How to Find the Height of a Triangle

Triangles each have three heights, each related to a separate base. Regardless of having up to three different heights, one triangle will always have only one measure of area. In some triangles, like right triangles, isosceles and equilateral triangles, finding the height is easy with one of two methods.

How to Find the Height of a Triangle

Every triangle has three heights, or altitudes, because every triangle has three sides. A triangle's height is the length of a perpendicular line segment originating on a side and intersecting the opposite angle.

In an equilateral triangle, like $△SUN$ below, each height is the line segment that splits a side in half and is also an angle bisector of the opposite angle. That will only happen in an equilateral triangle.

By definition of an equilateral triangle, you already know all three sides are congruent and all three angles are $60°$. If a side is labelled, you know its length.

Our bright little $△SUN$ has one side labelled $24 cm$, so all three sides are $24 cm$. Each line segment showing the height from each side also divides the equilateral triangle into two right triangles.

Height of a triangle formula

Your ability to divide a triangle into right triangles, or recognize an existing right triangle, is your key to finding the measure of height for the original triangle. You can take any side of our splendid $△SUN$ and see that the line segment showing its height bisects the side, so each short leg of the newly created right triangle is $12 cm$. We already know the hypotenuse is $24 cm$.

Knowing all three angles and two sides of a right triangle, what is the length of the third side? This is a job for the Pythagorean Theorem:

Using Pythagorean Theorem

Focus on the lengths; angles are unimportant in the Pythagorean Theorem. Plug in what you know:

$a^2 + b^2 = c^2$

${12}^2 + b^2 = {24}^2$

$144 + b^2 = 576 c{m}^2$

$b^2 = 432 c{m}^2$

$\sqrt{b^2} = \sqrt{432 c{m}^2}$

$b = 20.7846096908 cm$

Most people would be happy to say the height (side $b$) is approximately $20.78$, or $b \approx 20.78$.

You can decide for yourself how many significant digits your answer needs, since the decimal will go on and on. Do not forget to use linear measurements for your answer!

The Pythagorean Theorem solution works on right triangles, isosceles triangles, and equilateral triangles. It will not work on scalene triangles!

Using the area formula to find height

The formula for the area of a triangle is $\frac{1}{2} base \times height$, or $\frac{1}{2} bh$. If you know the area and the length of a base, then, you can calculate the height.

In contrast to the Pythagorean Theorem method, if you have two of the three parts, you can find the height for any triangle!

Here we have scalene $△ZIG$ with a base shown as $56 yards$ and an area of $987 square yards$, but no clues about angles and the other two sides!:

Recalling the formula for area, where $A$ means area, $b$ is the base and $h$ is the height, we remember

$A = \frac{1}{2} bh$

This can be rearranged using algebra:

$A = \frac{bh}{2}$

$h = 2 \left(\frac{A}{b}\right)$

Put in our known values:

$h = 2 \left(\frac{987 square yards}{56 yards}\right)$

$h = 2 (17.625 yards)$

$h = 35.25 yards$

Remember how we said every triangle has three heights? If we take $△ZIG$ and rotate it clockwise so side $GZ$ is horizontal, and construct a height up to $\angle I$, we can get the height for that side, too.

$h = 2 \left(\frac{A}{b}\right)$

$h = 2 \left(\frac{987 square yards}{57.255}\right)$

$h = 2 \left(17.2385\right)$

$h = 34.477$

Next Lesson:

Hypotenuse: Definition & Formula