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In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is = /, where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a ...
The process of drawing the altitude from a vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one-half of the product of an altitude's length and its base's length (symbol b) equals the triangle's area: A = h b /2 ...
Because each special triangle has area , a polygon of area will be subdivided into special triangles. [ 5 ] The subdivision of the polygon into triangles forms a planar graph , and Euler's formula V − E + F = 2 {\displaystyle V-E+F=2} gives an equation that applies to the number of vertices, edges, and faces of any planar graph.
The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle. In the Euclidean plane, area is defined by comparison with a square of side length , which has area 1. There are several ways to calculate the area of an arbitrary triangle.
Regular polygons; Description Figure Second moment of area Comment A filled regular (equiliteral) triangle with a side length of a = = [6] The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.
If is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude and bases , , and . Their combined area is A = 1 2 a r + 1 2 b r + 1 2 c r = r s , {\displaystyle A={\tfrac {1}{2}}ar+{\tfrac {1}{2}}br+{\tfrac {1}{2}}cr=rs,} where s = 1 2 ( a + b + c ...
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
For any interior point P, the sum of the lengths of the perpendiculars s + t + u equals the height of the equilateral triangle.. Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. [1]
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