Isosceles Triangle Height Calculator

Created by Steven Wooding
Reviewed by Rijk de Wet
Last updated: Jun 05, 2023

This isosceles triangle height calculator will calculate for you the two heights of an isosceles triangle. It will also tell you whether you have a valid isosceles triangle. Read on to learn about:

  • What is an isosceles triangle?
  • How to calculate the height of an isosceles triangle.
  • The isosceles triangle height formula.

What is an isosceles triangle?

An isosceles triangle is a triangle where two sides are of equal length. You can also argue that a triangle where all three sides are equal (an equilateral triangle) is a special case of an isosceles triangle.

Isosceles triangles date back to ancient Egypt, where it was used for decoration and in their architecture — even the face of a pyramid is an isosceles triangle.

Isosceles triangle height formula

We can derive the formula for the height of an isosceles triangle (as measured from the base to the vertex between the two equal legs) from Pythagoras's Theorem if we look for a right-angle triangle inside the isosceles triangle.

You can form one if you only consider half of an isosceles triangle, cutting at the midpoint of the base (side bb in the figure above the calculator). Then we can derive the height hh starting from Pythagoras's equation:

a2=h2+(b/2)2h2=a2(b/2)2h2=a2b24 h=a2b24\begin{align*} &a^2 = h^2 + (b/2)^2\\[1em] &h^2 = a^2 - (b/2)^2\\[1em] &h^2 = a^2 - \frac{b^2}{4}\\[1em] &\ \boxed{h = \sqrt{a^2 - \frac{b^2}{4}}} \end{align*}

How to use the isosceles triangle height calculator

Using this isosceles triangle height calculator is really straightforward. Here are the steps to follow:

  1. Enter a value for the length of each leg of the triangle (the two equal sides).
  2. Input the length of the base of your isosceles triangle.
  3. You'll instantly receive a value for the height of the triangle, as measured from the base to the vertex (hb).
  4. As a bonus, you'll also get the height as measured from the leg side to the opposite vertex at the end of the base (ha).

Note that if you want to calculate the lengths in different units, you can change the unit before entering a value. Changing the unit after entering a value, the calculator is also a length unit converter.

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How do I find the height of an isosceles triangle?

To find the height of an isosceles triangle, follow these steps:

  1. Square the length of the two equal sides.
  2. Square the unequal side length, divide by four, and subtract the result from Step 1.
  3. Take the square root of the result from Step 2.
  4. The result is the height of an isosceles triangle.

What is the height of an isosceles triangle with a base of 10 cm and a leg of 15 cm?

14.1 cm. To arrive at this answer, the following formula for the height of an isosceles triangle was used:

h = √[a² − (b²/4)]

Plugging in the numbers:

h = √[15² − (10²/4)] = √200 = 14.1 cm

Steven Wooding
isosceles triangle with sides, angles and heights marked.
Leg (a)
Base (b)
Height to apex (hb)
Height from leg (ha)
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