Area of Ellipse
Area of an ellipse is defined as the amount of region present inside an ellipse. Alternatively, the area of an ellipse is the total number of unit squares that can be fitted in it. You might have observed many ellipse-shaped shapes in our daily lives, for example, a cricket ground, a badminton racket, orbits of planets, etc.
The area of the ellipse is the product of π, the length of the semi-major axis, and the length of the semi-minor axis. Let us explore a bit more about this shape while discussing its area, and the formula for the area of the ellipse, with solved examples and FAQs.
| 1. | What Is the Area of Ellipse? |
| 2. | Area of Ellipse Formula |
| 3. | Proof of Formula of Area of Ellipse |
| 4. | Area of Ellipse Using Integration |
| 5. | How to Find the Area of Ellipse? |
| 6. | FAQs on Area of Ellipse |
What Is the Area of Ellipse?
Area of an ellipse is the area or region covered by the ellipse in two dimensions. The area of an ellipse is expressed in square units like in2, cm2, m2, yd2, ft2, etc. Ellipse is a 2-D 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 of ellipse. F1 and F2 are the two foci. 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. 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.
Ellipse is the locus of all the points, whose sum of distances from two fixed points on a plane is constant. For example, in the images given below the points P1, P2 are located in such a way that the sum of the distances of point P1 from the fixed points F1 and F2 is equal to the sum of the distances of point P2 from the fixed points F1 and F2. That means if we join all such points P1, P2, P3 , etc; we will get a shape called an ellipse.
\(\begin{equation} \mathbf{P}_{1} \mathbf{F}_{1}+\mathbf{P}_{1} \mathbf{F}_{2}=\mathbf{P}_{2} \mathbf{F}_{1}+\mathbf{P}_{2} \mathbf{F}_{2}=\mathbf{P}_{3} \mathbf{F}_{1}+\mathbf{P}_{3} \mathbf{F}_{2}=...=\text{constant} \end{equation}\)
where,
- \(F_1, F_2\) = Two fixed points on the plane
- \(P_1, P_2, P_3\) ... = Points forming locus of ellipse
Area of Ellipse Formula
Area of an ellipse formula can be calculated using a general formula, given the lengths of the major and minor axis. The formula to find the area of an ellipse is given by,
Area of ellipse = π a b
where,
- a = length of semi-major axis
- b = length of semi-minor axis
Proof of Formula of Area of Ellipse
Let E be an ellipse, with major axis of length 2a and minor axis of length 2b, aligned in a cartesian plane in the reduced form. Then, from the equation of ellipse in reduced form, we get,
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
Thus:
\(y = \pm \dfrac{b}{a} \sqrt{a^2 - x^2}\)
Consider a circle, C, of radius 'a' whose center is at the origin. From the equation of circle formula with origin at center, we have,
\(x^2 + y^2 = a^2\)
⇒ y = ± \(\sqrt{a^2 - x^2}\)
We observe from the formulas that each ordinate of the ellipse is \(b/a\) times the ordinate of the circle. The same thing is true for the vertical chords. We can thus also relate the areas of the two shapes as,
Area of ellipse = (b/a) × Area of circle
We know, area of a circle with the given equation, \(x^2 + y^2 = a^2\), is: A = πa2, where 'a' is the radius of the circle.
⇒ Area of ellipse = (b/a)(πa2) = πab
Area of an Ellipse Using Integration
Area of an ellipse formula can also be derived using integration. The general equation for an ellipse is given as, \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}} = 1\)
⇒ \(y = b.\sqrt{1-\left ( \dfrac{x}{a} \right )^{2}}\)
We observe that the ellipse is divided into four quadrants. So, we can calculate the area of 1 quadrant and multiply by 4 to calculate the area of an ellipse.
Area, \(A = 4. \int_{0}^{a} y.dx\)
\(A = 4. \int_{0}^{a} \dfrac{b}{a} \sqrt{a^{2}-x^{2}} dx\)
Substituting, x = a sin t, we get, dx = a cos t . dt
⇒ x = 0 changes to t = 0 and x = a changes to t = π/2
\(A = 4. \dfrac{b}{a}\int_{0}^{\dfrac{\pi}{2}} \sqrt{a^{2}-a^{2} \sin^{2}t} .a \cos t .dt\)
⇒ \(A = 4b \int_{0}^{\pi/2}\sqrt{a^{2}(1-sin^{2}t)} cost dt\)
⇒ \(A = 4ba \int_{0}^{\pi/2}\sqrt{cos^{2}t} cost dt\)
⇒ \(A = 4ba \int_{0}^{\pi/2}cos^{2}t dt\)
⇒ \(A = 4ba \int_{0}^{\pi/2}\dfrac{(1 + cos2t)}{2}dt\)
⇒ \(A = 4ab \left ( \dfrac{t}{2} + \dfrac{\sin 2t}{4} \right )_{0}^{\pi/2}\)
⇒ A = 4ab (π/4)
⇒ A = πab
where,
- a = length of semi-major axis
- b = length of semi-minor axis
How to Find the Area of Ellipse?
The area of an ellipse can be calculated using the following four steps and using the length of the major and minor axis.
- Step 1: Find the distance from the farthest point on the ellipse from the center ('a' i.e., length of the semi-major axis).
- Step 2: Find the distance from the closest point on the ellipse from the center ('b' i.e., length of the semi-minor axis).
- Step 3: Multiply the product of a and b with π.
- Step 4: Mention the area in square units.
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 = π a b
- Take the value of π as 3.14 or 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.
FAQs on Area of Ellipse
What Is the Area of an Ellipse?
The area of an ellipse is the total region inside an ellipse in the two-dimensional plane. It can be expressed in square units. Some common units used to express the area of an ellipse are in2, cm2, m2, yd2, ft2, etc.
What Is the Formula to Find Area of an Ellipse?
The formula to calculate the area of an ellipse is given as, area of ellipse, A = πab, where, 'a' is the length of the semi-major axis and 'b' is the length of the semi-minor axis.
How Do You Calculate Area of an Ellipse?
The area of an ellipse can be calculated using the following steps.
- Step 1: Note the length of the semi-major axis, 'a', and length of the semi-minor axis as 'b'.
- Step 3: Multiplication of the product of a and b with π.
- Step 4: Express the area in square units.
How to Derive the Formula for Area of an Ellipse?
The area of ellipse formula can be derived using integration. We can calculate the area of any one quadrant of the ellipse with the given major and minor axis and multiply the result by 4, to obtain the area of an ellipse. For detailed derivation, refer to section, Area of an Ellipse Using Integration.
What Is the Proof of Area of an Ellipse Formula?
The area of the ellipse formula can be proved using the analogy between the shape circle and ellipse. To see the detailed proof of formula using this method, refer to the section, Proof of Formula of Area of an Ellipse.
What Are the Units Used to Express the Area of an Ellipse?
In measurements, the area of an ellipse is expressed using square units or (unit)2. The common units that can be used are square meters, square inches, square yards, etc.