Ellipse Area Calculator

Enter the lengths of the semi-axes or the ellipse equation to calculate the area of the figure and get the step-by-step solution.

Major semi-axis (a)
Minor semi-axis (b)

Quick Examples

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How to Use This Calculator

This online ellipse area calculator is a mathematical tool designed for finding the exact surface area of this geometric figure (often referred to as an oval). Unlike a traditional solver, this system not only gives you the final answer but also generates a detailed step-by-step solution, explaining each stage of the procedure.

Use the top drop-down menu to specify what data you know from your math problem. The supported calculation modes are as follows:

  • Length of the semi-axes: Enter the measurement of the figure's semi-axes (the longest and shortest dimensions of the oval), and the area finder will calculate the surface using the formula A = a b π.
  • Ellipse equation: The algebraic engine is ready to interpret the geometric equation in any of its forms. You do not need to arrange it beforehand; you can enter the standard form, the general form, or an unsimplified expression. The solver will automatically deduce the length of the semi-axes to then calculate the area.

Upon running the calculation, you will get the exact result expressed analytically (keeping the irrational number π and simplifying roots) along with its decimal approximation. Remember that since it is a surface area, the final value represents square units (mm², cm², m², in², ft², etc.), which will depend directly on the original unit of measurement of your data.

Note: all input fields of the calculator support integers, decimals, fractions, and irrational numbers (such as square roots). These values will be kept purely symbolically throughout the step-by-step solution, guaranteeing an exact mathematical result free of rounding errors.

Solved Examples

The following are examples of problems solved by the calculator.

Calculate the area of the ellipse with semi-axes a = 5 and b = 3.

Area of the ellipse

$$ A = 15\pi \approx 47.12 $$

Since it is an area, it is measured in square units (mm2, cm2, m2, in2, ft2, etc.) depending on the units of the semi-axes.

Step-by-step solution

1. Identify the parameters.

We take the values of the entered semi-axes:

$$ \begin{array}{l} a = 5 \\[1.3em] b = 3 \end{array} $$

2. Calculate the area.

We use the ellipse area formula and substitute the values to obtain the final result:

$$ \begin{array}{l} A = ab\pi \\[1.2em] A = (5)(3)\pi \\[1.5em] A = 15\pi \approx 47.12 \end{array} $$
Find the surface area of the ellipse with semi-axes \(a = \sqrt{5}\) and \(b = \sqrt{2}.\)

Area

$$ A = \sqrt{10}\pi \approx 9.93 $$

Step-by-step solution

1. Identify the parameters.

We take the values of the entered semi-axes:

$$ \begin{array}{l} a = \sqrt{5} \\[1.3em] b = \sqrt{2} \end{array} $$

2. Calculate the area.

We use the ellipse area formula and substitute the values to obtain the final result:

$$ \begin{array}{l} A = ab\pi \\[1.2em] A = (\sqrt{5})(\sqrt{2})\pi \\[1.5em] A = \sqrt{10}\pi \approx 9.93 \end{array} $$
Determine the area of the ellipse with the equation \(\dfrac{x^2}{16} + \dfrac{y^2}{9}=1.\)

Area of the ellipse

$$ A = 12\pi \approx 37.7 $$

Step-by-step solution

1. Extract the denominators.

From the standard form, we extract the denominator values, which represent the square of the semi-axes:

$$ \begin{array}{l} a^2 = 16 \\[1em] b^2 = 9 \end{array} $$

2. Calculate the semi-axes.

We apply the square root to the obtained values to find the lengths of the semi-axes:

$$ \begin{array}{l} a = \sqrt{16} = 4 \\[1em] b = \sqrt{9} = 3 \end{array} $$

3. Calculate the area.

We use the ellipse area formula and substitute the values to obtain the final result:

$$ \begin{array}{l} A = ab\pi \\[1.2em] A = (4)(3)\pi \\[1.5em] A = 12\pi \approx 37.7 \end{array} $$
Find the area of the ellipse whose general equation is \( 25x^2+9y^2-50x-200=0. \)

Area of ​​the oval

$$ A = 15\pi \approx 47.12 $$

Step-by-step solution

1. Convert the equation to standard form.

We transform the general equation into standard form by completing the square. The resulting equation is:

$$ \dfrac{\left(x - 1\right)^2}{9} + \dfrac{y^2}{25} = 1 $$

2. Extract the denominators.

From the standard form, we extract the denominator values, which represent the square of the semi-axes:

$$ \begin{array}{l} a^2 = 25 \\[1em] b^2 = 9 \end{array} $$

3. Calculate the semi-axes.

We apply the square root to the obtained values to find the lengths of the semi-axes:

$$ \begin{array}{l} a = \sqrt{25} = 5 \\[1em] b = \sqrt{9} = 3 \end{array} $$

4. Calculate the area.

We use the ellipse area formula and substitute the values to obtain the final result:

$$ \begin{array}{l} A = ab\pi \\[1.2em] A = (3)(5)\pi \\[1.5em] A = 15\pi \approx 47.12 \end{array} $$

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Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

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