Ellipse Calculator — Center, Foci, Vertices & Eccentricity

Use this free ellipse calculator to analyze an ellipse in standard form. Enter the center and semi-axes to get the vertices, co-vertices, foci, eccentricity, and area, with a diagram and steps.

Key parts of an ellipse

With semi-major and semi-minor axes, the focal distance is c = √(major² − minor²); the foci sit c units from the center along the major axis, and eccentricity is e = c/major.

How to use it

1. Enter the center (h, k) and semi-axes a and b.
2. Press Analyze.

Related: the Circle Equation Calculator and more free math calculators.

Frequently asked questions

What is the formula for an ellipse?

An ellipse centered at (h, k) has the standard form (x − h)²/a² + (y − k)²/b² = 1, where a and b are the semi-axes. If a > b, the major axis is horizontal; if b > a, it is vertical. For x²/25 + y²/9 = 1, the center is the origin, the semi-major axis is 5 and the semi-minor axis is 3.

How do you calculate ellipse area?

Multiply π by the two semi-axes: area = πab. For an ellipse with a semi-major axis of 5 and a semi-minor axis of 3, the area is π × 5 × 3 = 15π ≈ 47.12 square units. Be sure to use half-lengths; if you are given full axis lengths of 10 and 6, divide each by 2 first.

How do you calculate ellipse circumference?

There is no simple exact formula, so people use an approximation. A very accurate one, from Ramanujan, is C ≈ π[3(a + b) − √((3a + b)(a + 3b))]. For a = 5 and b = 3, this gives π[24 − √252] ≈ π × 8.125 ≈ 25.53 units. When a = b, the formula reduces to the circle formula 2πr.

How do you calculate ellipse foci?

Find c with c² = major² − minor², then place the foci c units from the center along the major axis. For x²/25 + y²/9 = 1, c = √(25 − 9) = √16 = 4, so the foci are at (−4, 0) and (4, 0). The foci always sit inside the ellipse, between the center and the vertices.

How do you calculate ellipse eccentricity?

Divide the focal distance by the semi-major axis: e = c/major, where c = √(major² − minor²). For semi-axes 5 and 3, c = 4, so e = 4/5 = 0.8. Eccentricity is always between 0 and 1 for an ellipse; 0 is a perfect circle and values near 1 mean a long, flat shape.

How do you calculate the radius of an ellipse?

An ellipse has no single radius like a circle. Instead it has two semi-axes: the semi-major axis, from the center to the farthest point on the curve, and the semi-minor axis, from the center to the nearest point. If an ellipse is 16 units wide and 4 units tall, the semi-axes are 8 and 2. The distance from the center to the edge changes between those two values as you move around.

How to use the Ellipse Calculator for homework

The Ellipse Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the calculator only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Ellipse Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the calculator to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Ellipse Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Ellipse Calculator result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator’s result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the calculator, you are much closer to owning the skill.