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Lecture

Triangle Geometry: Area Formula

Description

This lecture covers the formula for calculating the area of a triangle using the base and height, as well as the determinant of the triangle's sides in a given coordinate system. It also explores similar formulas by permuting the vertices of the triangle.

Official source

Ontological neighbourhood

Mathematics

Geometry: Euclidean geometry

Related concepts (25)

Triangle

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted . In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane.

Area

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.

Heron's formula

In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths a, b, c. If is the semiperimeter of the triangle, the area A is, It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work Metrica, though it was probably known centuries earlier. Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. This triangle's semiperimeter is and so the area is In this example, the side lengths and area are integers, making it a Heronian triangle.

Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring.

Shoelace formula

The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like threading shoelaces. It has applications in surveying and forestry, among other areas.

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