Conic Sections Calculator - Circle, Ellipse, Parabola

What Is a Conic Sections Calculator?

A conic sections calculator finds the eccentricity and standard equation of a circle, ellipse, parabola, or hyperbola from a few measurements. These four curves appear when a flat plane cuts through a cone at different angles.

The Four Conic Sections

Slicing a cone straight across produces a circle. Tilting the plane produces an ellipse. Cutting parallel to the side of the cone produces a parabola. Cutting through both halves of a double cone produces a hyperbola.

Each curve has an eccentricity, a number that shows how much it differs from a perfect circle. The eccentricity alone identifies the type of conic:

• Circle: $e = 0$
• Ellipse: $0 < e < 1$
• Parabola: $e = 1$
• Hyperbola: $e > 1$

How to Use This Calculator

1. Choose a conic section type: circle, ellipse, parabola, or hyperbola.
2. Enter the required measurements.
   • Circle: radius ($r$)
   • Ellipse: semi-major axis ($a$) and semi-minor axis ($b$)
   • Parabola: focal length ($f$)
   • Hyperbola: semi-transverse axis ($a$) and semi-conjugate axis ($b$)
3. The eccentricity and standard equation appear automatically as soon as valid, positive inputs are entered for the selected conic type; there is no Calculate button to click.

All inputs must be positive numbers. For an ellipse, the semi-major axis must be at least as long as the semi-minor axis, because the semi-major axis is defined as the longer one. A hyperbola has no such rule: its semi-transverse and semi-conjugate axes can be any positive lengths, including equal lengths.

Circle Formula

For a circle with radius $r$ centered at the origin:

$$x^2 + y^2 = r^2$$

Eccentricity: $e = 0$.

A circle is a special case of an ellipse where both axes are equal length and both foci sit at the center.

Ellipse Formula

For an ellipse centered at the origin with semi-major axis $a$ and semi-minor axis $b$, where $a \geq b > 0$:

$$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ $$e = \sqrt{1 - \left(\dfrac{b}{a}\right)^2}$$

This follows from $e = c/a$, where $c$ is the distance from the center to a focus and $c^2 = a^2 - b^2$ for an ellipse.

Example: $a = 5$, $b = 3$.

$$e = \sqrt{1 - \left(\dfrac{3}{5}\right)^2} = \sqrt{1 - 0.36} = \sqrt{0.64} = 0.8$$

Equation: $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$

When $a = b$, the ellipse becomes a circle and $e = 0$.

Parabola Formula

For a parabola opening to the right, with focal length $f$ (the distance from the vertex to the focus):

$$y^2 = 4fx$$

Eccentricity: $e = 1$, always.

Example: $f = 2$. Equation: $y^2 = 8x$.

Hyperbola Formula

For a hyperbola centered at the origin with semi-transverse axis $a$ and semi-conjugate axis $b$, where $a$ and $b$ are any positive numbers:

$$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$$ $$e = \sqrt{1 + \left(\dfrac{b}{a}\right)^2}$$

For a hyperbola, $e = c/a$ again, but $c^2 = a^2 + b^2$, so $e$ is always greater than 1.

Unlike an ellipse, a hyperbola does not require $a$ to be larger than $b$. The two semi-axes are independent lengths, so $b$ can be smaller than, equal to, or larger than $a$. When $a = b$, the curve is called a rectangular hyperbola, with eccentricity $\sqrt{2} \approx 1.4142$.

Example: $a = 5$, $b = 3$.

$$e = \sqrt{1 + \left(\dfrac{3}{5}\right)^2} = \sqrt{1 + 0.36} = \sqrt{1.36} \approx 1.1662$$

Equation: $\dfrac{x^2}{25} - \dfrac{y^2}{9} = 1$

Real-World Uses of Conic Sections

Planets and comets follow orbits shaped like conic sections. Kepler's first law states that a planet's orbit is an ellipse with the Sun at one focus. Some comets follow parabolic or hyperbolic paths and never return.

Parabolas focus parallel rays to a single point, so engineers use parabolic shapes in satellite dishes, telescope mirrors, and car headlights. Hyperbolic curves appear in cooling tower walls, where the shape gives strength while using less material. Ellipses show up in arches, whispering-gallery ceilings, and gear design.

A Short History of Conic Sections

The Greek mathematician Menaechmus (about 380 to 320 BCE) studied conic sections while trying to solve the problem of doubling a cube. Apollonius of Perga (about 262 to 190 BCE) wrote a detailed treatise called Conics. It introduced the names ellipse, parabola, and hyperbola and described their properties. Centuries later, Johannes Kepler used the ellipse to describe planetary motion, and Isaac Newton used conic sections to explain gravity.

Frequently Asked Questions

What are the four types of conic sections?

Circle, ellipse, parabola, and hyperbola. Each forms when a plane cuts through a cone at a different angle, and each has a different eccentricity range.

How do you calculate eccentricity?

Eccentricity is fixed for a circle ($e = 0$) and a parabola ($e = 1$). For an ellipse, $e = \sqrt{1 - (b/a)^2}$, using the semi-major axis $a$ and semi-minor axis $b$. For a hyperbola, $e = \sqrt{1 + (b/a)^2}$, using the semi-transverse axis $a$ and semi-conjugate axis $b$.

Does a hyperbola's transverse axis have to be longer than its conjugate axis?

No. Unlike an ellipse's semi-major and semi-minor axes, a hyperbola's semi-transverse axis ($a$) and semi-conjugate axis ($b$) are independent measurements. Both must be positive, but either one can be the larger.

Can an ellipse become a circle?

Yes. When the semi-major axis equals the semi-minor axis, the ellipse is a circle and its eccentricity is 0.

Why do satellite dishes use a parabolic shape?

A parabola reflects rays traveling parallel to its axis to a single focus point. Satellite dishes and telescope mirrors use this property to concentrate signals or light at one spot.

What is a rectangular hyperbola?

A rectangular hyperbola is one where the semi-transverse axis equals the semi-conjugate axis ($a = b$). Its asymptotes are perpendicular, and its eccentricity is $\sqrt{2} \approx 1.4142$.

References

1. OpenStax. Calculus Volume 2, 7.5 Conic Sections. Conic Sections — eccentricity ranges for each conic, $e = c/a$, $c^2 = a^2 - b^2$ (ellipse) and $c^2 = a^2 + b^2$ (hyperbola), and the reflective property of a parabolic dish.
2. NASA Science. Orbits and Kepler's Laws. Orbits and Kepler's Laws — planets move in elliptical orbits with the Sun at one focus.
3. MacTutor History of Mathematics, University of St Andrews. Menaechmus. Menaechmus biography — dates, and conic sections studied to solve the problem of duplicating the cube.
4. MacTutor History of Mathematics, University of St Andrews. Apollonius of Perga. Apollonius biography — dates, and Conics introducing the terms parabola, ellipse and hyperbola.