Ellipse Calculator

How to Calculate Ellipse Properties

An ellipse is a closed curve on a plane, defined as the locus of points such that the sum of the distances from two fixed points (called foci) is constant. Understanding how to calculate various properties of an ellipse is crucial in geometry, astronomy, physics, and engineering. Here's a comprehensive guide on how to perform these calculations:

Ellipse Formulas

The key formulas for calculating ellipse properties are:

• Area (A) = \(\pi ab\)
• Circumference (C) ≈ \(\pi(3(a + b) - \sqrt{(3a + b)(a + 3b)})\)
• Foci Distance (c) = \(\sqrt{a^2 - b^2}\)
• Eccentricity (e) = \(\frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}\)
• Standard Form Equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Where:

• a is the length of the semi-major axis
• b is the length of the semi-minor axis
• c is the distance from the center to a focus
• π (pi) is approximately 3.14159

Calculation Steps

1. Identify the semi-major axis (a) and semi-minor axis (b) of the ellipse.
2. Calculate the area using A = πab.
3. Compute the circumference using the approximation formula C ≈ π(3(a + b) - √((3a + b)(a + 3b))).
4. Find the foci distance using c = √(a² - b²).
5. Calculate the eccentricity using e = c/a.
6. Determine the vertices (±a, 0) and co-vertices (0, ±b).
7. Write the standard form equation using a and b.

Example Calculation

Let's calculate the properties of an ellipse with a semi-major axis of 5 units and a semi-minor axis of 3 units:

1. Given: a = 5 units, b = 3 units
2. Area: A = π × 5 × 3 ≈ 47.12 square units
3. Circumference: C ≈ π(3(5 + 3) - √((3×5 + 3)(5 + 3×3))) ≈ 25.53 units
4. Foci Distance: c = √(5² - 3²) = 4 units
5. Eccentricity: e = 4/5 = 0.8
6. Vertices: (±5, 0), Co-vertices: (0, ±3)
7. Standard Form Equation: x²/25 + y²/9 = 1

Visual Representation

This diagram illustrates an ellipse with semi-major axis a = 5 and semi-minor axis b = 3. The x and y axes are shown, along with labels for the semi-major and semi-minor axes.