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) In geometry, two figures are said to be congruent if one can be transformed into the other via a distance preserving map. (2) In number theory, two integers are said to be congruent if their difference is divisible by a given modulus. |
Geometry |
Geometry is 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. |
Linear Function |
(1) In geometry, a linear function is a function of the form y = m x + b. (2) In calculus, a linear function is a function whose graph is a straight line. (3) In linear algebra, a linear function is a function that satisies f(x + y) = f(x) + f(y) and f(a x) = a f(x). |
Similar |
In geometry, two figures are said to be similar when their corresponding angles are all equal and their distances are all scaled by the same ratio. |
High-Dimensional Solids
High-Dimensional Solid: |
A high-dimensional solid generalization of a solid such as a cube or a sphere to more than three dimensions. |
Hypercube: |
A hypercube is a generalization of a cube to more than three dimensions. |
Hyperplane: |
A hyperplane is a generalization of a plane to more than two dimensions. |
Hypersphere: |
A hypersphere is a generalization of a sphere to more than three dimensions. |
Polytope: |
A polytope is a generalization of a polyhedron to more than three dimensions. |
Plane Geometry
Acute Angle: |
An acute angle is an angle that measures less than 90 degrees. |
Altitude: |
An altitude of a triangle is a line segment from one of its vertices which meets the opposite side at a right angle. |
Angle: |
An angle is a measure of 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: |
Area is a measure of the amount of material that would be needed to "cover" a surface completely. |
Circle: |
A circle is the set of points in a plane that are equidistant from a given center point. |
Circumference: |
The circumference of a circle is the length of its perimeter. |
Collinear: |
Three or more points are said to be collinear if they lie on the same straight line. |
Complementary Angles: |
Complementary angles are a pair of angles whose measures add up to 90 degrees. |
Diameter: |
(1) In plane geometry, the diameter is the straight-line distance between two points opposite one another across the center of a circle. (2) In solid geometry, the diameter is the straight-line distance between two antipodal points on a sphere. |
Geometric Construction: |
A geometric construction is a construction of a geometric figure using only straightedge and compass, as originally studied by the ancient Greeks. |
Golden Ratio: |
The golden ratio φ is a mathematical constant obtained as the ratio of longest-to-shorted side lengths in a rectangle constructed so that when it is partitioned into a square and new rectangle, the new rectangle has the same ratio of side lengths as the original. The golden ratio has value of approximately 1.618. |
Golden Rectangle: |
A golden rectangle is a rectangle in which the ratio of the longest to shortest sides is equal to the golden ratio, give approximately by 1.618. Such rectangles appear particularly promnentl in art and architecture. |
Hypotenuse: |
The hypotenuse is the longest side of of a right triangle, which is always opposite the right angle. |
Midpoint: |
The midpoint is the point on a line segment that divides it into two segments of equal length. |
Obtuse Angle: |
An obtuse angle is 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 are said to be parallel. 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 perimeter is a 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 are said to be perpendicular. |
Pi: |
Pi is the mathematical constant defined as the ratio of the circumference of a circle to its diameter with value of approximately 3.14159. |
Plane Geometry: |
Plane geometry is the portion of geometry dealing with figures in a plane, as opposed to solid geometry. |
Point: |
A point is a zero-dimensional mathematical object that can be specified in n-dimensional space using n coordinates. |
Radius: |
The radius of a circle is the distance from its center to its circumference or from the center of a sphere to its surface. The radius is equal to half the diameter. |
Supplementary Angles: |
Supplementary angles are pairs of angles whose measures sum to 180 degrees. |
Triangle Inequality: |
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. |
Polygons
Decagon: |
A decagon is a 10-sided polygon. |
Dodecagon: |
A dodecagon is a 12-sided polygon. |
Equilateral Triangle: |
A triangle in which all three sides are of equal length. In such a triangle, the angles are all equal as well. |
Equilateral Triangle: |
A triangle in which all three sides are of equal length. In such a triangle, the angles are all equal as well. |
Hendecagon: |
A hendecagon is an 11-sided polygon. |
Heptagon: |
A heptagon is a 7-sided polygon. |
Hexagon: |
A hexagon is a 6-sided polygon. |
Isosceles Triangle: |
An isosceles triangle is a triangle having (at least) two sides of equal length, and therefore also with (at least) two equal angles. |
Nonagon: |
A nonagon is a 9-sided polygon. |
Octagon: |
An octagon is an 8-sided polygon. |
Parallelogram: |
A parallelogram is a quadrilateral with opposite sides parallel and therefore opposite angles equal. |
Pentagon: |
A pentagon is a 5-sided polygon. |
Polygon: |
A polygon is a two-dimensional figure that consists of a collection of line segments, joined at their ends. |
Quadrilateral: |
A quadrilateral is a four-sided polygon. |
Rectangle: |
A rectangle is a quadrilateral with opposite sides of equal length and with four right angles. |
Regular Polygon: |
A regular polygon is a polygon in which the sides are all the same length and the angles all have the same measure. |
Right Triangle: |
A right triangle is a triangle that has a right angle. The Pythagorean Theorem is a relationship among the sides of a right triangle. |
Square: |
A square is a polygon with four sides of equal length and at right angles to each other. |
Trapezoid: |
A trapezoid is a quadrilateral with two sides parallel. |
Triangle: |
A triangle is a three-sided (and three-angled) polygon. |
Solid Geometry
Cone: |
A cone is a pyramid with a circular cross section. |
Convex Hull: |
The convex hull of a set of points S is the intersection of all convex sets containing S. |
Cross Section: |
The cross section of a solid is a plane figure obtained by the intersection of that solid with a plane. |
Cube: |
The cube is the Platonic solid comprised of six equal square faces that meet each other at right angles, eight vertices, and twelve edges. |
Cylinder: |
A cylinder is a solid of circular cross section in which the centers of the circles all lie on a single line. |
Dodecahedron: |
(1) A general dodecahedron is any polyhedron having 12 faces. (2) The regular dodecahedron is the Platonic solid comprised of 12 pentagonal faces, 20 vertices, and 30 edges. |
Icosahedron: |
(1) A general icosahedron is any polyhedron having 20 faces. (2) The regular icosahedron is the Platonic solid comprised of 20 equilateral triangles, 12 vertices, and 30 edges. |
Octahedron: |
(1) A general octahedron is any polyhedron having eight faces. (2) The regular octahedron is the Platonic solid comprised of eight equilateral triangular faces, eight edges, and six vertices. |
Platonic Solid: |
The Platonic solids are the five convex solids composed of identical regular polygons. |
Polyhedron: |
A polyhedron is a three-dimensional solid that consists of a collection of polygons, joined at their edges. |
Prism: |
A prism is a polyhedron with two congruent polygonal faces and with all remaining faces parallelograms. |
Pyramid: |
A pyramid is 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: |
Solid geometry is that portion of geometry dealing with solids, as opposed to plane geometry. |
Sphere: |
A sphere is the set of all points in three-dimensional space that are located at a fixed distance from a given point. |
Surface: |
A surface is a two-dimensional piece of three-dimensional space. |
Surface Area: |
Surface area is the area of a surface that lies in three-dimensional space, or the total area of all surfaces that bound a solid. |
Tetrahedron: |
(1) A general tetrahedron is any polyhedron having four faces. (2) The regular tetrahedron is the Platonic solid comprised of four equilateral triangles, four vertices, and six edges. |
Volume: |
In mathematics, volume is the amount of space occupied by a closed three-dimensional object. |
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