Glossary of Geometrical Terms

Have you ever imagined a geometrical shape, and then had trouble describing it?  Many shapes have names, but they're fairly obscure, so a glossary might help.

General notes: planes, lines, cones, and cylinders extend infinitely in both (or all) directions.  A cone, in particular, is not shaped like an "ice cream cone" but rather like two such objects, attached at their pointy ends, and extending infinitely in both directions.  Shapes that are bounded are called "line segments", "frustums", "solid cones", "prisms", etc.  So read the definitions carefully.  (See also thesaurus and mathworld)

acute angle - an angle whose measure is greater than 0 but less than 90 degrees

adjacent angles - 2 nonstraight and nonzero angles that have a common side in the interior of the angle formed by the noncommon sides

alternate exterior angles - exterior angles on alternate sides of the transversal (not on the same parallel line)

alternate interior angles - interior angles on alternate sides of the transversal (not on the same parallel line)

angle - the union of 2 rays that have the same endpoint; measured in degrees or radians; the five types of angles are zero, acute, right, obtuse, and straight

angle bisector - a ray that is in the interior of an angle and forms two equal angles with the sides of that angle

annulus - a ring; the area bounded by two concentric circles

apothem - in a regular polygon, the perpendicular distance from the center to a side; in a circle with a chord, the distance from the midpoint of a chord to the circle's center

arbelos - a plane region bounded by three mutually tangent semicircles whose diameters are collinear.

arc - a continuous portion (as of a circle or ellipse) of a curved line

arc length - the distance between an arc's endpoints along the path of the circle

area - the surface enclosed within a closed plane figure; the measure of the surface, expressed in equivalent square units, such as square inches.

axis - a straight line about which a body or a geometric figure rotates or may be supposed to rotate; a straight line with respect to which a body or figure is symmetrical -- called also axis of symmetry; one of the reference lines of a coordinate system

ball - a "solid sphere"; the interior of a sphere (open ball); a sphere and its interior (closed ball);
 

Borromean rings - A set of three rings joined in such a way that no pair is interlinked, but the three cannot be separated. 

bundle - the set of planes through a point, in projective geometry.

cage - regular graph that has as few vertices as possible for its girth.  Formally, an (r,g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g.  The (r,3)-cage is a complete graph K_r+1 on r+1 vertices, and the (r,4)-cage is a complete bipartite graph K_r,r on 2r vertices.   Other cages listed in Wikipedia include the Moore graphs: (3,5)-cage: Petersen graph, (3,6)-cage: Heawood graph, (3,8)-cage: Tutte�Coxeter graph, (7,5)-cage: Hoffman�Singleton graph.

Cartesian plane - a coordinate plane

central angle of a circle - an angle whose vertex is the center of the circle

cevian - a line segment connecting a vetex of a triangle with a point on the opposite side of the triangle.  The condition for three general Cevians from the three vertices of a triangle to concur (at a cevian point) is known as Ceva's theorem.

cevian point - the point of concurrence of three cevians

cevian triangle - Given a point P and a triangle ABC, the Cevian triangle A'B'C' is defined as the triangle composed of the endpoints of the cevians through the Cevian point P.

chord - In general, a straight line joining two points on a curve; often, chord is used to mean a straight line segment joining and included between two points on a circle; 

chord of a circle - a segment whose endpoints are on a circle

circle - a closed plane curve every point of which is equidistant from a fixed point, called the center (the center is not part of the circle)

circular - having the shape of a circle; a circular cylinder is one in which its defining shape is a circle, and most often is used to mean a right circular cylinder; a circular cone is one in which its defining shape is a circle, and most often is used to mean a right circular cone;

complementary angles - 2 angles whose measures, when added together, equal 90 degrees

complex polygon - a polygon having at least one pair of intersecting edges; as opposed to simple; reflex polygon.

concave set - a set of points in which not all segments connecting points of the set lie entirely in the set; nonconvex.

concentric - having the same center, as concentric circles; having the same axis, as concentric cylinders

cone - in general, a cone is the locus of (i.e. surface traced by) the surface formed by lines joining every point of the boundary of a fixed planar closed curve (the base) to a common vertex; commonly, a right circular cone; a "solid cone" is a solid (or the space) bounded by the planar closed curve, called the base, and the line segments connecting the base to the vertex.  The area of a solid cone is (1/3)Ah, where A is the area of the base, and h is the height.

congruent - equal; exactly the same (size, shape, etc.).  Two figures are congruent when one is the image of the other under a reflection or composite of reflections

conic section - the figure formed by the intersection of a plane and a (right circular) cone

consecutive sides - sides of a polygon that share an endpoint

consecutive vertices - endpoints of a single side of a polygon

constant width - of a bounded figure: having the same width in every direction.  A Reuleaux triangle is an example of a figure of constant width.  Other examples include the shape formed by an equilateral triangle of side s with circles of radius r around each vertex, and arcs of radius s+r centered on the "far" vertices that join these circles.  The same construction can be made with an equilateral (but not necessarily equiangular n-sided star, with n an odd number.   See Cut-the-knot Shapes of Constant Width and  Cut-the-knot Barbier's Theorem for more information.

convex set - a set of points in which all segments connecting points of the set lie entirely in the set; There are three things one can do to see if a figure is convex - look for "dents", extend the segments (they shouldn't enter the figure), and connect any two points within the figure with a segment (if any part of the segment lies outside the figure, it's concave)

corresponding angles - any pair of angles in similar locations with respect to a transversal

coterminal angles - two angles that have the same terminal side; two angles that differ by an integer multiple of 360 degrees.

cyclic quadrilateral - a quadrilateral whose four vertices lie on a circle.

cyclogon - a curve traced by any vertex of a regular polygon that rolls without sliding on a straight line.  The area under one arch of the cyclogon (pink, in this diagram) is equal to the area of the polygon plus twice the area of the circle that circumscribes the polygon.  If the "vertices" of the cyclogon were linked by straight lines instead of curved ones, then the area under one arch of that figure would be three times the area of the regular polygon.
 

cycloid - a curve traced by any point on the circumference of a circle that rolls without sliding on a straight line.  The area under one arch of the cycloid (blue, in the diagram) is exactly three times the area of the circle.  A curtate cycloid is a curve generated in the same way, except that the point is inside the circle rather than on the circle.  A prolate cycloid is a curve generated in the same way, except that the point is rigidly fixed outside the circle.  All cycloids, including curtate and prolate cycloids, are trochoids.

cylinder - in general, a cylinder is the locus of (i.e. surface traced by) a straight line moving parallel to a fixed straight line and intersecting a fixed planar closed curve (the base); commonly, a right circular cylinder

diagonal - a line segment whose endpoints are 2 nonconsecutive vertices of a polygon.

diameter of a circle (or sphere) - the line segment whose endpoints are points on a circle (or sphere) that contains the center of the circle as its midpoint; the length of that line segment

disk - the interior of a circle (an open disk); a circle and its interior (a closed disk)

ellipsoid - a squashed or stretched sphere in which each of the three axes can be of different lengths.  (Contrast to a spheroid, in which two of the three axes have the same length.)  An ellipsoid has the equation x²/a² + y²/b² + z²/c² = 1

epicycloid - a curve generated by the motion of a point on the circumference of a circle that rolls externally, without slipping, on a fixed circle. Equation: x = (a + b) cos(θ) − b cos[(a + b)θ/b] and y = (a + b) sin(θ) − b sin[(a + b)θ/b]. . . . . . . diagram needs rework.

epitrochoid - a curve generated by the motion of a point rigidly connected with -- but not on the circumference of -- a circle that rolls externally, without slipping, on a fixed circle.
 

equianglular - having angles of the same measure

equidistant - the same distance from something

equilateral - having sides that are equal in length

even node - a node that has an even number of arcs

exterior angle of a polygon - angle formed by one side and the extension of an adjacent side.

exterior angles - angles outside of two lines cut by a transversal.

face - a polygonal region of a surface

frustum - the basal part of a solid cone or pyramid formed by cutting off the top by a plane parallel to the base; the part of a solid intersected between two usually parallel planes

graph - abstract representation of a set of objects (nodes) where some pairs of the objects are connected by links (edges).  See also cage, snark, and lattice.  Graphs listed in Wikipedia include Bull, Chv�tal, Clebsch, Desargues,D�rer, Folkman, Foster, Franklin, Frucht, Gray, Gr�tzsch, Hall�Janko, Harries, Harries-Wong, Heawood, Higman�Sims, Hoffman�Singleton, McGee, M�bius�Kantor, Null, Pappus, Petersen, Rado, Shrikhande, Tietze's, Tutte�Coxeter; Cages: Balaban 10-cage, Balaban 11-cage; Snarks: Flower, Szekeres; Lattices: Young�Fibonacci

great circle - the circle formed by the intersection of a sphere and a plane that contains its center and that divides the sphere into two hemispheres

hemisphere - half of a sphere

hypocycloid - a curve generated by the motion of a point on the circumference of a circle that rolls internally (on the concave side), without slipping, on a fixed circle.

hypotenuse - the side opposite the right angle in a right triangle

hypotrochoid - a curve generated by the motion of a point rigidly connected with -- but not on the circumference of -- a circle that rolls internally (on the concave side), without slipping, on a fixed circle.

icosahedron - a 20-sided solid; each side is in the shape of a triangle

included angle - the interior angle made by two consecutive sides of a polygon

included side - the side between two consecutive angles in a polygon

initial side - the side that the measurement of an angle starts from

interior angles - angles between two lines cut by a transversal

isosceles trapezoid - a trapezoid that has a pair of equiangular base angles.  (classification of quadrilaterals)

isosceles triangle - a triangle having two sides of equal length

kite - a quadrilateral that has two distinct pairs of consecutive equilateral sides

klein bottle - the 3-dimensional analog of the Mobius Strip.  The figure is impossible in three dimensions, because one part of its surface passes through another part of its surface without touching it.  More on Wikipedia and Mathworld.

lattice - a regular geometrical arrangement of points over an area or in space

lattice graph - a graph whose nodes form a lattice

lattice points - points in the coordinate plane having integer coordinates

leg of a right triangle - a side of a right triangle that include the 90 degree angle

lemniscate - a figure that looks like the "infinity" symbol; a lazy 8; an 8 on its side.  This is a cross-section of a torus (a toric section) taken on a plane tangent to the inner circle of the torus.  In polar coordinates, the equation of a lemniscate is r=sqrt(cos(2θ)).  See mathworld.

linear pair of angles - two supplementary adjacent angles whose noncommon sides form a line

locus - the set of points that satisfy a given condition or equation

major arc - an arc whose endpoints form an angle over 180 degrees with the center of the circle; as opposed to minor arc;

major axis (ellipse) - the line segment containing the foci of the ellipse and whose endpoints are on the ellipse.

measure of an angle - the size of an angle, typically measured in degrees

measure of an arc - The measure of the central angle subtended by an arc.

median - the segment connecting the vertex of an angle in a triangle to the midpoint of the side opposite it

midpoint - the point M on line segment AB such that AM = AB.

minor arc - an arc whose endpoints form an angle less than 180 degrees with the center of the circle.

minor axis (ellipse) - the perpendicular bisector of the line segment whose endpoints are the foci of the ellipse.

M�bius Strip (sometimes spelled Moebius or Mobius) - a surface having only one side and only one edge.  It can be formed from a rectangle by giving it a half twist, then gluing the edges together.

mouhefanggai - the solid formed as intersection of two perpendicular right circular cylinders of equal diameter.  This is also known as a Steinmetz Solid or a bicylinder.  Cross sections of a mouhefanggai are square if they are taken parallel to the plane of the axes of the two intersecting cylinders.

mouhefanggoid

mouhefanggoid - the solid formed as the intersection of two perpendicular cylinders of elliptical cross-section and equal major or minor axes aligned.  The multi-colored diagram to the right illustrates half a mouhefanggoid formed by the intersection of a circular cylinder whose axis is parallel to the y axis with an elliptical cylinder whose axis is parallel to the x axis.  The major axis of the ellipse is twice the minor axis, so cross-sections of this half- mouhefanggoid taken parallel to the x-y plane are squares.

n-gon - a polygon having n sides.

nonagon - a nine-sided polygon.

nonconvex set - a set of points in which not all segments connecting points of the set lie entirely in the set; concave

non-Euclidean geometry - geometry in which given a line and a point on the line, there is either no line (elliptic geometry) or infinitely many lines (hyperbolic geometry) parallel to the given line that pass through the given point.

non-included side - the side of a triangle that is not included by 2 given angles.

normal - perpendicular.  In particular, a line is normal to a plane

oblique prism or cylinder - a non-right prism or cylinder.

obtuse angle - an angle whose measure is greater than 90 but less than 180 degrees.

octagon - an eight-sided polygon

opposite rays - two rays having a common endpoint that form a line.

parallelepiped - a prism whose opposite faces are all parallelograms and congruent (in pairs)

parallelogram - a quadrilateral such that both pairs of opposite sides parallel

pencil - the set of lines through a point, in projective geometry.

pentagon - a five-sided polygon

perimeter of a polygon - the sum of the lengths of the sides of the polygon.

perpendicular bisector - the bisector of a segment perpendicular to it.

perpendicular lines (line segments; rays) - 2 lines (line segments; rays) that meet at a 90 degree angle

plane - a flat surface of such nature that a straight line joining two of its points lies wholly in the surface; a plane figure (or a planar figure) is a figure that is entirely within a plane

polygon - a union of 3 or more line segments where each segment intersects 2 other segments, one at each endpoint.

polygonal region - the union of a polygon and its interior.

polyhedron - a closed surface formed by polygonal plane faces, connected at the edges;  a "solid polyhedron" is a solid (or the space) enclosed by a polyhedron.

pyramid - a polyhedron having for its base a polygon and for faces triangles with a common vertex (the vertex is not in the plane of the base); a solid pyramid is a solid cone having a polygonal base, so its volume is (1/3)Ah.

quadrangle - a four-sided polygon; another name for quadrilateral

quadrilateral - a four-sided polygon; special quadrilaterals include rhombus, parallelogram, square, rectangle, trapezoid, isosceles trapezoid, kite.

radius - of a circle or sphere, the line segment (or distance) from the center to the circle or sphere; of a regular polygon, the line segment from the center to a vertex, which is the radius of the circumscribed circle.

ratio of similitude - the ratio of a line segment in figure B to the corresponding line segment in figure A; the ratio of the length of an image to the length of the preimage; ratio of proportionality.

ray - a one-dimensional figure that consists of one endpoint A, one point B, all of the points on AB, and all points for which B is between them and A

rectangle - a quadrilateral whose angles are all right angles

rectangular solid - the union of a parallelepiped (box) and its interior.

reference angle - the angle of less than 360 degrees that corresponds to an angle of over 360 degrees; In order to get the reference angle, you must subtract 360 degrees from the given angle until there is less than 360 degrees left.

reflex polygon - a polygon for which 2 or more of its sides intersect each other; complex polygon.

regular polygon - a convex polygon whose angles and sides are all congruent.

regular pyramid - a pyramid whose base is a regular polygon.

rhombus - a quadrilateral having four equilateral sides; equivalently, a parallelogram having four equilateral sides.

right - having a measure of 90 degrees: a right angle; perpendicular; having a right angle: a right triangle; a right cylinder is one in which the straight lines that compose it are normal to its base; a right cone (meaningful only when the base has a center) is one in which the straight line connecting the vertex to the center of the base is normal to the base; a right pyramid (meaningful only when the base has a center) is one in which the straight line connecting the center of the base is normal to the base; a right prism is a prism whose faces meet at right angles.

roulette - a curve traced by a point on a radius or an extension of the radius of a given curve that rolls, without slipping, on a curve, another circle, or a straight line.  Cf: trochoid, in which the given curve is a circle.

sagitta - the radius minus the apothem

scalene triangle - a triangle having no equilateral sides.

secant to a circle - a line that intersects the circle in two points

section - the intersection of a three-dimensional figure with a plane. e.g. a conic section is the intersection of a cone with a plane.

sector - the interior points of a circle bounded by an arc and two radii

segment (of a circle) - the interior points of a circle bounded by a circular arc and a chord

semicircle - an arc whose endpoints are a diameter of the circle.

septagon - a seven-sided polygon.

simple polygon - a polygon whose edges don't cross one another; as opposed to complex;

skew lines - non-coplanar lines.  Lines that are neither parallel nor intersecting.

slant height - the length of a lateral edge of a cone

snark - is a connected, bridgeless cubic graph with chromatic index equal to 4.  In other words, it is a graph in which every vertex has three neighbors, and the edges cannot be colored by three colors without two edges of the same color meeting at a point.  Snarks listed in Wikipedia include Petersen graph, Flower snark, Szekeres snark, Tietze's graph, Blanu�a snarks, Descartes' snark, double star snark.

solid - the union of the surface and the region of space enclosed by a 3-D figure; examples: conic solid, cylindric solid, rectangular solid

solid geometry - the study of figures in three-dimensional space.

sphere - a closed surface in three-dimensional space, every point of which is equidistant from a fixed point, called the center (the center is not part of the sphere)

sphericon - a "cone with a twist".  It is created by starting with two identical right circular solid cones (of a particular shape) joined at the base, then slicing them along a plane through both vertices, and then rotating one of the two pieces a quarter turn, and gluing them together.

spheroid - an ellipsoid in which two of the three axes are equal.  (Contrast to an ellipsoid, in which all three axes may have different lengths.)  A spheroid has the equation (x²+y²)/a² + z²/c² = 1.  An oblate spheroid has a polar axis that is shorter than the diameter of the equatorial circle, and can be formed by rotating an ellipse about its minor axis.  A prolate spheroid has a polar axis that is longer than the diameter of the equatorial circle, and can be formed by rotating an ellipse about its major axis.

square - an equilateral and equiangular quadrilateral

straight angle - an angle whose measure is 180 degrees; an angle whose sides are collinear.

straight line - a collection of points of such a nature that if one picks any three of them, A, B, and C, the distances AB, AC, and BC will be such that the sum of two of them will equal the third.

supplementary angles - two angles whose measures sum to 180 degrees.

surface - the boundary of a 3-D figure.

symmetry diagonal - the diagonal of a kite (a kind of quadrilateral) that perpendicularly bisects the other.

symmetry line - the line of reflection in a reflection-symmetric figure.

terminal side - the side that the measurement of an angle ends at.

tesselation - a covering of a plane with congruent copies of the same region with no holes or overlaps.

tesseract - the four-dimensional analog of the cube; a 4-D hypercube.

tetragon - a four-sided polygon; another (rather unusual) word for quadrilateral

three dimensional figure - a set of points in space; examples: box, cone, cylinder, parallelpiped, prism, pyramid, regular pyramid, right cone, right cylinder, right prism, sphere.

torus - a surface or solid shaped like a doughnut and formed by revolving a circle about a line in its plane that does not intersect it.

transversal - a line that intersects two other (usually parallel) lines.

trapezoid - a quadrilateral that has at least one pair of parallel sides

triangle - a polygon having three sides.

trochoid - a curve traced by a point on a radius or an extension of the radius of a circle that rolls, without slipping, on a curve, another circle, or a straight line. Equation: x = aθ − b sin θ, y = a − b cos θ;  any of a class of curves including the cycloid, epicycloid, epitrochoid, hypocycloid, and hypotrochoid.  Cf: roulette, in which the rolling circle is replaced by any curve.

vertex angle - the angle formed by the equilateral sides of an isosceles triangle.

vertex of a polygon - an endpoint of a segment in a polygon.

vertex of an angle - the common endpoint of the two rays.

vertical angles - 2 angles that share a common vertex and whose sides form 2 lines.

width - of a bounded shape: the distance between two parallel lines, each touching the boundary of the shape but not its interior.  This is called "the width of the shape in the direction of the lines".  Note that figures of constant width (such as a Reuleaux triangle) have the same width in every direction.

zero angle - an angle whose measure is 0. In a zero angle, both the initial and terminal sides are the same.

Internet references

Cut-the-knot Shapes of Constant Width has an applet showing how the Reuleaux triangle has constant width.  A "Reuleaux drill" can't make a square hole, but it can make a hole that's almost square, carving out almost 99% of the area of the square.

Cut-the-knot Barbier's Theorem for more applets and a general method of making shapes of constant width.  Barbier's Theorem states that the circumference of any shape of constant width is equal to πD, where D is the width of the shape.

Number Patterns Fun with Curves and Topology for an animated Reuleaux Triangle.

Mathematical Recreations describes the "cone with a twist", i.e. the sphericon.

Model of a half mouhefanggoid shape with a circular base that has square cross sections.

Thinkquest: Geometry Glossary

Related pages in this website

Number Theory Glossary

Statistics Glossary

Topology Glossary

The webmaster and author of this Math Help site is Graeme McRae.