How to Find the Height of a Triangle

Co-authored by David Jia

Last Updated: August 24, 2023 Fact Checked

To calculate the area of a triangle you need to know its height. To find the height follow these instructions. You must at least have a base to find the height.

Method 1

Method 1 of 3:

Using Base and Area to Find Height

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1
Recall the formula for the area of a triangle. The formula for the area of a triangle is
A=1/2bh.
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- A = Area of the triangle
- b = Length of the base of the triangle
- h = Height of the base of the triangle
You already know the area, so assign that value to A. You should also know the value of one side length; assign that value to "'b'".
Any side of a triangle can be the base,
regardless of how the triangle is drawn. To visualize this, just imagine rotating the triangle until the known side length is at the bottom.

Example
If you know that the area of a triangle is 20, and one side is 4, then:
A = 20 and b = 4.

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First multiply the base (b) by 1/2, then divide the area (A) by the product. The resulting value will be the height of your triangle!

Example
20 = 1/2(4)h Plug the numbers into the equation.
20 = 2h Multiply 4 by 1/2.
10 = h Divide by 2 to find the value for height.

Method 2

Method 2 of 3:

Finding an Equilateral Triangle's Height

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1
Recall the properties of an equilateral triangle. An equilateral triangle has three equal sides, and three equal angles that are each 60 degrees. If you
cut an equilateral triangle in half, you will end up with two congruent right triangles.
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- In this example, we will be using an equilateral triangle with side lengths of 8.
Recall the Pythagorean Theorem. The Pythagorean Theorem states that for any right triangle with sides of length a and b, and hypotenuse of length c:
a² + b² = c².
We can use this theorem to find the height of our equilateral triangle!^{[3]
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3
Break the equilateral triangle in half, and assign values to variables a, b, and c. The hypotenuse c will be equal to the original side length. Side a will be equal to 1/2 the side length, and side b is the height of the triangle that we need to solve.
- Using our example equilateral triangle with sides of 8, c = 8 and a = 4.
^{[4]
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Research source} First square c and a by multiplying each number by itself. Then subtract a² from c².

Example
4² + b² = 8² Plug in the values for a and c.
16 + b² = 64 Square a and c.
b² = 48 Subtract a² from c².
5
Find the square root of b² to get the height of your triangle! Use the square root function on your calculator to find Sqrt(². The answer is the height of your equilateral triangle!
- b = Sqrt (48) = 6.93

Method 3

Method 3 of 3:

Determining Height With Angles and Sides

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1
Determine what variables you know. The height of a triangle can be found if you have 2 sides and the angle in between them, or all three sides. We'll call the sides of the triangle a, b, and c, and the angles, A, B, and C.
- If you have all three sides, you'll use
  Heron's formula
  , and the formula for the area of a triangle.
- If you have two sides and an angle, you'll use the formula for the area given two angles and a side.
  A = 1/2ab(sin C).^{[5]
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Heron's formula has two parts. First, you must find the variable
s, which is equal to half of the perimeter of the triangle.
This is done with this formula:
s = (a+b+c)/2.^{[6]
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Heron's Formula Example
For a triangle with sides a = 4, b = 3, and c = 5:
s = (4+3+5)/2
s = (12)/2
s = 6
Then use the second part of Heron's formula, Area = sqr(s(s-a)(s-b)(s-c). Replace Area in the equation with its equivalent in the area formula: 1/2bh (or 1/2ah or 1/2ch).
Solve for h. For our example triangle this looks like:
1/2(3)h = sqr(6(6-4)(6-3)(6-5).
3/2h = sqr(6(2)(3)(1)
3/2h = sqr(36)
Use a calculator to calculate the square root, which in this case makes it 3/2h = 6.
Therefore, height is equal to 4, using side b as the base.
Replace area in the formula with its equivalent in the area of a triangle formula: 1/2bh. This gives you a formula that looks like 1/2bh = 1/2ab(sin C). This can be simplified to
h = a(sin C)
, thereby eliminating one of the side variables.^{[7]
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Research source} Note that angle C and side a are both positioned across from the height that you need to find (both on the right side from it, or both on the left side).

Finding Height with 1 Side and 1 Angle Example
For example, with a = 3, and C = 40 degrees, the equation looks like this:
h = 3(sin 40)
Use your calculator to finish the equation, which makes h roughly 1.928.

Practice Problems and Answers

Practice Problems and Answers to Find the Height of a Triangle

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Question

How do I find the area of an equilateral triangle when only the height is given?

Community Answer

H = height, S = side, A = area, B = base. You know that each angle is 60 degrees because it is an equilateral triangle. If you look at one of the triangle halves, H/S = sin 60 degrees because S is the longest side (the hypotenuse) and H is across from the 60 degree angle, so now you can find S. The base of the triangle is S because all the sides are the same, so B = S. Using A = (1/2)*BH, you get A = (1/2)*SH, which you can now find.

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Question

How do I calculate the height of a right triangle, given only the length of the base and the interior angle at the base?

Donagan

Top Answerer

Look up the tangent of the angle in a trigonometry table. Multiply the tangent by the length of the base.

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Question

How do I determine the height of a triangle when I know the length of all three sides?

Community Answer

You already know the base, so calculate the area by Heron's formula. Then, substitute the values you know in the formula. Area=1/2 * base * height or height=2 * Area/base and find your answer.

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