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.

Three heights of 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

a2 + b2 = c2

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

a2 + b2 = c2

122 + b2 = 242

144 + b2 = 576 cm2

b2 = 432 cm2

b2 = 432 cm2

b = 20.7846096908 cm

Find height of a triangle using pythagorean theorem

Most people would be happy to say the height (side b) is approximately 20.78, or b  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 12 base × height, or 12 bh. If you know the area and the length of a base, then, you can calculate the height.

A = 12 bh

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!:

Using area formula to find height of a triangle

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

A = 12 bh

This can be rearranged using algebra:

A = bh2

h = 2 (Ab)

Put in our known values:

h = 2 (987 square yards56 yards)

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 I, we can get the height for that side, too.

How to find the height of a triangle example

h = 2 (Ab)

h = 2 (987 square yards57.255)

h = 2 (17.2385)

h = 34.477

Next Lesson:

Hypotenuse: Definition & Formula

What you'll learn:

After working your way through this lesson and video, you will be able to:

  • Identify and define the height of a triangle
  • Recall and apply two different methods for calculating the height of a triangle, using either the Pythagorean Theorem or the area formula
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
Tutors online
Ashburn, VA

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