Isosceles Triangle Height Calculator

Enter the equal leg length and base of an isosceles triangle to get its height (h = √(a² − (b/2)²)), area, perimeter, and base angles.

Quick Facts

Height formula
h = √(a² − (b/2)²)
a is the equal leg length, b is the base; comes straight from the Pythagorean theorem.
Area formula
A = ½ × b × h
Same base-times-height rule used for any triangle.
Validity condition
a > b / 2
The leg must exceed half the base or no triangle can close.

Your Results

Calculated
Height
-
h = √(a² − (b/2)²)
Area
-
A = ½ × base × height
Perimeter
-
P = 2 × leg + base
Base Angles
-
Each base angle; apex angle noted below

Ready

Enter the leg length and base, then press Calculate.

Formula and Method for Isosceles Triangle Height

An isosceles triangle has two sides of equal length (the legs, a) and a third side (the base, b) that can differ. Because the two legs are equal, the perpendicular dropped from the apex (the vertex between the two legs) to the base always lands exactly at the midpoint of the base. That single fact turns the height calculation into a straightforward application of the Pythagorean theorem: h = √(a² − (b/2)²).

How the calculation works

Drawing the altitude from the apex to the base splits the isosceles triangle into two congruent right triangles. Each right triangle has the leg a as its hypotenuse, half the base (b/2) as one side, and the height h as the other side. By the Pythagorean theorem, a² = h² + (b/2)², which rearranges to h = √(a² − (b/2)²). Once you have the height, area follows from the standard triangle formula A = ½ × b × h, perimeter is P = 2a + b, and each base angle is θ = arccos((b/2) / a), with the apex angle equal to 180° − 2θ.

Common mistakes

  • Using the full base instead of half: the right triangle formed by the altitude uses b/2, not b — forgetting to halve the base is the most common error.
  • Confusing the two side types: "leg" (a) refers to one of the two equal sides, not the base; swapping them into the formula gives a nonsensical or invalid result.
  • Ignoring the triangle inequality: if the leg is not longer than half the base (a ≤ b/2), the two legs cannot meet above the base and no real triangle — and no real height — exists.

Real-world applications

  • Roof trusses and A-frame structures are commonly isosceles, and the height determines the peak clearance for a given span
  • Gable ends, pediments, and decorative triangular panels use the height to calculate material area and cut angles
  • Surveying and land plots that form triangular parcels use the height and base to compute usable area
  • Engineering drawings and CAD layouts use the base-angle formula to set precise cut or fold angles for symmetric triangular parts

Frequently Asked Questions

What is the formula for the height of an isosceles triangle?
h = √(a² − (b/2)²), where a is the length of each equal leg and b is the base. The altitude from the apex to the base bisects the base into two segments of length b/2, forming a right triangle with the leg as the hypotenuse, so the Pythagorean theorem gives the height.
What if I know a base angle instead of the leg length?
If you know the base angle θ (the angle between the base and a leg), the height is h = (b/2) × tan(θ). This calculator asks for the leg and base directly, but you can first find the leg with a = (b/2) / cos(θ) and enter that value.
How do I find the area once I have the height?
Area = ½ × base × height, the same formula used for any triangle, since the base and the perpendicular height are all that area requires.
What condition must the leg and base satisfy for a valid triangle?
The leg length must be greater than half the base (a > b/2), which is the triangle inequality (2a > b) applied to an isosceles triangle. If a ≤ b/2, the two legs cannot meet above the base and no triangle exists.

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