How to Calculate Altitude in a Triangle

To calculate the area of a triangle, you need to know its altitude. If the test doesn’t give these numbers, you can still easily find a high based on what you know! This article will show you two different ways to find the altitude of a triangle, based on the information you have in the problem.

Table of Contents

Use base and area to find height

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Repeat the formula for calculating the area of a triangle. To find the area of a triangle, we have the formula A=1/2bh . ^{[1] X Research Source}

A = area of the triangle
b = length of the base of the triangle
h = altitude from bottom edge

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Look at the triangle and identify the variables that you already know. In this case, you have the area to assign to the value of the quantity A . You also know the side length; assign that value to the quantity “‘b'”. If you don’t have both the area and the length of one side, you’ll have to use another method.

Any side of the triangle can become the base, depending on how you draw it. To visualize this, just imagine you rotating the triangle in multiple directions until the side of known length is at the base.
For example, the area of that triangle is 20 and one side is 4, we have: A = 20 and b = 4 .

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Substitute your numbers into the expression A=1/2bh and do the math. First, multiply the value (b) by 1/2, then divide the area (A) by the product you just found. The result of this calculation will be the height of the triangle!

In this example, we have: 20 = 1/2(4)h
20 = 2h
10 = h

Find the altitude of an equilateral triangle

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Recall the properties of an equilateral triangle. An equilateral triangle has three equal sides and three equilateral angles of 60 degrees. If you divide this triangle in half, you will get two similar right triangles. ^{[2] X Research Source}

In this example, we will find the altitude of an equilateral triangle with side length 8.

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Remember the Pythagorean theorem. According to the Pythagorean theorem, any right triangle with two sides a , b and a hypotenuse c then: a ² + b ² = c ² . We can use this theorem to find the altitude of an equilateral triangle! ^{[3] X Research Sources}

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Draw a line that bisects an equilateral triangle, then assign the values a , b , and c to the figure. The hypotenuse c will be equal to the length of the side of the equilateral triangle, while the side a will be 1/2 the length of the side of the equilateral triangle and the side b is the altitude of the triangle we are looking for.

Going back to the example of an equilateral triangle with side 8, we have c = 8 and a = 4 .

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Substitute those values into the Pythagorean theorem and calculate b ² . First, we square c and a by multiplying each number by itself. Then subtract a ² from c ² .

4 ² + b ² = 8 ²
16 + b ² = 64
b ² = 48

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Calculate the square root of b ² to find the altitude of the triangle! Use the square root calculator function to find the square root of b ² . The result is the altitude of the equilateral triangle!

b = 48 = 6.93

Find altitudes with angles and sides

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Identify what values you have. We can calculate the altitude of a triangle in the following cases: if you have an angle and a side; if you have a bottom edge, a side edge, and an angle between them; if you have all three edges. Let’s call the sides of the triangle a, b, c and the angles A, B, C.

If there are three sides, you can use Heron’s formula and the formula for the area of a triangle.
If there are two sides and one angle, you can use the formula to calculate the area of a triangle with two angles and one side. A = 1/2ab(sin C). ^{[4] X Research Sources}

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Apply Heron’s formula if you have three sides of a triangle. This recipe has two parts. First, you have to find the variable p, i.e. the half-perimeter of the triangle. We have the formula: p = (a+b+c)/2. ^{[5] X Research Sources}

For a triangle with three sides a = 4, b = 3 and c = 5, the half-perimeter p = (4+3+5)/2. = (12)/2. We have p = 6.
Next, you apply the second part of Heron’s formula, which is the area A = √(p(pa)(pb)(pc)). Replace the A value in the equation with the equivalent expression: 1/2bh (or 1/2ah or 1/2ch) from the area formula.
Do the math to find h. In this example, we have 1/2(3)h = √((6(6-4)(6-3)(6-5)) So 3/2h = √((6(2)() 3)(1)) Continuing the calculation, we have 3/2h = √36. Using a calculator to calculate the square root, the expression becomes 3/2h = 6. So, by using edge b as the base, we find out that the altitude of this triangle is equal to 4.

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Use the formula for the area with two sides and an angle if the problem gives you the length of one side and one angle. Substitute the area into the formula with the equivalent expression: 1/2bh. You will have 1/2bh = 1/2ab(sin C). Simplifying the expression by removing the same variables, we get h = a(sin C). ^{[6] X Research Sources}

Solve the problem using the variables you have. For example, for a = 3, C = 40 degrees, the expression becomes: h = 3(sin 40). Use a calculator to find the answer, in this example, the rounded h will be 1.928.

Steps

Use base and area to find height

Find the altitude of an equilateral triangle

Find altitudes with angles and sides

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