How To Calculate the Three Heights' Lengths of a Triangle, Given the Sides' Lengths

• The heights of a triangle

• Definition: The height of a triangle is a straight line segment drawn from a vertex of the triangle, perpendicular to the opposite side (at a 90-degree angle). Every triangle has three heights. All three heights always cross each other at one single point, which is called the orthocenter.
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  The three heights of the triangle

• In the triangle △ABC h_a is the height starting from the vertex A, perpendicular to the side BC, h_a ⊥ BC. h_b is the height starting from the vertex B, perpendicular to side AC, h_b ⊥ AC. h_c starts from the vertex C, perpendicular to the side AB, h_c ⊥ AB. They all intersect at point H, the orthocenter.

• How to calculate the area of a triangle

• When the three sides' lengths are given, the area of a triangle can be calculated using Heron's Formula:

• Area² = s × (s - a) × (s - b) × (s - c)
• Where:
• a, b and c are the lengths of the three sides of the triangle
• s is the semiperimeter of the triangle
• s = ^{(a + b + c)} / ₂

» Click here to see How to calculate the area of a triangle using Heron's Formula

• How to calculate the heights of the triangle

• Knowing the value of the area and the sides' lengths of the triangle, we can calculate the heights h_a, h_b and h_c from vertices A, B and C to the sides a, b and c respectively.
• Starting from the classic formula of the area of a triangle:

• Example: Calculate the three heights of a triangle

• Let's calculate the heights of a triangle with the sides' lengths a = 18, b = 12 and c = 15.

1. First, calculate the semiperimeter of the triangle:

2. Second, calculate the area of the triangle, Heron's Formula:

3. Third, calculate the heights' lengths of the triangle: