Geometry

A point has position but no size. A line is the shortest path between two points. A plane is a 2-dimensional surface with length and width.

An angle is formed when two lines meet at a point. The measure of an angle indicates how far apart the lines are.

A polygon with three sides. The sum of interior angles is always 180°.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

The area of a triangle is half the product of its base and height.

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

Two triangles are congruent if they have exactly the same shape and size. All corresponding sides and angles are equal.

A polygon with four sides. The sum of interior angles is 360°.

The set of all points in a plane that are at a fixed distance from a center point.

An arc is a portion of the circle between two points. A sector is the region bounded by two radii and an arc.

The straight-line distance between two points in a coordinate plane, derived from the Pythagorean theorem.

A formula to find the coordinates of the midpoint of a line segment between two points.

A measure of how steep a line is, calculated as the ratio of vertical change to horizontal change.

Curves formed by intersecting a cone with a plane: circle, ellipse, parabola, and hyperbola.

Three-dimensional shapes with volume and surface area.

A symmetric (0,2)-tensor defining distances and angles on a manifold. Used to raise and lower indices.

Coordinate expression of the Levi-Civita connection. Used to define geodesic equations and covariant derivatives.

A method of differentiating tensor fields in curved spaces. Related to parallel transport, providing coordinate-independent differentiation.

A (1,3)-tensor measuring intrinsic curvature of a manifold. Defined by the non-commutativity of covariant derivatives.

A symmetric (0,2)-tensor obtained by contracting the Riemann tensor. Central component of Einstein field equations.