Area of ellipse:
Like a circle or any other closed figure, an ellipse is also a closed figure. Therefore it is possible to find its area. Area of an ellipse can be found using the values of a and b. But what are these a and b? That can be understood using the general equation of an ellipse.
General equation of an ellipse:
[(x-h)/a]^2 + [(y-k)/b]^2, where a is the semi major axis and b is the semi minor axis. (h,k) is the centre of the ellipse.
Area of ellipse formula:
From the above general equation we know that the semi major axis is a and the semi minor axis is b. The formula for the area of an ellipse can now be written as follows:
Area = A = pi * ab. The semi major axis can also be termed as the distance between the centre and the vertex, which is ‘a’ here. The semi minor axis can also be terms as the distance between the centre and the co-vertex, which is ‘b’ here. See the picture below:
Implicit equation of an ellipse:
Another method for finding the area of an ellipse is using its implicit equation. The general form of the implicit equation of an ellipse is:
Px^2 + Qxy + Ry^2 = 1
To calculate area of an ellipse if the equation of the ellipse is given in this form, we use the following formula:
Area = A = 2 pi/√(4PR – Q^2)
Let us now try to understand how to calculate the area of an ellipse using an example.
Example 1:
Find the area of an ellipse with centre at (2,3), length of major axis being 6 cm and length of minor axis being 4 cm.
Solution:
In this problem, major axis = 6, therefore semi major axis = a = 6/2 = 3 cm
Minor axis = 4, therefore semi minor axis = b = 4/2 = 2 cm
Using the formula for area of ellipse,
A =pi* ab = pi * 3 * 2 = 6 pi cm^2
Like a circle or any other closed figure, an ellipse is also a closed figure. Therefore it is possible to find its area. Area of an ellipse can be found using the values of a and b. But what are these a and b? That can be understood using the general equation of an ellipse.
General equation of an ellipse:
[(x-h)/a]^2 + [(y-k)/b]^2, where a is the semi major axis and b is the semi minor axis. (h,k) is the centre of the ellipse.
Area of ellipse formula:
From the above general equation we know that the semi major axis is a and the semi minor axis is b. The formula for the area of an ellipse can now be written as follows:
Area = A = pi * ab. The semi major axis can also be termed as the distance between the centre and the vertex, which is ‘a’ here. The semi minor axis can also be terms as the distance between the centre and the co-vertex, which is ‘b’ here. See the picture below:
Implicit equation of an ellipse:
Another method for finding the area of an ellipse is using its implicit equation. The general form of the implicit equation of an ellipse is:
Px^2 + Qxy + Ry^2 = 1
To calculate area of an ellipse if the equation of the ellipse is given in this form, we use the following formula:
Area = A = 2 pi/√(4PR – Q^2)
Let us now try to understand how to calculate the area of an ellipse using an example.
Example 1:
Find the area of an ellipse with centre at (2,3), length of major axis being 6 cm and length of minor axis being 4 cm.
Solution:
In this problem, major axis = 6, therefore semi major axis = a = 6/2 = 3 cm
Minor axis = 4, therefore semi minor axis = b = 4/2 = 2 cm
Using the formula for area of ellipse,
A =pi* ab = pi * 3 * 2 = 6 pi cm^2
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