How to Calculate the Area of an Ellipse

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The area of an ellipse is a fundamental concept in geometry, widely used in various fields including mathematics, physics, and engineering. An ellipse is a closed curve that resembles a stretched circle, characterized by having two axes of symmetry – the major axis and the minor axis. Calculating the area of an ellipse is not as straightforward as finding the area of a regular shape like a square or a circle. It requires using specialized formulas and understanding the properties of an ellipse. In this guide, we will explore the step-by-step process of calculating the area of an ellipse, providing clarity and practical examples to help you grasp this concept easily. Whether you’re a student studying geometry or an individual who requires this knowledge for everyday applications, this guide will equip you with the necessary tools to confidently calculate the area of an ellipse.

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This article was co-written by David Jia. David Jia is a tutoring teacher and founder of LA Math Tutoring, a private tutoring facility based in Los Angeles, California. With over 10 years of teaching experience, David teaches a wide variety of subjects to students of all ages and grades, as well as college admissions counseling and prep for SAT, ACT, ISEE, etc. scoring 800 in math and 690 in English on the SAT, David was awarded a Dickinson Scholarship to the University of Miami, where he graduated with a bachelor’s degree in business administration. Additionally, David has worked as an instructor in online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

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In 2D geometry, the ellipse looks like a long, flattened circle. It is very easy to calculate the area of an ellipse if you know the lengths of the major and minor axes.

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Table of Contents

Steps

The area

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Calculate the length of the major axis of the ellipse. This is the distance from the center to the furthest point on the edge of the ellipse. You can think of this as the radius of the “flat” part of the ellipse. Measure or mark on the picture. We call this value a .

• You can also call this value “semi-major axis”. ^{[1] X Research Source}

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Find the length of the minor axis. It’s not hard to guess, the minor axis is the distance from the center to the nearest point on the edge of the ellipse. ^{[2] X Research Source} This value is denoted by b .

• This axis is perpendicular to the major axis, but you don’t need to calculate any angles to find the area.
• You might call this “semi-baby shaft.”

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Multiply by pi. The area of the ellipse will be equal to a x b x π. Since you are multiplying two lengths, the final answer will be in squared units. ^{[3] X Research Sources}

• For example, if the ellipse has a major axis with a length of 5 mm and a minor axis with a length of 3 mm, the area will be equal to 3 x 5 x π or 47 mm ² .
• If you don’t have a calculator or a calculator without the π symbol, replace it with “3.14”.

Prove the formula for the area of an ellipse

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Remember the formula for calculating the area of a circle. The area of a circle is equal to π r ² or π x r x r . Try to calculate the area of the circle using the formula of the ellipse. Measure the radius of the circle, we have: r . Continue to measure another radius at an angle of 90º from the previous radius, still r . When we substitute the formula for the area of an ellipse, we get: π xrxr! So the circle is a special ellipse with the major and minor axes equal. ^{[4] X Research Sources}

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Draw a distorted circle. Imagine a circle being squeezed into an ellipse. When squeezed, one radius will get shorter and shorter and the other will get longer and longer. The area of the shape remains the same, only the shape is different. As long as we still use both of these radii in the formula, the “squeeze” and “flatten” will cancel each other out and the area remains unchanged.

Advice

• If you want to rigorously prove the formula for the area of an ellipse, you need to use the integral operator calculus. ^{[5] X Research Sources}

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This article was co-written by David Jia. David Jia is a tutoring teacher and founder of LA Math Tutoring, a private tutoring facility based in Los Angeles, California. With over 10 years of teaching experience, David teaches a wide variety of subjects to students of all ages and grades, as well as college admissions counseling and prep for SAT, ACT, ISEE, etc. scoring 800 in math and 690 in English on the SAT, David was awarded a Dickinson Scholarship to the University of Miami, where he graduated with a bachelor’s degree in business administration. Additionally, David has worked as an instructor in online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

This article has been viewed 3,332 times.

In 2D geometry, the ellipse looks like a long, flattened circle. It is very easy to calculate the area of an ellipse if you know the lengths of the major and minor axes.

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In conclusion, calculating the area of an ellipse involves simple mathematical formulas and can be accomplished using various methods. The most common approach is to use the equation A = π * a * b, where A represents the area, a represents the semi-major axis, and b represents the semi-minor axis. Another method involves dividing the ellipse into a series of infinitely small sectors, summing up their areas, and taking the limit as the sectors become infinitesimally small. Alternatively, one can use numerical approximation techniques such as the Monte Carlo method to estimate the area of an ellipse. Regardless of the method used, understanding the concept and variables involved in calculating the area of an ellipse helps to solve geometric problems and accurately measure the enclosed space.

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