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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 perpendicular from the vertex opposite the base onto the line containing the base.
The area of a triangle is its half of the product of the base times the height (length of the altitude). For a triangle with opposite sides ,,, if the three altitudes of the triangle are called ,,, the area is: = = =. Given a fixed base side and a fixed area for a triangle, the locus of apex points is a straight line parallel to the base.
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.
Given a triangle with sides of length a, b, and c, if a 2 + b 2 = c 2, then the angle between sides a and b is a right angle. For any three positive real numbers a, b, and c such that a 2 + b 2 = c 2, there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.
The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} , where a is the length of one leg, b is the length of ...
Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero. Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices,
This formula can be derived by partitioning the n-sided polygon into n congruent isosceles triangles, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height. The following formulations are all equivalent:
Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1. The fact that the remaining leg AD has length √ 3 follows immediately from the Pythagorean theorem.