Introduction to Area of Ellipse
An ellipse just seems to be like an ordinary oval shape, resulting when a cone is snipped off by an oblique plane in a way that yields a closed curve which does not bisect the base. The area of ellipse is the same as the area of a circle i.e. A = π*r² or also written as: A= π · r * r. If a circle becomes flat it transforms into the shape of an ellipse and the semi-axes (OA and OB) of such an ellipse will be the stretched and compressed radii.
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Formulas for Ellipse
In an ellipse, if ‘a’ stands for OA and ‘b’ stands for OB, it can be proved that the area of the ellipse can be calculated by substituting ‘ab’ for ‘r r’ in the formula for the area of the circle, which then provides the below given:
Formula for the area of an ellipse: A = π · a · b
Where ‘a’ (horizontal segment) = major axis [semi-major axis or ½ of the major axis]
‘b’ (vertical segment) = minor axis [semi-minor axis or ½ the minor axis]
\[\text{Formula for Perimeter of the ellipse is: P = } 2 \pi \sqrt{\frac{a^{2} + b^{2}}{2}}\]
\[\text{Formula for volume of the ellipse is: V = } \frac{\pi (R_{1} + R_{2} + R_{3})}{3}\]
Major and Minor Axes
The Major Axis is said to be the longest diameter of an ellipse. This line segment travels from one side of the ellipse, through the center, to the other side, at the broadest part of the ellipse. And the Minor Axis is known to be the shortest diameter (at the narrowest part of the ellipse).
The half of the Major Axis is called the Semi-major Axis, and half of the Minor Axis is the Semi-minor Axis.
Below is a clear depiction of major and minor Axis with formulas to calculate:
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Special Case of Ellipse in a Circle's Area
A circle is a unique case of an ellipse. In correspondence to the fact that a square is a kind of rectangle, a circle is also a special case of an ellipse. The formula used to calculate the area of a circle is π r². However, seeing that a circle is an ellipse with equal minor and major axes, the formula for the area of the ellipse is the same as the formula for area of a circle.
Purpose of Calculating Area of Ellipse
Area of ellipse can be used to calculate a number of figures and fields such as:-
Find the surface area of a pond or an oval pool cover.
Computing the volume of a large table or an oval table top.
Compute the surface area of an oval structure like a dome.
Computing the area of a Football field.
Constructing an agricultural tunnel or an Elliptical Pipe.
To set a comparison between round and oval furnace hole airflow.
Used in laying flooring.
Solved Examples
Example 1:
A billiards table has horizontal radius of measurement 16.5 cm and vertical radius 10.5 cm. Calculate the area of the ellipse.
Solution1:
A billiards table is in the shape of an ellipse.
Given that a billiards table has:
Horizontal radius (a) = 16.5 cm
Vertical radius (b) = 10.5 cm
Area of ellipse = π × a × b
Thus,
A = π × 16.5 × 10.5
= 22/7 × 173.25
=544.5
Therefore,
A = 544.5cm2
Note: we will take the value of π as 22/7 unless, otherwise stated.
Example 2:
Determine the volume, area and perimeter of an ellipse having the radius R1, R2 & R3 of 11 cm, 8 cm, and 3 cm respectively?
Solution2:
Following are the given values:-
radius R1 = 11 cm
radius R2 = 8 cm
radius R3 = 3 cm
Step by step calculation
Using the Formula to calculate the volume = (4π/3) x R1 x R2 x R3
Now substituting the values
= (4π/3) x 11 x 8 x 3
= 1105.28cm3
Using the Formula to calculate area = π R1 R2
Now substituting the values
= π x 11 x 8
= 276.57 cm2
Using the Formula to calculate the perimeter = \[2 \pi (\sqrt{\frac{(R_{1} ^{2} + R_{2} ^{2})}{2}})\]
Now substituting the values
= 2π x \[\sqrt{\frac{(11 + 8)}{2}}\]
= 59.66cm2.
FAQs on Area of Ellipse
1. What is an ellipse in simple terms?
An ellipse is a closed, oval-shaped curve that is a fundamental shape in geometry, classified as a conic section. It is defined by two points called foci. For any point on the ellipse, the sum of its distances from the two foci is constant. An ellipse has a major axis (its longest diameter) and a minor axis (its shortest diameter), which are perpendicular to each other.
2. What is the standard formula to calculate the area of an ellipse?
The formula for the area of an ellipse is A = πab. In this formula, 'A' is the total area, 'π' (pi) is a mathematical constant (approximately 3.14159), 'a' is the length of the semi-major axis, and 'b' is the length of the semi-minor axis.
3. What do 'a' and 'b' represent in the ellipse area formula?
In the area formula A = πab, the variables represent the following dimensions of the ellipse:
- 'a' is the length of the semi-major axis, which is the distance from the center of the ellipse to the farthest point on its curve. It is half the length of the major axis.
- 'b' is the length of the semi-minor axis, which is the distance from the center to the nearest point on its curve. It is half the length of the minor axis.
4. How does the area of an ellipse relate to the area of a circle?
A circle is actually a special type of ellipse where both foci are at the same point (the center). This means its major and minor axes are equal. If we set a = b = r (the radius), the ellipse area formula A = πab becomes A = π(r)(r), which simplifies to A = πr². This shows that the formula for a circle's area is a specific instance of the more general ellipse area formula.
5. How is the area of an ellipse derived using integration, as per the CBSE syllabus?
As per the CBSE Class 12 syllabus, the area of an ellipse is a key application of definite integrals. Given the standard ellipse equation x²/a² + y²/b² = 1, we first solve for y to get y = (b/a)√(a² - x²). We then find the area of one quadrant by integrating this function from x=0 to x=a. The integral of (b/a)√(a² - x²) dx over this interval is (πab)/4. Since an ellipse has four identical quadrants, we multiply this result by 4 to get the total area: 4 * (πab)/4 = πab.
6. How does calculating the area of an ellipse differ from finding its perimeter (circumference)?
There is a significant difference in complexity. The area of an ellipse has a simple, exact formula (A = πab). However, there is no simple formula for the exact perimeter of an ellipse. Calculating the precise circumference requires complex mathematics involving elliptic integrals. While there are several formulas that provide good approximations, they are not exact, unlike the straightforward area formula.
7. How would you calculate the area of a semi-ellipse?
A semi-ellipse is exactly half of a full ellipse. To find its area, you first calculate the area of the complete ellipse using the formula A = πab. Then, you simply divide that area by two. Therefore, the area of a semi-ellipse is (πab) / 2, regardless of whether it is cut along its major or minor axis.
8. What are some important real-world examples where the concept of an ellipse's area is used?
The concept of an ellipse and its area appears in many real-world applications, including:
- Astronomy: The orbits of planets, moons, and comets are elliptical. Calculating the area an orbit sweeps in a given time is fundamental to understanding orbital mechanics (Kepler's Second Law).
- Architecture and Acoustics: The design of 'whispering galleries' relies on the elliptical shape to focus sound waves from one focus to another.
- Engineering: Elliptical gears are used in machinery to produce variable rotational speeds or torque from a constant input speed.
- Medical Technology: Lithotripsy machines use an elliptical reflector to focus shock waves to break up kidney stones without invasive surgery.
