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)

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

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.

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; 

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;

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 

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.

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

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.

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.

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

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.

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 closed plane figure bounded by straight lines; 

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;
A pyramid is a solid cone with a polygonal base, so its volume is (1/3)Ah.

quadrilateral - a polygon of four sides

radius - of a circle, the distance from the center to the circle; of a regular polygon, the distance from the center to a vertex, which is the radius of the circumscribed circle

right - having a 90 degree angle; perpendicular;
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.

segment - a portion of a disk bounded by a circular arc and the chord that cuts it.

sagitta - the radius minus the apothem

sector - A wedge bounded by an arc and the line segments connecting the arc's center to its endpoints

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

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.

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.

triangle - a polygon having three sides.

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.

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 pD, 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.

Related Pages in this Website

Number Theory Glossary

Statistics Glossary

Topology Glossary

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