102 LearnersLast updated on August 30, 2025
Surface Area of Ellipse
An ellipse is a two-dimensional shape that resembles a stretched circle. The surface area of an ellipse, more commonly referred to as the area of an ellipse, is the total region enclosed by its boundary. In this article, we will learn about the area of an ellipse.
What is the Surface Area of an Ellipse?
The surface area of an ellipse is the total area enclosed by its boundary. It is measured in square units.
An ellipse looks like a squashed circle and is characterized by two axes: the major axis and the minor axis.
The major axis is the longest diameter of the ellipse, while the minor axis is the shortest.
The surface area of an ellipse can be calculated using a specific formula that takes into account both axes.
Area of an Ellipse Formula
An ellipse has a specific formula to calculate its surface area based on its axes.
Consider an ellipse with a major axis (2a) and a minor axis (2b).
The formula for calculating the area of an ellipse is given by: Area = πab square units
Where a is the semi-major axis (half of the major axis) and b is the semi-minor axis (half of the minor axis).
Calculation of Ellipse Area
The area of an ellipse is determined by its semi-major and semi-minor axes.
The formula for the area of an ellipse is:
Area = πab Here, a is the semi-major axis, and b is the semi-minor axis.
This formula shows how the product of the semi-axes and π gives the total area enclosed by the ellipse.
Visualizing the Area of an Ellipse
To visualize the area of an ellipse, imagine a circle that has been stretched along one of its diameters.
The area is not just a simple multiplication of the axes, but rather a more complex shape requiring the use of π.
The formula: Area = πab helps in understanding that the ellipse's area is influenced by both axes, with π adjusting the area to account for the elliptical shape.
Volume of a 3D Elliptical Cylinder
The volume of a 3D shape with an elliptical base, like an elliptical cylinder, can be calculated using the area of the ellipse as the base area and multiplying by the height of the cylinder.
The volume formula is: Volume = πabh (cubic units) where a is the semi-major axis, b is the semi-minor axis, and h is the height of the cylinder.
Confusion between Major and Minor Axes
Students may confuse the major and minor axes, leading to incorrect calculations.
Remember that the major axis is the longest diameter, and the minor axis is the shortest.
Always use half of these lengths (semi-major and semi-minor) in the formula.
Mistake 1
Incorrect Use of π
Some students mistakenly use an incorrect value of π, such as 22. Always use accurate values like 3.14 or 22/7 to ensure the correct calculation of the area.
Mistake 2
Forgetting to Halve the Axes
A common mistake is using the full lengths of the axes in the formula instead of halving them. Remember to use the semi-major axis and semi-minor axis in the formula: Area = πab.
Mistake 3
Using Diameter Instead of Axes
Students might incorrectly use the diameter instead of the axes. Ensure that the axes used are the lengths from the center to the edge (semi-major and semi-minor), not the full diameter.
Mistake 4
Assuming an Ellipse is a Circle
Some students mistakenly treat an ellipse as a circle and use the formula for the area of a circle. Remember, the area of an ellipse requires the specific formula: Area = πab.
Mistake 5
Solved Examples of Surface Area of Ellipse
Find the area of an ellipse with a semi-major axis of 4 cm and a semi-minor axis of 3 cm.
Area = 37.68 cm²
Problem 1
Given a = 4 cm, b = 3 cm. Use the formula: Area = πab = 3.14 × 4 × 3 = 37.68 cm²
Calculate the area of an ellipse with a semi-major axis of 5 cm and a semi-minor axis of 2 cm.
Explanation
Area = 31.4 cm²
Problem 2
Use the formula: Area = πab = 3.14 × 5 × 2 = 31.4 cm²
An ellipse has a semi-major axis of 6 cm and a semi-minor axis of 4 cm. Find the area.
Explanation
Area = 75.36 cm²
Problem 3
Use the formula: Area = πab = 3.14 × 6 × 4 = 75.36 cm²
Find the area of an ellipse with a semi-major axis of 7 cm and a semi-minor axis of 3.5 cm.
Explanation
Area = 76.93 cm²
Problem 4
Area = πab = 3.14 × 7 × 3.5 = 76.93 cm²
The area of an ellipse is 125.6 cm², and its semi-major axis is 8 cm. Find the semi-minor axis.
Explanation
Semi-minor axis = 5 cm
It is the total area enclosed by the boundary of the ellipse, calculated using its semi-major and semi-minor axes.
1.What are the axes in an ellipse?
2.How is the area of an ellipse calculated?
3.Is an ellipse the same as a circle?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in Calculating the Area of an Ellipse
Students often make mistakes while calculating the area of an ellipse, leading to incorrect answers. Below are some common mistakes and ways to avoid them.
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
