Area of an Ellipse

Have you ever observed :

  1. An egg
  2. A cricket ground
  3. An American football
  4. A badminton racket

Can you identify something common about them?

Yes! all are oval in shape or also known as elliptical.

elliptical shapes

Let us explore a bit more about this shape while discussing its area and its properties.

Lesson Plan


What Is an Ellipse? 

It is a shape obtained by connecting all the points which are at a constant distance from the two fixed points on the plane.

The fixed points are called foci. F1 and F2 are the two foci.

Locus of an ellipse

Ellipse is the locus of all the points, whose sum of distances from two fixed points on a plane is constant.

how to draw an ellipse

Definition of ellipse

For example, in the above image the points P1, Pare located in such a way that the sum of the distances of point P1 from the fixed point Fand F2 is equal to the sum of the distances of point P2 from the fixed point Fand F2.

That means if we join all such points  P1, P2,  P3 , etc; we will get a shape called ellipse.

\begin{equation}
\mathbf{P}_{1} \mathbf{F}_{1}+\mathbf{P}_{1} \mathbf{F}_{2}=\mathbf{P}_{2} \mathbf{F}_{1}+\mathbf{P}_{2} \mathbf{F}_{2}=\mathbf{P}_{3} \mathbf{F}_{1}+\mathbf{P}_{3} \mathbf{F}_{2}=\text{constant}
\end{equation}


How to Find the Area of an Ellipse?

 As an ellipse is not a perfect circle, the distance from the center of the ellipse to the points on the circumference is not constant.

So, an ellipse has two radii.

radii of an ellipse

semi-major and semi-minor axes

The longest chord in the ellipse is called the major axis of the ellipse.

The minor axis is the chord that is the perpendicular bisector to the major axis.

Area of an Ellipse

The formula to find the area of an ellipse is given by :

Area of ellipse = \(\pi \text{ a b}\)

where,

  • a = length of semi-major axis
  • b = length of semi-minor axis

Process of Finding the Area of an Ellipse

  1. Find the distance from the farthest point on the ellipse from the center ('a' i.e., length of the semi-major axis).
  2. Find the distance from the closest point on the ellipse from the center ('b' i.e., length of the semi-minor axis).
  3. Multiply the product of a and b with \(\pi\).
  4. Mention the area in square units.
 
important notes to remember
Important Notes
  • Ellipse is the locus of all the points, whose sum of distances from two fixed points on a plane is constant.
  • Area of ellipse = \(\pi\text{ a b}\)

Take the value of  \(\pi \) as 3.14 or \(\dfrac{22}{7}\)

Find the values of the semi-major axis (a) and the semi-minor axis (b) and then use the formula for the area of an ellipse.

Area of an Ellipse Calculator

Let us find the area of an ellipse with any given lengths of major and minor axes.

Type the length of the major axis and the minor axis respectively, for which you need to find the area in the boxes and press "calculate".  (\(\pi=3.1416\))


Solved Examples

Example 1

 

 

A whispering gallery is constructed in the shape of an ellipse. Find the area of the gallery, if the length of its major axis is 14 units and the length of its minor axis is 10 units.

whispering gallery in the shape of an ellipse

Solution

Given that,

  • length of major axis = 14m \(\implies\) length of semi-major axis is = \(\frac{1}{2}\times14=7\text{units}\)
  • length of minor axis = 12m \(\implies\) length of semi-minor axis is = \(\frac{1}{2}\times10=5\text{units}\)
  • a = 7 & b = 5

Area of the ellipse is \(\pi\text{ab}\).

\(\therefore\) Area of whispering gallery : \begin{eqnarray}\pi\text{ab} &= &\pi\times7\times5& \\&=&\dfrac{22}{7}\times7\times5\\&=& 22 \times5\\&=& 110\text{ square units}\end{eqnarray}

\(\therefore\)Area of whispering gallery is 110 sq.units
Example 2

 

 

Harry has to find the cross-sectional area of football with its semi-major axis as 5.5 inches and semi-minor axis as 3.5 inches.

Can you help him out?

area of ellipse - example football

Solution

American football is in the shape of an ellipse.

Length of semi-major axis = a = 5.5 inches

Length of semi-minor axis = b = 3.5 inches

Let us find out the area of football:\begin{eqnarray}
\pi\text{ab} &= &\pi\times5.5\times3.5& \\
 &=&\dfrac{22}{7}\times5.5\times3.5\\
&=& 11 \times5.5\\
&=& 60.5\text{ square inches}
\end{eqnarray}

\(\therefore\) The cross-sectional area of football is 60.5 sq. inches.
 

Interactive Questions on Area of an Ellipse

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of the area of an ellipse. The math journey around the area of ellipse starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs) 

1.What is the perimeter of an ellipse? 

The perimeter of an ellipse is in line with the perimeter of a circle.

We need to find req for the ellipse. req \(=\sqrt{\frac{a^{2}+b^{2}}{2}}\)

Perimeter of ellipse : P= \(=2\pi\sqrt{\frac{a^{2}+b^{2}}{2}}\)

2.What is the volume of an ellipse ?

Volume of ellipsoid: V= \(\pi\text{ abc }\)

3.What is A & B in Ellipse?

A represents the length of the major axis.

B represents the length of the minor axis.

4.How do you calculate the area of an oval?

Area of ellipse = \(\pi \text{ a b}\)

where \(a\), \(b\) are lengths of semi-major axis and semi-minor axis respectively.

5.What is a and b in an ellipse?

\(a\) is the length of the semi-major axis i.e., distance from center to end of the major axis.

\(b\) is the length of the semi-minor axis i.e., distance from center to end of the minor axis.

6.Is circle an ellipse?

The circle and ellipse belong to the family of conic sections.

The ellipse is a squashed circle.

7.What does an ellipse equation look like?

The equation of an ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

8.What is eccentricity of ellipse?

Eccentricity is denoted by \(e\).

It is the ratio of \(c\) and \(a\). i.e., \(e=\frac{c}{a}\)

\(c\) is the distance of focus from the center.

\(a\) is the distance of the vertex from the center.

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