About

In geometry and structural engineering, the altitude (or height) of a triangle is more than just a line segment—it is a critical vector determining stability, area distribution, and orthographic projection. Whether you are an architect designing a truss system or a student solving complex trigonometric problems, precision is non-negotiable. A miscalculation in altitude can lead to errors in area estimation or structural load balancing.

This tool eliminates the guesswork by offering two distinct calculation modes. It handles the standard Base & Area approach for quick estimates, and employs Heron’s Formula reverse engineering to solve for all three altitudes when only the side lengths are known. The integrated visualizer plots the triangle and its orthocenter, providing immediate geometric context to your numerical results.

Formulas

When the area is unknown but side lengths are available, we first calculate the semi-perimeter s and the Total Area A using Heron’s Formula:

s = a + b + c2

A = √s⋅(s − a)⋅(s − b)⋅(s − c)

Once the Area is established, the altitude h corresponding to any base b is derived from the standard area formula:

h_b = 2⋅Ab

Reference Data

Triangle Type	Side Inputs (a, b, c)	Area (A)	Altitude to Base a (h_a)
Equilateral Unit	1, 1, 1	0.433	0.866
Egyptian Triangle (Right)	3, 4, 5	6	4 (Leg is height)
Isosceles	5, 5, 6	12	4.8
Scalene (Heron)	7, 24, 25	84	24
Obtuse Scalene	4, 7, 9	13.416	6.708
Golden Triangle	1.618, 1.618, 1	0.786	1.572

Frequently Asked Questions

Yes. Mathematically, the formula remains the same regardless of the angles. However, visually, the altitude of an obtuse triangle may lie outside the triangle itself (dropping onto the extended base). This calculator solves the numerical value correctly for all planar triangles.

The triangle inequality theorem states that the sum of any two sides must be greater than the third side (a + b > c). If you input invalid dimensions, Heron's formula will produce a square root of a negative number (imaginary), and the calculator will alert you that the geometry is impossible.

Every triangle has three bases (its three sides). Consequently, there is a unique altitude corresponding to each base. The product of any base and its corresponding height is constant (double the area). In an equilateral triangle, all three heights are equal; in a scalene triangle, they differ.