Area of Ellipse

An ellipse is a geometric figure that resembles a stretched circle, characterized by its two axes: the semi-major axis (a) and the semi-minor axis (b). The area of an ellipse  is determined using the formula:

Area = πab

This formula shows that the area is proportional to both the lengths of the semi-major and semi-minor axes, making it crucial for solving problems in geometry and various real-world applications where elliptical shapes are involved.

1.0Ellipse Definition

An ellipse is the locus of points in the plane for which the sum of their distance from two fixed points (foci) is constant.

2.0Equations of the Ellipse

Central equations concern an ellipse whose axes meet at right angles at the origin of the orthogonal coordinate system.

Two cases occur: the one for the ellipse whose major axis coincides with the y-axis and one for major axis coincide with x-axis.

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

center at the origin major axis coinciding with the x-axis. 

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

center at the origin, major axis coinciding with the y-axis,

where 2a and 2b are the axes of the ellipse.

3.0What is the Area of Ellipse?

The area of an ellipse can be thought of as the amount of space enclosed within its boundary. Unlike a circle, which has a constant radius, an ellipse has two distinct axes: the major axis (the longest diameter) and the minor axis (the shortest diameter). The lengths of these axes are critical in determining the ellipse's area.

4.0Area of Ellipse 

Formula for Area of Ellipse is 

Area = πab

Where a and b are the semi-major axis semi-minor axis respectively.

5.0Proof of Formula of Area of Ellipse

The standard equation of an ellipse centered of the origin with semi-major axis a and semi-minor axis b is:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

In terms of y

$\mathrm{y}=\pm \mathrm{b}\sqrt{1-\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}}$

We can see that the ellipse is divided into 2 parts so the area can be found by integrating the top half of the ellipse and doubling the result.

Area = $2{\int }_{-a}^{a}b\sqrt{1-\frac{x^2}{a^2}}dx$

$=2b{\int }_{-a}^a\sqrt{1-\frac{x^2}{a^2}}dx$

Substitution x = asinθ, dx = acosθdθ.

$Area=2b{\int }_{-\pi /2}^{\pi /2}\sqrt{1-{\sin }^2\theta }\cdot a\cos \theta \mathrm{d}\theta$

$=2\mathrm{ab}{\int }_{-\pi /2}^{\pi /2}\sqrt{{\cos }^2\theta }\cdot \cos \theta \mathrm{d}\theta$

$=2\mathrm{ab}{\int }_{-\pi /2}^{\pi /2}{\cos }^2\theta \mathrm{d}\theta$

$=4\mathrm{ab}{\int }_0^{\pi /2}{\cos }^2\theta \mathrm{d}\theta$

$=4\mathrm{ab}{\left(\frac{\theta }{2}+\frac{\sin 2\theta }{4}\right)}_0^{\pi /2}$

$=4\mathrm{ab}\left(\frac{\pi }{4}\right)$

= π.a.b

6.0Semi Ellipse Area Formula 

A semi ellipse is half of an ellipse, either split along the major or minor axis. The area of a semi ellipse can be calculated by halving the area of the full ellipse:

Area of semi ellipse $=\frac{1}{2}\times \pi \cdot a\cdot b$

$=\frac{\pi }{2}\cdot a\cdot b.$

7.0Solved Examples on Area of Ellipse

Example 1: Consider an ellipse with a semi major axis a = 10 units and a semi minor axis b = 6 units. Find the area of the ellipse.

Solution:

Using the area formula for an ellipse.

Area = π × a × b.

Area = π × 10 × 6 

= 60 π square units

So, the area of the ellipse is 60π square units.

Example 2: A semi-ellipse has a semi-major axis of 12 units and a semi minor axis of 7 units. What is the area of the semi ellipse?

Solution:

Area of semi ellipse = $\frac{\pi }{2}\cdot a\cdot b$

$=\frac{\pi }{2}\times 12\times 7$

= 42π square units.

So, the area of the semi-ellipse is 42π square units.

Example 3: An ellipse has an area of 150π square units and a semi-minor axis of 5 units. Find the length of the semi-major axis.

Solution:

Using the area of ellipse formula.

Area = π × a × b.

Area = 150 π, b = 5 units, a =?

150 π = π × a × 5.

$a=\frac{150\pi }{5\pi }$

a = 30 units

So, the semi major axis a is 30 units.

Example 4: Find the area of an ellipse with a semi-major axis of 7 units and a semi-minor axis of 4 units.

Solution:

Using the area of ellipse formula.

Area = π × a × b.

= π × 7 × 4

= 28π

Thus, the area of the ellipse is 28π square units.

8.0Practice Questions on Area of Ellipse

1. Calculate the area of an ellipse with a = 6 and b = 2.

2. A semi-ellipse has a semi-major axis of 8 and a semi-minor axis of 5. Find its area.

3. If the area of an ellipse is 50π and the semi-major axis is 10, find the length of the semi-minor axis.

4. If the area of a semi ellipse is 24π square units and the semi-major axis is 8 units. Find the semi-minor axis.

9.0Sample Question on Area of an Ellipse

1. What is the formula for the area of an ellipse?

Ans: The area of an ellipse is calculated using the formula:

$\text{ Area }=\pi \times a\times b$, where a and b are the semi-major axis semi-minor axis.