What is a conic section?

In geometry, a conic section is any curve produced when a plane intersects a right circular cone. The angle of the plane relative to the cone determines the type of conic section. The conic sections are circles , ellipses , hyperbolas , and parabolas . When the plane passes through the apex of the cone, special cases arise, resulting in a single point or one or two intersecting straight lines.

How are conic sections formed?

Conic sections are curves formed when a plane intersects a right circular cone. The type of curve depends on the angle of the plane relative to the cone. The possibilities are: * If the plane is parallel to the cone's surface, the result is a parabola . * If the plane intersects the cone at an angle, the result is an ellipse . A circle is a special case of an ellipse where the plane is perpendicular to the cone's axis. * If the plane intersects both halves of the cone, the result is a hyperbola . There are also special cases where the plane passes through the apex of the cone, resulting in a single point, one straight line, or two intersecting straight lines.

What are the main types of conic sections?

In geometry, a conic section is a curve produced by the intersection of a plane and a right circular cone. The main types of conic sections are the circle, ellipse, hyperbola, and parabola. Depending on the angle of the plane relative to the cone, the intersection can result in different shapes. If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola. There are also special cases where the plane passes through the apex of the cone, resulting in a single point, one straight line, or two intersecting straight lines.

How do the angles at which a plane intersects a cone create different conic sections?

A conic section is a curve created when a plane intersects a right circular cone. The type of curve formed depends on the angle of the plane relative to the cone.

Where do we see conic sections in real life or nature?

Conic sections appear in various real-world scenarios and in nature: Planetary Motion : Planets travel in elliptical orbits with the sun at one focus, according to Kepler's first law of planetary motion. Trajectories : Projectiles follow a parabolic path , a principle applied in ballistics, sports, and space exploration. Architecture : Parabolic arches are used in bridges and buildings due to their efficient distribution of forces. Optics : Parabolic reflectors are used in telescopes and satellite dishes to focus light or electromagnetic waves. Whispering Galleries : Some structures, like those in the U.S. Capital and St. Paul’s Cathedral, are ellipsoidal, allowing whispers to be heard at one focus from the other.

conic section

References & Edit History Quick Facts & Related Topics

Images

Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated.

Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape.

Illustration showing how a parabolic satellite dish antenna focuses incoming transmission signals onto a single point, with red lines representing signal paths reflecting off the dish surface toward the feedhorn. Insets provide a side view and a three-quarter view, highlighting the role of the paraboloid shape in signal reception.

A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked.

At a Glance

conic section summary

Quick Facts

Also called:: conic

Key People:: Blaise Pascal; Apollonius of Perga; Girard Desargues; Colin Maclaurin; Conon of Samos

Related Topics:: parabola; eccentricity; directrix; latus rectum; focus

On the Web:: Atlantic Canada Pressbooks Network - Algebra and Trigonometry OpenStax - Conic Sections in Polar Coordinates (Nov. 07, 2026)

See all related content

conic section

geometry

Also known as: conic

Written by

Christian Marinus Taisbak

Fact-checked by

Britannica Editors

Last updated

Nov. 7, 2026 •History

Top Questions

What is a conic section?
In geometry, a conic section is any curve produced when a plane intersects a right circular cone. The angle of the plane relative to the cone determines the type of conic section.

The conic sections are circles, ellipses, hyperbolas, and parabolas. When the plane passes through the apex of the cone, special cases arise, resulting in a single point or one or two intersecting straight lines.
6 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How are conic sections formed?
Conic sections are curves formed when a plane intersects a right circular cone. The type of curve depends on the angle of the plane relative to the cone.

The possibilities are:
* If the plane is parallel to the cone's surface, the result is a parabola.
* If the plane intersects the cone at an angle, the result is an ellipse. A circle is a special case of an ellipse where the plane is perpendicular to the cone's axis.
* If the plane intersects both halves of the cone, the result is a hyperbola.

There are also special cases where the plane passes through the apex of the cone, resulting in a single point, one straight line, or two intersecting straight lines.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
What are the main types of conic sections?
In geometry, a conic section is a curve produced by the intersection of a plane and a right circular cone. The main types of conic sections are the circle, ellipse, hyperbola, and parabola.

Depending on the angle of the plane relative to the cone, the intersection can result in different shapes. If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola. There are also special cases where the plane passes through the apex of the cone, resulting in a single point, one straight line, or two intersecting straight lines.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
What shapes do circles, ellipses, parabolas, and hyperbolas have?
In geometry, circles, ellipses, parabolas, and hyperbolas are types of conic sections. A conic section is any curve produced by the intersection of a plane and a right circular cone.

Depending on the angle of the plane relative to the cone, the intersection is a circle, an ellipse, a hyperbola, or a parabola.
- A parabola is an open curve.
- A hyperbola is a two-branched open curve.
- An ellipse is a closed curve, and a circle is a special case of an ellipse.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How do the angles at which a plane intersects a cone create different conic sections?
A conic section is a curve created when a plane intersects a right circular cone. The type of curve formed depends on the angle of the plane relative to the cone.
- If the plane is parallel to an element of the cone, a parabola is formed.
- If the plane is not parallel to the base, axis, or an element of the cone, an ellipse is formed.
- When the plane cuts both nappes of the cone, a hyperbola is the result.
- If the plane is perpendicular to the cone’s axis, the image of the circle will be a circle.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Where do we see conic sections in real life or nature?
Conic sections appear in various real-world scenarios and in nature:
- Planetary Motion: Planets travel in elliptical orbits with the sun at one focus, according to Kepler's first law of planetary motion.
- Trajectories: Projectiles follow a parabolic path, a principle applied in ballistics, sports, and space exploration.
- Architecture: Parabolic arches are used in bridges and buildings due to their efficient distribution of forces.
- Optics: Parabolic reflectors are used in telescopes and satellite dishes to focus light or electromagnetic waves.
- Whispering Galleries: Some structures, like those in the U.S. Capital and St. Paul’s Cathedral, are ellipsoidal, allowing whispers to be heard at one focus from the other.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

conic section, in geometry, any curve produced by the intersection of a plane and a right circular cone. Depending on the angle of the plane relative to the cone, the intersection is a circle, an ellipse, a hyperbola, or a parabola. Special (degenerate) cases of intersection occur when the plane passes through only the apex (producing a single point) or through the apex and another point on the cone (producing one straight line or two intersecting straight lines). See the figure.

The basic descriptions, but not the names, of the conic sections can be traced to Menaechmus (flourished c. 350 bc), a pupil of both Plato and Eudoxus of Cnidus. Apollonius of Perga (c. 262–190 bc), known as the “Great Geometer,” gave the conic sections their names and was the first to define the two branches of the hyperbola (which presuppose the double cone). Apollonius’s eight-volume treatise on the conic sections, Conics, is one of the greatest scientific works from the ancient world.

Analytic definition

Conics may also be described as plane curves that are the paths (loci) of a point moving so that the ratio of its distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant, called the eccentricity of the curve. If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola. See the figure.

Britannica Quiz

Numbers and Mathematics

Every conic section corresponds to the graph of a second degree polynomial equation of the form Ax² + By² + 2Cxy + 2Dx + 2Ey + F = 0, where x and y are variables and A, B, C, D, E, and F are coefficients that depend upon the particular conic. By a suitable choice of coordinate axes, the equation for any conic can be reduced to one of three simple r forms: x²/a² + y²/b² = 1, x²/a² − y²/b² = 1, or y² = 2px, corresponding to an ellipse, a hyperbola, and a parabola, respectively. (An ellipse where a = b is in fact a circle.) The extensive use of coordinate systems for the algebraic analysis of geometric curves originated with René Descartes (1596–1650). See History of geometry: Cartesian geometry.

Greek origins

The early history of conic sections is joined to the problem of “doubling the cube.” According to Eratosthenes of Cyrene (c. 276–190 bc), the people of Delos consulted the oracle of Apollo for aid in ending a plague (c. 430 bc) and were instructed to build Apollo a new altar of twice the old altar’s volume and with the same cubic shape. Perplexed, the Delians consulted Plato, who stated that “the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt for geometry.” Hippocrates of Chios (c. 470–410 bc) first discovered that the “Delian problem” can be reduced to finding two mean proportionals between a and 2a (the volumes of the respective altars)—that is, determining x and y such that a:x = x:y = y:2a. This is equivalent to solving simultaneously any two of the equations x² = ay, y² = 2ax, and xy = 2a², which correspond to two parabolas and a hyperbola, respectively. Later, Archimedes (c. 290–211 bc) showed how to use conic sections to divide a sphere into two segments having a given ratio.

Diocles (c. 200 bc) demonstrated geometrically that rays—for instance, from the Sun—that are parallel to the axis of a paraboloid of revolution (produced by rotating a parabola about its axis of symmetry) meet at the focus. Archimedes is said to have used this property to set enemy ships on fire. The focal properties of the ellipse were cited by Anthemius of Tralles, one of the architects for Hagia Sophia Cathedral in Constantinople (completed in ad 537), as a means of ensuring that an altar could be illuminated by sunlight all day.

Post-Greek applications

Conic sections found their first practical application outside of optics in 1609 when Johannes Kepler derived his first law of planetary motion: A planet travels in an ellipse with the Sun at one focus. Galileo Galilei published the first correct description of the path of projectiles—a parabola—in his Dialogues of the Two New Sciences (1638). In 1639 the French engineer Girard Desargues initiated the study of those properties of conics that are invariant under projections (see projective geometry). Eighteenth-century architects created a fad for whispering galleries—such as in the U.S. Capital and in St. Paul’s Cathedral in London—in which a whisper at one focus of an ellipsoid (an ellipse rotated about one axis) can be heard at the other focus, but nowhere else. From the ubiquitous parabolic satellite dish (see the figure) to the use of ultrasound in lithotripsy, new applications for conic sections continue to be found.