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171 LearnersLast updated on September 15, 2025
Area of Ellipse
Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of the ellipse.
Area of the Ellipse Formula
To find the area of the ellipse, we use the formula: π × a × b, where a and b are the semi-major and semi-minor axes, respectively.
Derivation of the formula:- An ellipse is essentially a stretched circle, and its area can be thought of as a scaled version of a circle's area. The area of a circle is πr². For an ellipse, the radius is replaced by the semi-major and semi-minor axes. Therefore, the area of the ellipse = π × a × b
How to Find the Area of Ellipse?
We can find the area of the ellipse using the formula where the semi-major and semi-minor axes are commonly used. The area of the ellipse is calculated as follows:
Method Using the Semi-Major and Semi-Minor Axes
If the semi-major axis a and the semi-minor axis b are given, we find the area of the ellipse using the formula: Area = π × a × b
For example, if a and b are 5 cm and 3 cm, respectively, what will be the area of the ellipse? Area = π × a × b = π × 5 × 3 = 15π The area of the ellipse is 15π cm²
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Unit of Area of Ellipse
We measure the area of an ellipse in square units. The measurement depends on the system used:
In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²)
In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²)
Special Cases or Variations for the Area of Ellipse
There are no special formulas for the area of an ellipse since it is calculated using the semi-major and semi-minor axes. However, understanding the orientation and the axes can be critical:
Case 1: Circular Shape If the ellipse becomes a circle (a = b), the area is calculated using the formula for a circle, Area = πr², where r is the radius.
Case 2: Rotated Ellipse If the ellipse is rotated but the axes are given, use the formula Area = π × a × b for the calculation.
Tips and Tricks for Area of Ellipse
To ensure that you get correct results while calculating the area of the ellipse, here are some tips and tricks you should know about:
- The semi-major and semi-minor axes are distinct and should not be confused.
- Ensure you use the correct units when calculating the area, and convert if necessary to maintain consistency. π is approximately 3.14159, but using a calculator with a π function is more precise.
Common Mistakes and How to Avoid Them in Area of Ellipse
It is common for students to make mistakes while finding the area of the ellipse. Let’s take a look at some mistakes made by students.
Mistake 1
Using diameter instead of semi-major and semi-minor axes
To find the area of the ellipse, use the semi-major and semi-minor axes, not the full diameters.
Remember, a and b are half the lengths of the major and minor axes.
Mistake 2
Measuring the area using the wrong units
For the area to be in square meters (m²), both a and b should be in meters.
If one is in cm and the other is in m, convert the cm to m before calculating the area.
Mistake 3
Confusing a Circle and an Ellipse
A circle is a special case of an ellipse where the semi-major and semi-minor axes are equal.
For a general ellipse, these values differ.
Mistake 4
Incorrect Calculation of π
Using an incorrect value for π can lead to inaccurate results.
Use a precise value or a calculator with a π function.
Mistake 5
Misidentifying axes lengths
Ensure you correctly identify the longer axis as the major axis and the shorter one as the minor axis to avoid calculation errors.
Area of Ellipse Examples
Problem 1
The semi-major axis a and semi-minor axis b of an elliptical garden are given as 7 m and 4 m. What will be the area?
We will find the area as 28π m²
Explanation
Here, the semi-major axis a is 7 m and the semi-minor axis b is 4 m.
The area of the ellipse = π × a × b = π × 7 × 4 = 28π m²
Problem 2
What will be the area of the ellipse if the semi-major axis is 8 cm and the semi-minor axis is 5 cm?
We will find the area as 40π cm²
Explanation
If the semi-major and semi-minor axes are given, we use the formula, area of the ellipse = π × a × b.
Here, a and b are 8 cm and 5 cm.
Hence, the area will be π × 8 × 5 = 40π cm²
Problem 3
The area of the ellipse is 36ฯ mยฒ and the length of the semi-major axis a is 6 m.
We find the length of the semi-minor axis b as 6 m
Explanation
To find the semi-minor axis, use the formula area = π × a × b.
Here, the area of the ellipse is given as 36π m², and the length of semi-major axis a is 6 m.
Substitute the values: 36π = π × 6 × b 36 = 6 × b b = 36/6 = 6
Problem 4
Find the area of the ellipse if its semi-major axis is 10 cm and the semi-minor axis is 7 cm.
We will find the area as 70π cm²
Explanation
The given semi-major axis is 10 cm and the semi-minor axis is 7 cm.
If the axes are given, we find the area of the ellipse using the formula π × a × b.
Substituting the values in the formula: Area = π × 10 × 7 = 70π cm²
Problem 5
Help Linda find the area of the ellipse if the semi-major axis is 9 m and the semi-minor axis is 3 m.
We will find the area as 27π m²
Explanation
The semi-major axis is 9 m and the semi-minor axis is 3 m.
We calculate the area using the formula π × a × b.
Hence, we find the area of the ellipse as π × 9 × 3 = 27π m²
FAQs on Area of Ellipse
1.Is it possible for the area of the ellipse to be negative?
2.How to find the area of an ellipse if the lengths of the axes are given?
3.How to find the area of an ellipse if only the diameter is given?
4.How is the circumference of the ellipse calculated?
5.What is meant by the area of the ellipse?
Glossaries of Terms Related to Area of Ellipse
- Area: The total space occupied by a shape. It tells us how much surface a shape covers.
- Ellipse: A geometrical shape that resembles a stretched circle, defined by two axes.
- Semi-Major Axis: The longest radius of an ellipse, half of the major axis.
- Semi-Minor Axis: The shortest radius of an ellipse, half of the minor axis.
- Pi (π): A mathematical constant approximately equal to 3.14159, used in calculations involving circles and ellipses.
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
