Topics in a Geometry Course

To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page.

General

Congruent	(1) A property of two geometric figures if one can be transformed into the other via a distance preserving map. (2) A property of two integers whose difference is divisible by a given modulus.
Geometry	The branch of mathematics that studies figures, objects, and their relationships to each other. This contrasts with algebra, which studies numerical quantities and attempts to solve equations.
Similar	A property of two figures whose corresponding angles are all equal and whose distances are all increased by the same ratio.

High-Dimensional Solids

High-Dimensional Solid:	A generalization of a solid such as a cube or a sphere to more than three dimensions.
Hypercube:	The generalization of a cube to more than three dimensions.
Hyperplane:	The generalization of a plane to more than two dimensions.
Hypersphere:	The generalization of a sphere to more than three dimensions.
Polytope:	A generalization of a polyhedron to more than three dimensions.

Plane Geometry

Acute Angle:	An angle that measures less than 90 degrees.
Altitude:	A line segment from a vertex of a triangle which meets the opposite side at a right angle.
Angle:	The amount of rotation about the point of intersection of two lines or line segments that is required to bring one into correspondence with the other.
Area:	The amount of material that would be needed to "cover" a surface completely.
Circle:	The set of points in a plane that are equidistant from a given center point.
Circumference:	The perimeter of a circle.
Collinear:	Three or more points are said to be collinear if they lie on the same straight line.
Complementary Angles:	A pair of angles whose measures add up to 90 degrees.
Diameter:	(1) The maximum distance between two opposite points on a circle. (2) The maximum distance between two antipodal points on a sphere.
Geometric Construction:	A construction of a geometric figure using only straightedge and compass. Such constructions were studied by the ancient Greeks.
Golden Ratio:	Generally represented as φ. Given a rectangle having sides in the ratio 1:φ, partitioning the original rectangle into a square and new rectangle results in the new rectangle having sides with the ratio 1:φ. φ is approximately equal to 1.618.
Golden Rectangle:	A rectangle in which the ratio of the sides is equal to the golden ratio. Such rectangles have many visual properties and are widely used in art and architecture.
Hypotenuse:	The longest side of a right triangle (i.e., the side opposite the right angle).
Midpoint:	The point on a line segment that divides it into two segments of equal length.
Obtuse Angle:	An angle that measures greater than 90 degrees and less than 180 degrees.
Parallel:	In two-dimensional Euclidean space, two lines that do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines.
Perimeter:	The length around the boundary of a closed two-dimensional region. The perimeter of a circle is called its circumference.
Perpendicular:	Two lines, vectors, planes, etc. that intersect at a right angle.
Pi:	The ratio of the circumference of a circle to its diameter. It is equal to 3.14159....
Plane Geometry:	The portion of geometry dealing with figures in a plane, as opposed to solid geometry.
Point:	A zero-dimensional mathematical object that can be specified in n-dimensional space using n coordinates.
Radius:	The distance from the center of a circle to its perimeter, or from the center of a sphere to its surface. The radius is equal to half the diameter.
Supplementary Angles:	For a given angle, the angle that when added to it totals 180 degrees.
Triangle Inequality:	The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Polygons

Equilateral Triangle:	A triangle in which all three sides are of equal length. In such a triangle, the angles are all equal as well.
Isosceles Triangle:	A triangle with (at least) two sides of equal length, and therefore also with (at least) two equal angles.
Parallelogram:	A quadrilateral with opposite sides parallel and therefore opposite angles equal.
Polygon:	A two-dimensional figure that consists of a collection of line segments, joined at their ends.
Quadrilateral:	A four-sided polygon.
Rectangle:	A quadrilateral with opposite sides of equal lengths, and with four right angles.
Regular Polygon:	A polygon in which the sides are all the same length and the angles all have the same measure.
Right Triangle:	A triangle that has a right angle. The Pythagorean Theorem is a relationship among the sides of a right triangle.
Square:	A polygon with four sides of equal length and at right angles to each other.
Trapezoid:	A quadrilateral with two sides parallel.
Triangle:	A three-sided (and three-angled) polygon.

Solid Geometry

Cone:	A pyramid with a circular cross section.
Convex Hull:	For a set of points S, the intersection of all convex sets containing S.
Cross Section:	The plane figure obtained by a solid's intersection with a plane.
Cube:	A Platonic solid consisting of six equal square faces that meet each other at right angles. It has 8 vertices and 12 edges.
Cylinder:	A solid of circular cross section in which the centers of the circles all lie on a single line.
Dodecahedron:	A Platonic solid consisting of 12 pentagonal faces, 30 edges, and 20 vertices.
Icosahedron:	(1) A 20-sided polyhedron. (2) The Platonic solid consisting of 20 equilateral triangles.
Octahedron:	A Platonic solid consisting of eight triangular faces, eight edges, and six vertices.
Platonic Solid:	A convex solid composed of identical regular polygons. There are exactly five Platonic solids.
Polyhedron:	A three-dimensional solid that consists of a collection of polygons, joined at their edges.
Prism:	A polyhedron with two congruent polygonal faces and with all remaining faces parallelograms.
Pyramid:	A polyhedron with one face (known as the "base") a polygon and all the other faces' triangles meeting at a common polygon vertex (known as the "apex").
Solid Geometry:	That portion of geometry dealing with solids, as opposed to plane geometry.
Sphere:	The set of all points in three-dimensional space that are located at a fixed distance from a given point.
Surface:	A two-dimensional piece of three-dimensional space.
Surface Area:	The area of a surface that lies in three-dimensional space, or the total area of all surfaces that bound a solid.
Tetrahedron:	A Platonic solid consisting of four equilateral triangles.
Volume:	The amount of space occupied by a closed three-dimensional object. MathWorld