Area of Ellipse

Introduction

Area of an ellipse is the area or region covered by the ellipse in two dimensions. An ellipse appears to be a regular oval shape, formed when a cone is snipped off by an oblique plane in such a way that a closed curve that does not bisect the base is produced.

An ellipse is the locus of all the points in a plane whose distances from two fixed points in the plane are constant. The fixed points, which are surrounded by the curve, are known as foci (singular focus). The constant ratio is the eccentricity of the ellipse and the fixed line is the directrix. Eccentricity is an ellipse factor that shows elongation and is denoted by the letter 'e.'

Many ellipse-shaped shapes can be found in our daily lives, such as a cricket ground, a badminton racket, planet orbits, and so on. In this tutorial, we will discuss the area of the ellipse.

Ellipse

Ellipse is a conic section component with features similar to a circle. An ellipse, unlike a circle, has an oval shape. An ellipse has an eccentricity less than one and represents the locus of locations whose distances from the ellipse's two foci are a constant value.

Characteristics of Ellipse

• There are several characteristics that help identify an ellipse from other similar shapes. These elliptical attributes are as follows −

• An ellipse is formed when a plane intersects a cone at its base angle.

• Every ellipse has two foci or focal points. The sum of the distances between any two locations on the ellipse is a constant value.

• All ellipses have a center as well as a major and a minor axis.

• All ellipses have eccentricity values of less than one.

Area of an Ellipse

The area of an ellipse is the two-dimensional area or region covered by the ellipse. Ellipse is a 2-D shape formed by connecting all the points on the plane that are a constant distance apart. The fixed points are known as ellipse foci.

The major axis of the ellipse is the longest chord in the ellipse. The chord that is the perpendicular bisector to the major axis is known as the minor axis.

Given the lengths of the major and minor axes, the area of an ellipse formula can be calculated using a general formula. The formula for calculating the area of an ellipse is as follows −

$$\mathrm{Area\: of\: ellipse = π a b}$$

a = semi-major axis length

b = semi-minor axis length

Length of the latus rectum of an ellipse

The latus rectum is a line that passes through the focal point of the ellipse and is perpendicular to the horizontal axis. The ellipse has two latus rectums because it has two focal points.

This is a graph of an ellipse whose equation is

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

where 2a is the length of the major axis and 2b is the length of the minor axis.

CD and AB are the latus rectum of the ellipse and the coordinate of A is $\mathrm{(ae,\frac{b^2}{a})}$, the coordinate of B is $\mathrm{(ae,-\frac{b^2}{a})}$, the coordinate of C is $\mathrm{(-ae,\frac{b^2}{a})}$, and the coordinate of D is $\mathrm{(-ae,-\frac{b^2}{a})}$ where e is the eccentricity of an ellipse.

Hence the length of the latus rectum of the ellipse is

$$\mathrm{|\frac{b^2}{a}|+|\frac{b^2}{a}|=\frac{2b^2}{a}}$$

Note:

• The locus of all points whose sum of distances from two fixed points on a plane is constant is an ellipse.

• Area of ellipse = π a b

• Consider the value of π as 3.14 or 22/7

• Determine the values of the semi-major axis (a) and the semi-minor axis (b), and then apply the ellipse area formula.

Solved Examples

1) Find the area of the ellipse whose length of the major and minor axis is 2 and 4, respectively.

Answer: We know that the formula for calculating the area of the ellipse is Area of ellipse = π a b if the length of the minor and major axis is given.

$$\mathrm{Area\: of\: ellipse = π a b}$$

$$\mathrm{Area\: of\: ellipse = π ×2×4}$$

$$\mathrm{Area\: of\: ellipse = 8π }$$

2) Find the area of the ellipse whose length of the major and minor axis is 8 and 3, respectively.

Answer: We know that the formula for calculating the area of the ellipse is Area of ellipse = π a b if the length of the minor and major axis is given.

$$\mathrm{Area\: of\: ellipse = π a b}$$

$$\mathrm{Area\: of\: ellipse = π ×8×3}$$

$$\mathrm{Area\: of\: ellipse = 24π }$$

3) If the equation of the ellipse is $\mathrm{\frac{x^2}{16}+\frac{y^2}{144}=1}$ then find the length of the latus rectum.

Answer: Given the equation of the ellipse is $\mathrm{\frac{x^2}{16}+\frac{y^2}{144}=1}$

$$\mathrm{\Rightarrow \frac{x^2}{4^2}+\frac{y^2}{12^2}=1}$$

Now compare this equation with $\mathrm{\Rightarrow \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$

$$\mathrm{\Rightarrow a=4 and b=12}$$

We know that length of the latus rectum is $\mathrm{\frac{2b^2}{a}}$

Hence the length of the latus rectum = $\mathrm{\frac{2(12)^2}{4}=72}$

4) If the equation of the ellipse is $\mathrm{\frac{x^2}{16}+\frac{y^2}{121}=1}$ then find the length of the latus rectum.

Answer: The given equation is $\mathrm{\frac{x^2}{16}+\frac{y^2}{121}=1}$

$$\mathrm{\Rightarrow \frac{x^2}{4^2} +\frac{y^2}{11^2} =1}$$

Now compare this equation with $\mathrm{\frac{x^2}{a^2}+\frac{y^2}{11^2}=1}$

$$\mathrm{\Rightarrow a=4\: and\: b=11}$$

We know that length of the Latus rectum = $\mathrm{\frac{2b^2}{a}}$

Hence the length of the latus rectum of an ellipse $\mathrm{\frac{2(11)^2}{4}=\frac{121}{2}}$

5) What is the area of the given ellipse $\mathrm{\frac{x^2}{16}+\frac{y^2}{25}=1}$?

Answer: It is given that the equation of the ellipse is $\mathrm{\frac{x^2}{16}+\frac{y^2}{25}=1}$

$$\mathrm{\Rightarrow \frac{x^2}{4^2}+\frac{y^2}{5^2}=1}$$

Now compare this equation with $\mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$

$$\mathrm{\Rightarrow a=4\: and\: b=5}$$

Now apply the formula

$$\mathrm{Area\: of\: ellipse = π a b}$$

$$\mathrm{Area\: of\: ellipse = π ×4×5}$$

$$\mathrm{Area\: of\: ellipse = 20π }$$

Conclusion

The amount of region present inside an ellipse is defined as its area. Alternatively, an ellipse's area is the total number of unit squares that can fit inside it.

The area of an ellipse is the product of π and the lengths of the semi-major and semi-minor axes.

FAQs

1. What do you mean by the area of the ellipse?

The amount of region present inside an ellipse is defined as its area.

2. What is the length of the latus rectum of the ellipse?

The length of the latus rectum of an ellipse is $\mathrm{\frac{2b^2}{a}}$.

3. What do you mean by the Latus rectum?

The latus rectum is a line that passes through the focal point of the conic

section and is parallel to the direction of the conic section.

4. What is an ellipse?

An ellipse, unlike a circle, has an oval shape. An ellipse has an eccentricity less than one and represents the locus of locations whose distances from the ellipse's two foci are a constant value.

5. What is the formula for the area of the ellipse?

The formula for calculating the area of an ellipse is as follows −

$$\mathrm{Area\: of\: ellipse = π a b}$$

a = semi-major axis length

b = semi-minor axis length