axis.

Ramanujan’s Formula

The well-known Indian mathematician Srinivasa Ramanujan gave a more accurate version of the formula for people who wanted more accuracy:

P ≈ π [ 3(a + b) - √((3a + b)(a + 3b)) ]

This appears scarier, but it works the same way: you use the two radii (a and b) to locate the line that separates the two shapes.

Read More - Perimeter of Rectangle: Definition, Formulas, Solved Examples

Perimeter of Ellipse Types

We group them by how "eccentric" they are, or how flat they look, even though they are still ellipses. This changes how we look at the perimeter. 

1. High Eccentricity Ellipse: These are exceedingly long and narrow. There is a big difference between "a" and "b." The perimeter calculation here illustrates how far the form is from being a circle.

2. Low Eccentricity Ellipse: These look almost like circles. If 'a' and 'b' were perfectly equal, the ellipse would become a circle, and the perimeter of the ellipse would simply become $2\pi r$.

3. Horizontal Ellipse: The major axis lies along the x-axis.

4. Vertical Ellipse: The major axis lies along the y-axis.

Regardless of the orientation, the values of 'a' and 'b' remain the primary tools for measurement.

Perimeter of Ellipse Examples

Let's learn how to use the values in the formula to find the perimeter of an ellipse.

Example 1: Finding the Boundary

Question: Find the perimeter of an ellipse with a semi-major axis (a) of 5 cm and a semi-minor axis (b) of 3 cm. (Use 3.14 for π)

Solution:

1. Identify the values: a = 5, b = 3.

2. Use the formula: P ≈ 2π √[(a² + b²) / 2].

3. Square the values: a² = 25, b² = 9.

4. Add them: 25 + 9 = 34.

5. Divide by 2: 34 / 2 = 17.

6. Calculate the square root: √17 ≈ 4.12.

7. Final calculation: P ≈ 2 × 3.14 × 4.12 ≈ 25.87 cm.

Example 2: Mental Maths Challenge

Question: An ellipse has a major axis that is 20 cm long and a minor axis that is 12 cm long. Find the perimeter of the ellipse.

Solution:

1. Note: The question gives the full axes. We need the semi-axes.

2. a = 20 / 2 = 10 cm.

3. b = 12 / 2 = 6 cm.

4. Apply formula: P ≈ 2π √[(10² + 6²) / 2].

5. P ≈ 2π √[(100 + 36) / 2] = 2π √(136 / 2) = 2π √68.

6. √68 is roughly 8.25.

7. P ≈ 2 × 3.14 × 8.25 ≈ 51.81 cm.

Read More - Perimeter of a Triangle: Definition, Formulas, and Solved Examples

Important Tips for Perimeter of Ellipse

When students work on problems that involve the perimeter, they typically make tiny mistakes. Keep these things in mind:

• Always check the units: Ensure both 'a' and 'b' are in the same units (e.g., both cm or both m) before starting.

• Don't confuse Radius and Diameter: Often, a problem will give you the "major axis" length. Don't forget to divide by 2 to find 'a'.

• Order of Operations: Before you multiply by $2\pi$, always do the squares and the division inside the square root.

• Approximation: Since the ellipse doesn't have a "perfect" simple perimeter formula, your answer may be a little different depending on whether you choose 3.14 or 22/7 for Pi.

Why Learn about Perimeter of Ellipse?

You might wonder where this shape appears in real life. The perimeter of an ellipse is a vital calculation in astronomy. Most planetary orbits are not perfect circles; they are elliptical. Engineers also use these calculations when designing machine parts, arches in architecture, and even in the design of spectacles and mirrors. Understanding the ellipse perimeter helps you grasp how rounded shapes behave when they are stretched.

Summary Table: Circle vs Ellipse

This table highlights the key differences between a circle and an ellipse based on their features and formulas:

Solving Complex Perimeter of Ellipse Problems

As you progress in mental maths, you might encounter problems where the perimeter of an ellipse is given, and you must find a missing axis.

Example: If the perimeter is approximately 31.4 cm and the semi-major axis is 5 cm, what is the semi-minor axis?

You may find out how "flat" the shape is by changing the fundamental formula or trying different values until you find the right one. This kind of analysis is what helps pupils do well on math tests that are hard to pass.

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