Isosceles Triangle Height Calculator

Find the height of an isosceles triangle: h = sqrt(a^2 - (b/2)^2). Calculate from equal side and base, area and base, or angle and side.

Select Input Method

Result

Height (h)
--
units
Equal Side (a) --
Base (b) --
Area --
Perimeter --

Step-by-Step Solution

h = sqrt(a^2 - (b/2)^2)

How to Find the Height of an Isosceles Triangle

The height (or altitude) of an isosceles triangle is the perpendicular distance from the apex (the vertex between the two equal sides) to the base. Because the triangle is symmetric, this altitude bisects the base into two equal segments, creating two congruent right triangles.

This geometric property is the foundation for the primary height formula: h = sqrt(a2 - (b/2)2), which is derived directly from the Pythagorean theorem.

Height Formulas

From Equal Side and Base

The most common formula using the Pythagorean theorem.

h = sqrt(a2 - (b/2)2)

From Area and Base

Rearrange the area formula to solve for height.

h = 2A / b

From Base and Base Angle

Use the tangent function with the base angle.

h = (b/2) tan(alpha)

From Side and Apex Angle

Use cosine of half the apex angle.

h = a cos(beta/2)

Understanding the Derivation

When you draw the altitude from the apex to the base of an isosceles triangle, it creates two right triangles. In each right triangle:

  • The hypotenuse is the equal side a.
  • One leg is half the base: b/2.
  • The other leg is the height h.

By the Pythagorean theorem: a2 = h2 + (b/2)2. Solving for h: h = sqrt(a2 - (b/2)2).

Special Cases

  • Equilateral triangle (a = b): h = (sqrt(3)/2) a, approximately 0.866a.
  • Right isosceles (apex = 90 degrees): h = a / sqrt(2), and b = a sqrt(2).
  • Very flat triangle (b approaches 2a): Height approaches zero.

Practical Applications

Finding the height of an isosceles triangle is essential for calculating roof pitch in construction, determining the depth of triangular cross-sections in engineering, computing the altitude of triangular sails in sailing, and solving geometry problems in education. The height is also critical when computing the area, as A = 1/2 b h.