Let us understand what an ellipse is, and then look at the methodology to calculate the area of an ellipse.
What is an ellipse?
Ellipse is a shape with two dimensions and in simpler terms can be defined as a flat and elongated circle. An ellipse has two distinct focal points and the summation of the distance from these two focal points to any point on the circumference of the curve remains constant. It can be seen as a simple circle, but with the two focal points on the same place. The geometrical shape of an ellipse can be explained by the eccentricity which can range from anywhere between 1 and 0. It is to be noted that it is symmetrical along both the axes.
Image Credits: Wikipedia
What is the equation of an ellipse?
The equation of an ellipse can be given by x2 / a2 + y2 / b2 = 1
Where a is the semi-major axis (half of the longest diameter) and y is the semiminor axis (half of the shortest diameter)
In such a case, what will be the area of an ellipse?
In such a case, the area of the ellipse can be written as πab
Aellipse = πab
While that of half a circle would be πr2 / 2
For example, if the length of the semi-major axis is 16 inches and the length of the semi-minor axis is 4 inches, then, the area of the ellipse will be π x 16 x 4 =64 π
In order to find the area of a circle from the given data on an ellipse, the formula that can be used is the area of a circle = ( b / a ) x Area of the ellipse.
In integral calculus the formula that can be used to find the area of an ellipse when
Step by step process to find the area of an ellipse
For the formula area = π x a x b
1. First, find the major radius or the length of the semi-major axis. This can be defined as the length or distance from the center of the figure to the farthest point on the circumference. This can be denoted by ‘a’.
- Second, find the minor radius or the length of the semi-minor axis. This can be defined as the length or distance from the center of the figure to the closest point on the circumference of the ellipse. This can be denoted by ‘b’.
- Third, multiply both the values by pi. Thus, the area of the ellipse is pi x a x b. The area will be in the unit square.
For the formula of integration
- Express the equation in terms of y variable.
- Simplify to get it in the form of a and b.
- Integrate with respect to x.
- Multiply the same by 4 to get the area of the full ellipse.
- we can either substitute the value of pi to be 3.14 or leave it as it is.
Why does this work?
An ellipse can be thought of as a circle. The area of a circle is π r2 . In simpler terms, it can be written as π x r x r. if we had to find the area of the ellipse, we would simply measure the distance between the center and the circumference at two distinct points, keeping them at right angles, we would then put this in the formula to arrive at the value of the area. Hence, a circle is just a type of ellipse.
Some examples of the area formula of an ellipse
- If a = 3 cm and b = 7 cm, then what is the area of the ellipse?
Answer: According to the formula, area of the ellipse is π x a x b
Here, a = 3 and b = 7
Therefore, area = π x 3 x 7
Or, Area = 21 x 3.14
= 65.94 cm 2
- Calculate the area of the ellipse whose major radius is 2 cm and minor radius is 3 cm
Answer: area of the ellipse = π x a x b
Or, π x 2 x 3
= 6 π cm 2
- The ellipse has horizontal radius 8 cm and vertical radius 5 cm. Find the area of an ellipse.
Answer: Given that:
Horizontal radius (a) = 8 cm
Vertical radius (b) = 5 cm
A = π · a · b
A = π · 8 · 5
A = 125.6 cm2
- Dimensions of a rectangular metal paper are 15 inch x 10 inch. find the area, if the largest possible ellipse is made out from the rectangular paper.
Answer: The largest possible ellipse has its axes 15 inch and 10 inch.
Area of the ellipse = π / 4 ×15 × 10
= 117.75 inch squared
∴ required area of the ellipse is 117.75 inch 2
- Find the area of the ellipse if the length of the major axes is 7 cm and the length of the minor axes is cm.
Answer: the equation of the ellipse is x2 / a2 + y2 / b2 = 1
By terminating y we get y2 = b2 – b2 x2 / a2 = (b2 / a2)(a2 – x2)
Or y = b/a √ (a2 – x2)
To find the area, we integrate y from –a to a and multiply it with 2
Area = 2 ∫ b/a √ (a2 – b2) dx, –a <x <a
=2b / a ∫ √(a2 – b2)dx
= b/a [ x √(a2-x2) + a2 sin-1(x/a)], –a<x<a
= b/a [ a √ (a2 – a2) + a2 sin-1 (a/a) – a √(a2 – (-a)2) – a2 sin-1(-a/a)]
= b/a [ 0 + a2 π/2 – 0 – a2 (-π/2)]
= b/a [ 2a2 π/2]
= a . b . π
= (7/2) . (4/2) . π
= 22 cm2
Hence, the required area is 22 cm square.