Geometry

The Pythagorean theorem relates the three sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides. It is one of the oldest and most-proved theorems in mathematics, with hundreds of distinct known proofs, and underlies the definition of distance in geometry, trigonometry, and beyond.

A triangle is a polygon with three vertices and three sides, the simplest possible polygon and one of the basic shapes of geometry. Its three vertices are zero-dimensional points, and its three sides are line segments joining them in pairs. A triangle has three interior angles, and a foundational fact of Euclidean geometry is that these three angles always sum to 180° (a straight angle). Triangles are classified by side lengths (scalene, isosceles, equilateral) and by angles (acute, right, obtuse), and the area of any triangle equals one-half the product of a base and the corresponding height.

Two figures are congruent if one can be mapped exactly onto the other using only rigid motions — translations, rotations, and reflections — which preserve both shape and size. Two figures are similar if one can be mapped onto the other by a rigid motion combined with a uniform scaling (a dilation): similar figures have the same shape but not necessarily the same size, with all corresponding angles equal and all corresponding sides in the same fixed ratio (the scale factor). Congruence is really similarity with scale factor exactly 1, so every congruent pair of figures is automatically similar.

The triangle inequality states that, for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Equivalently, three positive lengths a, b, c can form a triangle if and only if each one is less than the sum of the other two. The theorem captures a basic fact about straight-line distance: the direct path between two points is never longer than any path that detours through a third point. It underlies not just triangle geometry but the very definition of 'distance' (a metric) used throughout mathematics.

A median of a triangle is a segment from a vertex to the midpoint of the opposite side; an angle bisector is a segment from a vertex that splits that vertex's angle into two equal halves. Every triangle has three medians and three angle bisectors, and each set meets at a single special point: the three medians meet at the centroid (the triangle's balance point), and the three angle bisectors meet at the incenter (the center of the inscribed circle). These are two of the four classical 'triangle centers' — points defined purely by a triangle's shape, independent of any coordinate system.

Points, lines, and planes are the undefined primitive objects of Euclidean geometry — every other geometric idea is built from them. A point marks a location with no size; a line is an infinitely long, straight, one-dimensional object with no width, extending endlessly in two opposite directions; a plane is a flat, two-dimensional surface extending infinitely in all directions. A line may be thought of as an idealization of a taut string or a ray of light, and a line segment is the portion of a line between two endpoints.

An angle is the geometric figure formed by two rays (called the sides of the angle) that share a common endpoint (called the vertex). Angles measure the amount of rotation or 'opening' between the two rays, typically in degrees (°) or radians (rad). Angles are classified by size (acute, right, obtuse, straight, reflex) and by the relationships between pairs of angles (complementary, supplementary, vertical, adjacent), and they underlie the study of triangles, polygons, circles, and trigonometry.

A circle is the set of all points in a plane that are at a fixed distance — the radius — from a fixed point called the center. The distance across the circle through the center is the diameter, exactly twice the radius. A circle encloses a two-dimensional region called a disc. Circles are governed by the constant π (pi), the ratio of a circle's circumference to its diameter, and they connect directly to angle measure, since a full rotation around a circle's center is 360° (2π radians).

Circle theorems are a family of results describing the relationships between angles, chords, tangents, and arcs of a circle. They build directly on the inscribed angle theorem — that an inscribed angle is always half the central angle subtending the same arc — and extend it to cyclic quadrilaterals, tangent lines, and intersecting chords. These theorems let you deduce unknown angles and lengths in circle diagrams without any measurement, using only the circle's defining property that every radius has the same length.

A locus (plural: loci) is the set of all points satisfying a given geometric condition — for example, 'all points a fixed distance from a center' defines a circle, and 'all points equidistant from two fixed points' defines the perpendicular bisector of the segment joining them. A compass-and-straightedge construction is a method for drawing exact geometric figures using only an unmarked straightedge (for drawing lines through two points) and a compass (for drawing circles/arcs of a given radius from a given center), following the rules laid out by Euclid over two thousand years ago. Constructions are really loci made physical: bisecting a segment, for instance, is done by intersecting two circular loci centered at its endpoints.

Perimeter is the total distance around the boundary of a two-dimensional shape; area is the measure of the region enclosed by that boundary — how much surface it covers. Area can be thought of as the amount of paint needed to cover a shape with a single coat, while perimeter is the length of fencing needed to enclose it. Every standard polygon and the circle has a well-known formula for each, derived from decomposing the shape into simpler pieces (typically triangles and rectangles) or, for the circle, from a limiting process.

Volume is the amount of three-dimensional space a solid occupies, measured in cubic units; surface area is the total area of all the faces (or curved surface) enclosing that solid, measured in square units. Just as area and perimeter generalize the two-dimensional measure of a shape, surface area and volume generalize the same idea to three dimensions — surface area is 'how much wrapping paper' a solid needs, and volume is 'how much it can hold' or 'how much space it displaces.' Every standard solid — prism, cylinder, cone, pyramid, sphere — has known formulas derived either by decomposing it into simpler pieces or, for curved solids, by a limiting (calculus-based) process.

A prism is a polyhedron with two parallel, congruent polygonal bases connected by rectangular (or parallelogram) lateral faces; a pyramid has a single polygonal base with triangular lateral faces that all meet at one apex point. Both are named after their base shape (triangular prism, pentagonal pyramid, and so on). A prism's volume is simply base area times height, the same 'stack the cross-section' idea used for a cylinder, while a pyramid's volume is exactly one-third of the prism with the same base and height, mirroring the cone-to-cylinder relationship — because the cross-sectional area shrinks toward zero as you approach the apex.

A geometric proof is a deductive argument establishing that a geometric statement must be true, built from previously established axioms, definitions, and theorems using accepted rules of logical inference. Unlike checking a statement in many specific examples (inductive/empirical evidence), a valid proof guarantees the statement holds in every possible case. Classical geometric proofs are usually written as a chain of statements, each justified by a definition, postulate, previously proved theorem, or a valid rule of logic — the two most common formats being two-column proofs (statement/reason table) and paragraph proofs (flowing prose).

A coordinate geometry proof (analytic proof) establishes a geometric statement by placing the figure on the Cartesian plane and using algebra — the distance formula, midpoint formula, and slope — instead of the synthetic (axiom-and-theorem) style of classical proof. This approach, pioneered by Descartes and Fermat, converts geometric claims like 'these two segments are equal' or 'these lines are perpendicular' into algebraic equations that can be verified by direct computation, often making proofs that are hard to construct synthetically almost mechanical.

A quadrilateral is a polygon with four vertices and four sides. Unlike a triangle, a quadrilateral is not rigid — its shape can flex even when all four side lengths are fixed, like a hinge — so classifying quadrilaterals requires extra conditions on angles, parallel sides, or diagonals. The family forms a nested hierarchy: every square is a rectangle and a rhombus, every rectangle and rhombus is a parallelogram, and every parallelogram is a trapezoid (under the inclusive definition). A foundational fact shared by all simple quadrilaterals is that their four interior angles always sum to 360°.

A polygon is a plane figure bounded by a finite chain of straight line segments (its sides) closing to form a loop, meeting only at their shared endpoints (the vertices). Triangles and quadrilaterals are the polygons with the fewest sides; the family continues with pentagons (5 sides), hexagons (6), heptagons (7), octagons (8), and generally an n-gon for n sides. A polygon is convex if every interior angle is less than 180° (no vertex 'caves in'); it is regular if all sides and all interior angles are equal. The interior angle sum of any simple polygon grows linearly with the number of sides, which is the single most important structural fact about polygons.

A tessellation (or tiling) is a covering of the plane using one or more repeated shapes, called tiles, with no gaps and no overlaps. A tessellation is regular if it uses only one type of regular polygon, meeting edge-to-edge; only three regular polygons tessellate the plane this way: equilateral triangles, squares, and regular hexagons. A tessellation is semi-regular if it mixes two or more types of regular polygons, arranged identically at every vertex. The key numerical requirement behind every tessellation is that the interior angles meeting at any single vertex must sum to exactly 360°, since the tiles must fit together with no gap and no overlap all the way around that point.

A geometric transformation is a function that moves every point of the plane (or space) to a new position according to a fixed rule, producing an image of the original figure. The four basic plane transformations are translation (sliding without rotating or resizing), reflection (flipping across a line), rotation (turning about a fixed point), and dilation (scaling by a fixed factor from a center point). Translations, reflections, and rotations are rigid motions (isometries): they preserve distances and angles, so the image is always congruent to the original. Dilation preserves angles and shape but not distance, producing a similar (not congruent) image.

Two polygons are similar when one is an enlargement or reduction of the other: corresponding angles are equal, and corresponding sides are all in the same ratio, called the scale factor. Similarity is a weaker condition than congruence — congruent figures are similar with scale factor 1, but similar figures need not be the same size, only the same shape. Because every linear measurement (side length, perimeter, diagonal) scales by the factor k, while area scales by k² and volume (for similar solids) scales by k³, scale factor is the single number that governs how all the other measurements of a shape change together.