"Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures."
The study of the properties, measurements, and relationships of points, lines, shapes, and solids.
Points, Lines, and Planes: This topic introduces the basic building blocks of geometry, including points (which have no dimensions), lines (which have infinite length and no width), and planes (which have infinite length and width but no thickness).
Angles: In geometry, angles are formed by two rays that start at a common point, and they are measured in degrees. This topic covers the different types of angles, how to measure angles, and how to calculate angle relationships.
Triangles: Triangles are one of the most important shapes in geometry, and this topic covers how to classify triangles by their sides and angles, how to find missing angles and side lengths, and the various theorems that apply to triangles.
Quadrilaterals: This topic covers the different types of quadrilaterals (such as rectangles, squares, parallelograms, and trapezoids), their properties, and how to solve problems involving them.
Circles: This topic covers the properties of circles, how to find the circumference and area of a circle, and the different parts of a circle (such as chords, radii, and diameters).
Similarity: When two shapes have the same shape (but possibly different sizes), they are said to be similar. This topic covers how to determine if two shapes are similar, how to find scale factors and ratios of similarity, and how to use similarity to solve problems.
Congruence: Congruent shapes have the same size and shape, and this topic covers how to determine if two shapes are congruent, the different types of congruence (such as side-angle-side and angle-side-angle), and how to use congruence to solve problems.
Polygons: Polygons are closed shapes with three or more straight sides, and this topic covers how to classify polygons by their number of sides, how to find missing angles and side lengths of polygons, and the properties of regular polygons.
Three-dimensional figures: This topic covers the properties and characteristics of three-dimensional figures such as spheres, cylinders, cones, and pyramids, and how to find their volumes and surface areas.
Coordinate Geometry: This topic combines algebra with geometry, and covers different systems of coordinates, how to graph shapes using coordinates, and how to use coordinate geometry to solve problems.
"Euclidean geometry includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts."
"The word 'geometry' is derived from the Ancient Greek words γῆ (gê) meaning 'earth, land', and μέτρον (métron) meaning 'a measure'."
"Geometry has applications in art, architecture, and other activities that are related to graphics."
"Methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem."
"One of the oldest discoveries is Carl Friedrich Gauss' Theorema Egregium ('remarkable theorem') that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space."
"During the 19th century, several discoveries enlarged dramatically the scope of geometry."
"Geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction."
"The geometry that underlies general relativity is a famous application of non-Euclidean geometry."
"The field has been split into many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc."
"Differential geometry is one of the subfields that depend on the underlying methods."
"Affine geometry omits the concept of angle and distance."
"This enlargement of the scope of geometry led to a change of meaning of the word 'space'."
"A geometric space, or simply a space, is a mathematical structure on which some geometry is defined."
"Geometry is a branch of mathematics."
"A mathematician who works in the field of geometry is called a geometer."
"Originally developed to model the physical world, geometry has applications in almost all sciences."
"Geometry is one of the oldest branches of mathematics."
"This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry."
"Geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics."