Ellipse Equation Calculator

What is an Ellipse?

An ellipse is a two-dimensional shape where the sum of the distances from any point on the ellipse to two fixed points called foci is constant. The longest diameter is called the major axis, and the shortest diameter is called the minor axis.

Standard Form Equation of an Ellipse

The standard form equation of an ellipse with a horizontal major axis is given by:

\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]

where:

• \( (h, k) \) is the center of the ellipse
• \( a \) is the semi-major axis
• \( b \) is the semi-minor axis

Formulas for Ellipses

Let \(a\) be the semi-major axis, \(b\) be the semi-minor axis, \(c\) be the distance from the center to each focus, \(e\) be the eccentricity, and \(A\) be the area. Then:

1. Semi-major axis: \(a\)
2. Semi-minor axis: \(b\)
3. Distance to foci: \(c = \sqrt{a^2 - b^2}\)
4. Eccentricity: \(e = \frac{c}{a}\)
5. Area: \(A = \pi a b\)
6. Perimeter: \(C \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right]\)

Step-by-Step Calculations

Let's calculate these properties for an ellipse with semi-major axis \(a = 5\) units and semi-minor axis \(b = 3\) units:

1. Semi-major axis: \[a = 5 \text{ units}\]
2. Semi-minor axis: \[b = 3 \text{ units}\]
3. Distance to foci: \[c = \sqrt{a^2 - b^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ units}\]
4. Eccentricity: \[e = \frac{c}{a} = \frac{4}{5} = 0.8\]
5. Area: \[A = \pi a b = \pi \times 5 \times 3 = 15\pi \approx 47.12 \text{ square units}\]
6. Perimeter: \[C \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right] = \pi \left[ 3(5 + 3) - \sqrt{(3 \cdot 5 + 3)(5 + 3 \cdot 3)} \right] \approx 25.13 \text{ units}\]

Visual Representation

This diagram illustrates the ellipse with the calculated dimensions and properties.