What is the Area of an Ellipse Formula with Solved Examples
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 Formula Derivation and Applications
1. What is the area of an ellipse?
The area of an ellipse is the total region enclosed by it and is calculated using the formula A = πab, where a is the semi-major axis and b is the semi-minor axis.
- a = half of the major axis length
- b = half of the minor axis length
- π ≈ 3.1416
2. What is the formula for the area of an ellipse?
The formula for the area of an ellipse is A = πab.
- a represents the semi-major axis
- b represents the semi-minor axis
3. How do you calculate the area of an ellipse step by step?
To calculate the area of an ellipse, use the formula A = πab and substitute the semi-axis values.
- Step 1: Identify the semi-major axis a
- Step 2: Identify the semi-minor axis b
- Step 3: Multiply π × a × b
4. Why is the area of an ellipse πab?
The area of an ellipse is πab because it is a stretched version of a circle whose area is πr².
- If a circle of radius r is stretched horizontally and vertically, r becomes a and b
- The scaling changes r² into ab
5. What is the area of an ellipse when the major and minor axes are given?
If the full major axis and minor axis are given, first divide them by 2 to get a and b, then use A = πab.
- Major axis = 10 → a = 5
- Minor axis = 6 → b = 3
- Area = π × 5 × 3 = 15π
6. How is the area of an ellipse different from the area of a circle?
The area of a circle is πr², while the area of an ellipse is πab.
- A circle has one radius (r = a = b)
- An ellipse has two different semi-axes (a and b)
7. Can you give an example of finding the area of an ellipse?
Yes, the area of an ellipse with a = 4 and b = 2 is 8π square units.
- Use the formula: A = πab
- Substitute values: π × 4 × 2
- Result: 8π ≈ 25.13 square units
8. What are the units of the area of an ellipse?
The units of the area of an ellipse are always square units, such as cm², m², or units².
- If a and b are in centimeters, area is in cm²
- If a and b are in meters, area is in m²
9. Does the orientation of an ellipse affect its area?
No, the orientation of an ellipse does not change its area as long as the semi-major axis a and semi-minor axis b remain the same.
- Horizontal or vertical alignment gives the same result
- Area depends only on a and b
10. What is the area of an ellipse in standard form equation?
If an ellipse is in standard form x²/a² + y²/b² = 1, its area is πab.
- Identify a from √(a²)
- Identify b from √(b²)
- Apply A = πab
