Free Conic Sections Calculator

Conic sections are curves obtained by cutting a double cone with a plane. The shape of the curve—circle, ellipse, parabola, or hyperbola—depends on the angle of the cut. With a Conic Sections Calculator (often called a Conic Equation Calculator), you can quickly determine the equation and all major parameters of any conic by providing a few numeric inputs. This free online tool also works as an Eccentricity Calculator and covers ellipse, hyperbola, and parabola types, making it perfect for students and professionals.

Formation of Conic Sections

Imagine a double cone placed tip to tip. When an intersecting plane is parallel to the base, the cross-section is a circle. As the plane tilts but remains less steep than the side of the cone, the resulting curve is an ellipse. If the plane is exactly parallel to the cone’s generating line, a parabola appears. For steeper tilts (up to vertical), the intersection becomes a hyperbola. These four curves belong to the family of conic sections.

Geometric Definitions

Each conic can be described using foci and directrices:

• Circle: determined by a center and radius; all points at a fixed distance from the center.
• Ellipse: two foci; the sum of distances from any point on the ellipse to the two foci is constant.
• Parabola: one focus and one directrix; any point is at equal distances from the focus and the directrix.
• Hyperbola: two foci; the absolute difference of distances from a point to the two foci is constant.

A circle can be treated as a special ellipse whose two foci coincide, and a parabola can be thought of as an ellipse whose second focus lies at infinity.

Standard Equations

When centered at the origin, conics take simple Cartesian equations. The table below lists the common standard forms with parameters $a$ and $b$ defining the axes:

Conic	Equation
Circle	$x^{2} + y^{2} = a^{2}$
Ellipse (horizontal)	$\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$
Ellipse (vertical)	$\dfrac{x^{2}}{b^{2}} + \dfrac{y^{2}}{a^{2}} = 1$
Parabola (right-opening)	$y^{2} = 4ax$
Hyperbola (horizontal)	$\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$
Hyperbola (vertical)	$\dfrac{y^{2}}{a^{2}} - \dfrac{x^{2}}{b^{2}} = 1$

For the vertical hyperbola in this table, the real axis runs along the $y$-direction, and $a$ is the semi-real axis.

Key Parameters

Conics are characterized by several numbers that reveal their shape and orientation:

• Eccentricity ($e$): measures how much the curve deviates from a circle. Ranges: circle $e = 0$, ellipse $0 < e < 1$, parabola $e = 1$, hyperbola $e > 1$.
• Linear eccentricity ($c$): distance from the center to a focus. Not defined for parabolas because they lack a center.
• Semi-latus rectum ($l$): half the length of the chord through a focus perpendicular to the major axis.
• Focal parameter ($p$): distance from a focus to the corresponding directrix. For circles the directrix is at infinity, so $p = \infty$.

These parameters together give a complete description of any conic’s shape and focusing behavior.

Parameter Formulas by Conic Type

The next table provides explicit formulas for each standard conic. All expressions are built into the Conic Equation Calculator, so you can instantly obtain these values.

Conic	Equation	$e$	$c$	$l$	$p$
Circle	$x^{2} + y^{2} = a^{2}$	$0$	$0$	$a$	$\infty$
Ellipse (horizontal)	$\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$	$\sqrt{1 - \dfrac{b^{2}}{a^{2}}}$	$\sqrt{a^{2} - b^{2}}$	$\dfrac{b^{2}}{a}$	$\dfrac{b^{2}}{\sqrt{a^{2} - b^{2}}}$
Ellipse (vertical)	$\dfrac{x^{2}}{b^{2}} + \dfrac{y^{2}}{a^{2}} = 1$	$\sqrt{1 - \dfrac{b^{2}}{a^{2}}}$	$\sqrt{a^{2} - b^{2}}$	$\dfrac{b^{2}}{a}$	$\dfrac{b^{2}}{\sqrt{a^{2} - b^{2}}}$
Parabola	$y^{2} = 4ax$	$1$	not defined	$2a$	$2a$
Hyperbola (horizontal)	$\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$	$\sqrt{1 + \dfrac{b^{2}}{a^{2}}}$	$\sqrt{a^{2} + b^{2}}$	$\dfrac{b^{2}}{a}$	$\dfrac{b^{2}}{\sqrt{a^{2} + b^{2}}}$
Hyperbola (vertical)	$\dfrac{y^{2}}{a^{2}} - \dfrac{x^{2}}{b^{2}} = 1$	$\sqrt{1 + \dfrac{b^{2}}{a^{2}}}$	$\sqrt{a^{2} + b^{2}}$	$\dfrac{b^{2}}{a}$	$\dfrac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Example: Ellipse with $a = 5$ and $b = 3$

Suppose you want to study an ellipse with semi-major axis $a = 5$ and semi-minor axis $b = 3$. Using the Eccentricity Calculator functionality of the conic tool, you would input these values and select “ellipse (horizontal)”. The calculator then returns:

• Eccentricity: $e = \sqrt{1 - (9/25)} = \sqrt{16/25} = 0.8$
• Linear eccentricity: $c = \sqrt{25 - 9} = 4$
• Semi-latus rectum: $l = 9/5 = 1.8$
• Focal parameter: $p = 9 / 4 = 2.25$

The same process works for hyperbolas, parabolas, and circles by simply changing the type and adjusting $a$ and $b$ accordingly.

How to Use the Conic Section Calculator

1. Select the conic type (circle, ellipse, parabola, or hyperbola).
2. For ellipses and hyperbolas, indicate the orientation (horizontal or vertical).
3. Enter the required parameters $a$ and $b$ (for parabolas only $a$ is needed).
4. Click “calculate.” The tool displays the standard equation, eccentricity, linear eccentricity, semi-latus rectum, and focal parameter.

All calculations use the formulas listed above, ensuring quick and accurate results.

Real-World Applications

Conic sections have far‑reaching applications beyond the classroom:

• Elliptical orbits – planets and moons follow elliptical paths (Kepler’s first law). Earth’s orbit has an eccentricity of only about 0.017, while comets often have much higher values.
• Parabolic reflectors – satellite dishes and car headlights exploit the reflective property of parabolas: parallel rays are focused onto a single point, enabling efficient signal reception and light projection.
• Hyperbolic trajectories – spacecraft that pass by a planet without entering orbit follow hyperbolic paths. The same geometry is used in some navigation systems.

From astronomy to antenna design, conic sections are essential. A Conic Sections Calculator puts all the necessary formulas in one place, making analysis straightforward.