How to find the Area of an Ellipse?

Ellipse is a 2-dimensional shape. It is an integral part of the conic section. It is a curve on a plane in which the sum of the distance to its two focal points or foci is always a constant quantity from the specified point. Ellipse is from a family of circles with two focal points. The eccentricity value of the ellipse is always less than one and the general equation of the ellipse is given by

 (x²/a²) + (y²/b²) = 1

Where a represents the length of the semi-major axis and b represents the length of the semi minor-axis.

Parts of Ellipse

Some of the important parts of the ellipse are:

• Focus: Ellipse has two foci or focal points whose coordinates are F1(q, 0), and F2(-q, 0) And the distance between them is 2q
• Center: It is the midpoint of the line joining two foci.
• Major axis: It is the longest diameter of the ellipse. Or we can say that a line segment that connects the two farthest points present on the ellipse passes through the centre of the ellipse.
• Minor axis: It is the shortest diameter of the ellipse. Or we can say that it is a line segment that joins the two nearest points present on the ellipse and passes through the centre of the ellipse.
• Latus Rectum: It is a line that is drawn perpendicular to the transverse axis and passes through the foci.

Area of Ellipse

An ellipse is created by connecting all the points that are present at a constant distance from two fixed points in the plane. Here the two fixed points are known as foci. So the area of the ellipse is known as the total amount of area present in it. It is represented by cm², in², m², etc. The area of the ellipse can be calculated using the length of the semi-major axis and the length of the semi-minor axis. The area of the ellipse is given by-product of the length of the semi-major axis and minor axis with π (22/7). The formula is given by

Area of ellipse A = π × m × n

Where m is the length of the semi-major axis and n is the length of the semi-minor axis.

Proof of Area of Ellipse formula 

Let us consider A be an ellipse, 2m is the major axis and 2n be the minor axis. It is aligned in the cartesian plane in the reduced form so the equation of the ellipse is

x²/m² + y²/n² = 1

So, 

y = ±n/m√m² – x² ……(1)

Now assume a circle of radius m and whose center lies at the origin. So the equation of the circle is 

x² + y² = m²

So, y = ±√m² – x²  ……(2)

On comparing equations (1) and (2) we conclude that the ellipse is n/m times the circle.

So we can write the area of the two shapes as:

Area of ellipse = n/m x Area of the circle

So, 

Area of ellipse = n/m x (πm²) = πnm

Hence proved

Steps to find the area of Ellipse

Step 1: Find the length of the semi-major axis or the distance from the farthest point from the center that is m.

Step 2: Find the length of the semi-minor axis or the distance from the nearest point from the center that is n.

Step 3: Now put all these values in the area formula to calculate the area.

Step 4: Now apply the units of area. 

Example: A floor of a room is constructed in the shape of an ellipse whose major axis length is 15 cm and minor axis length is 11 cm. Now find the area of the ellipse.

Solution:

Given that 

Major axis length(m) = 15 cm

minor axis length(n) = 11 cm

Area of ellipse (A) = π x m x n

= 22/7 x 15 x 11 

= 22/7 x 165 

= 518.57 cm²

Finding the area of an ellipse using integration

We can also find the area of an ellipse using integration. As we know the general equation of the ellipse is 

x²/m² + y²/n² = 1

y = ±n/m√m² – x² ……(1)

As we know that the ellipse is divided into four quadrants so we find the area or one quadrant then multiple with 4 to get the area of the full ellipse

So, A = 4

= 4

= 4n/m

Now put x = m sint, dx = m cost.dt and t = 0 and t = π/2 in the above equation

A = 4n/m

= 4n

= 4mn

= 4mn

= 4mn

= 4mn(π/4)

= πmn

Sample Questions

Question 1: What is the area of an ellipse if the length of the semi-major axis and the semi-minor axis is 6cm, 3cm respectively.

Solution:

Given

Length of semi major axis (a) = 6cm

Length of semi minor axis (b) = 3cm

Area of ellipse= π x a x b

                      = (22/7) x 6 x 3

                      = 56.57 cm²

So area of given ellipse is 56.57 cm².

Question 2: What is the area of ellipse if the length of the semi-major axis and semi-minor axis is 10cm, 4.5cm respectively.

Solution:

Given

Length of semi major axis (a) = 10cm

Length of semi minor axis (b) = 4.5cm

Area of ellipse= π x a x b

                      = (22/7) x 10 x 4.5

                      = 141.42 cm²

So area of given ellipse is 141.42 cm².

Question 3: What is the area of the ellipse if the length of major axis and minor axis is 10cm, 5cm respectively.

Solution:

Given

Length of major axis = 10cm

Length of minor axis = 5cm

So we need to find semi major axis and semi minor axis length. 

Length of semi major axis (a) = Length of major axis/2

                                               = 10/2 = 5cm

Length of semi minor axis (b) = Length of minor axis/2

                                               = 5/2 = 2.5cm

Area of ellipse = π x a x b

                        = (22/7) x 5 x 2.5

                        = 39.25 cm²

So area of given ellipse is 39.25 cm².

Question 4: Find the area of ellipse if the length of major axis and minor axis is 12cm, 4cm respectively.

Solution:

Given

Length of major axis = 12cm

Length of minor axis = 4cm

So we need to find semi major axis and semi minor axis length.

Length of semi major axis (a) = Length of major axis/2

                                               = 12/2 = 6cm

Length of semi minor axis (b) = Length of minor axis/2

                                                = 4/2 = 2cm

Area of ellipse = π x a x b

                        = (22/7) x 6 x 2

                        = 37.71 cm²

So area of given ellipse is 37.71 cm².

Question 5: Find the area of ellipse if the length of major axis and semi minor axis is 8cm, 2.5cm respectively.

Solution:

Given

Length of major axis = 8cm

Length of semi minor axis (b) = 2.5cm

So we need to find length of semi major axis 

Length of semi major axis (a) = Length of major axis/2

                                               = 8/2 = 4cm

Area of ellipse = π x a x b

                        = (22/7) x 4 x 2.5

                        = 31.43 cm²

So area of given ellipse is 31.43 cm².