The branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. Geometry is one of the oldest branches of mathematics, with roots dating back to ancient civilizations.
The study of geometry based on Euclid's axioms and postulates. Forms the foundation of classical geometry dealing with points, lines, planes, and solids in flat space.
A location in space with no size, represented by coordinates. The most fundamental object in geometry, having position but no dimensions.
A straight one-dimensional figure extending infinitely in both directions. Defined by two points and having length but no width.
A flat, two-dimensional surface extending infinitely in all directions. Defined by three non-collinear points or a point and a normal vector.
A polygon with three edges and three vertices. The most basic polygon with rich properties including angle sum of 180 degrees.
Read More →The set of all points in a plane equidistant from a center point. Perfect symmetry with constant curvature.
Read More →A closed plane figure bounded by straight line segments. Includes triangles, quadrilaterals, pentagons, and regular polygons.
A polygon with four edges and four vertices. Includes squares, rectangles, parallelograms, trapezoids, and rhombuses.
The figure formed by two rays sharing a common endpoint. Measured in degrees or radians, fundamental to geometric analysis.
Two figures are congruent if one can be transformed into the other through rigid motions: translation, rotation, or reflection.
Two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal, sides proportional.
The study of geometry using a coordinate system. Also called coordinate geometry, it bridges algebra and geometry.
A system that uses numbers to uniquely determine the position of points. Cartesian, polar, and other coordinate systems.
A formula derived from the Pythagorean theorem to calculate the distance between two points in a coordinate plane.
A measure of the steepness of a line, calculated as the ratio of vertical change to horizontal change between two points.
Curves obtained by intersecting a cone with a plane: circles, ellipses, parabolas, and hyperbolas.
A curve where the sum of distances from any point to two fixed foci is constant. Planetary orbits follow elliptical paths.
A U-shaped curve where any point is equidistant from a fixed point (focus) and a fixed line (directrix).
A curve where the difference of distances from any point to two fixed foci is constant. Two separate branches.
The study of three-dimensional geometric figures: polyhedra, spheres, cylinders, cones, and their properties.
A three-dimensional solid with flat polygonal faces, straight edges, and sharp vertices. Includes prisms and pyramids.
A perfectly round three-dimensional object where every point on the surface is equidistant from the center.
The amount of three-dimensional space enclosed by a closed surface. Measured in cubic units.
The total area of the surface of a three-dimensional object. Sum of the areas of all faces or surfaces.
The study of relationships between side lengths and angles of triangles. Essential for navigation, physics, and engineering.
In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: a² + b² = c².
Uses differential calculus to study problems in geometry. Studies curves, surfaces, and manifolds using calculus.
The study of properties preserved under continuous deformations: stretching, twisting, but not tearing or gluing.
The study of geometric transformations: translations, rotations, reflections, and dilations. Symmetry and patterns.
Creating geometric figures using only compass and straightedge. Classical problems of antiquity and their solutions.