In trigonometry, the height of a triangle can be determined in many different ways depending on whether it's a right triangle, isosceles triangle (a triangle with two equal sides), or equilateral triangle.
1. How To Find the Height of a Right Triangle
Before we start, here’s what you need to know about right triangles. A right triangle has three sides: the hypotenuse, height, and base of the triangle. The base and height of a right triangle are always the sides adjacent to the right angle, and the hypotenuse is the longest side.
The height of a right triangle can be determined with the area formula:
If the given area isn't known, you can use the Pythagorean theorem to solve for the height of a right triangle. Here’s what the Pythagorean theorem states, given c is the hypotenuse and a and b are the other two sides:
Let’s take the units from the figure above and plug in the length of the base and hypotenuse to solve for the missing height:
2. Finding the Height of a Non-Right Triangle
Unfortunately, you can’t use the Pythagorean theorem to find the height of an isosceles triangle or the height of an equilateral triangle (where all sides of the triangle are equal). Instead, you'll have to draw a perpendicular line through the base of the triangle to form a right angle:
This line represents the height of these non-right triangles. Once you've formed this line, you'll have to use Heron's formula to solve for the area of the entire triangle.
The first step of Heron's formula is calculating half the triangle’s perimeter. In this case, s represents half the perimeter and a, b, and c are the sides:
Once you've determined s, use the following formula to calculate the area of a triangle. Again, the two sides are a and b, and the longest side (the hypotenuse) is c:
Let’s plug in the side lengths from this isosceles triangle to find the area of the triangle:
Now, we’ll substitute s in the area formula for a non-right triangle.
Using Area To Find the Height of a Triangle
Now that you know the area of the triangle pictured above, you can plug it into triangle formula A=1/2bh to find the height of the triangle. In this case, the base would equal half the distance of five (2.5), since this is the shortest side of the triangle.
Master Triangle Height Formulas
Finding the height of a triangle is a multi-step process that can be confusing. However, mastering it helps you learn different types of area formulas, such as heron's formula and A=1/2bh. It also shows you how to use the Pythagorean theorem and triangle perimeter formulas to determine other quantities within a triangle.