TOPICS

# 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.