How to Find the Height of a Triangle

How to Find the Height of a Triangle

The height of a triangle can be useful in a variety of geometry calculations, like determining the area of a triangle or the volume of a triangular prism. While it may seem difficult at first to find the height of a triangle, especially with non-right and non-equilateral triangles, it is not hard with the correct process.

In this guide, you’ll learn everything you need to know to be able to find the height of any triangle you might come across in your studies.

Figure Out What You Know

Depending on which pieces of information you have about the triangle you’re dealing with, there are different strategies for determining its height. Based on the information you know, you’ll be able to select a technique for finding the height of the triangle that is guaranteed to work.

  • If you have the area and the base length of the triangle, you can work backwards using the area formula.
  • If the triangle is equilateral and you know the side lengths, you can use the Pythagorean theorem to find the height because the height of an equilateral triangle divides it into two identical right triangles.
  • If you have the lengths of all three sides but no angles, you can use Heron’s formula to find the area and work backwards to find the height with the area formula.
  • If you have the lengths of one side and an angle on either side of it, you can use the trigonometry formula to find the height, where c is the side and A is the angle.

Finding Height Given Area and Base Length

The area formula for a triangle is , where A is the area, b is the base length, and h is the height perpendicular to the base. If you know A and b, you can find h by plugging in A and b into the area formula and solving.

Finding Height of an Equilateral Triangle With Side Length

Equilateral triangles have the same length on all three sides and the same interior angle (60 degrees) throughout. Because of this, the height of an equilateral triangle divides the triangle into perfect halves.

how to find the height of a triangle

Use the Pythagorean theorem on one half to find the missing height, since you know two of the three side lengths in that half.

Finding Height With Three Sides

Heron’s formula allows you to find the area of a triangle given three side lengths.

Find the value s using this formula: s = (a + b + c)/2

Then, the area is √(s(s-a)(s-b)(s-c)) With this area and the bottom side as the base length, follow the “Finding Height Given Area and Base Length” instructions.

Finding Height With One Side and One Angle

Assuming that the angle is on either side of the side you know the length of, the height is equal to , where c is the side length and A is the angle.

Still Can’t Find It?

Determining the height of a triangle without at least knowing the base is impossible. However, you can rotate the triangle so that any side is the base and find the height relative to that base.

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