triangleheight

Triangle height, commonly called the altitude, is the perpendicular distance from a vertex to the opposite side (or to the line containing that side). A triangle has three altitudes, one from each vertex, usually denoted h_a, h_b, h_c corresponding to the opposite sides a, b, c.

The height is directly related to area. If the base is a, then the area Δ satisfies Δ = (1/2) a h_a, so h_a = 2Δ / a. Similarly, h_b = 2Δ / b and h_c = 2Δ / c. If the side lengths a, b, c form a triangle with semiperimeter s, then Δ can be found by Heron’s formula Δ = sqrt[s(s−a)(s−b)(s−c)], and the corresponding heights follow from h_a = 2Δ / a, etc. An alternative expression uses angles: h_a = b sin C = c sin B.

The foot of an altitude is the point where the altitude meets the opposite side (or its extension). In an acute triangle, all three feet lie on the sides themselves; in an obtuse triangle, the feet lie on extensions of the opposite sides; in a right triangle, the altitude from the right angle lies inside and ends on the hypotenuse.

Special cases and relations: in an equilateral triangle with side length s, the height is h = s√3/2. The altitude is closely tied to other triangle elements: the area can also be expressed as Δ = abc/(4R) where R is the circumradius, and also Δ = rs where r is the inradius and s is the semiperimeter.

Applications of triangle height include area computation, geometric constructions, and coordinate geometry, where the height is the perpendicular distance from a vertex to a line.