Calculate Height of a Triangle

Triangle Height Calculator

Enter the lengths of the three sides of the triangle to calculate its height relative to each side.

Understanding the Height of a Triangle

The height (or altitude) of a triangle is a fundamental geometric concept. It represents the perpendicular distance from a vertex to the opposite side (or its extension), which is called the base. Every triangle has three heights, one corresponding to each side chosen as the base.

Why is Triangle Height Important?

The height of a triangle is crucial for several reasons:

• Area Calculation: The most common formula for the area of a triangle is Area = (1/2) * base * height. Knowing the height is essential for finding the area.
• Geometric Properties: Heights are used in various geometric proofs and constructions.
• Engineering and Architecture: In practical applications, understanding the height can be important for structural stability, design, and calculations involving triangular components.

Methods to Calculate Triangle Height

There are several ways to determine the height of a triangle, depending on the information you have:

1. Given Area and Base

If you already know the area of the triangle and the length of one of its sides (which you designate as the base), you can easily find the corresponding height using the area formula:

Height = (2 * Area) / Base

Example: If a triangle has an area of 20 square units and a base of 8 units, its height relative to that base would be (2 * 20) / 8 = 40 / 8 = 5 units.

2. Given All Three Side Lengths (Using Heron's Formula)

When you only know the lengths of the three sides (let's call them a, b, and c), you can first calculate the area of the triangle using Heron's formula, and then use the area to find the height relative to each side.

Steps:

1. Calculate the semi-perimeter (s): This is half the perimeter of the triangle.
   s = (a + b + c) / 2
2. Calculate the Area (A) using Heron's Formula:
   A = sqrt(s * (s - a) * (s - b) * (s - c))
3. Calculate the Height for each side: Once you have the area, you can find the height corresponding to each side by treating that side as the base.
   • Height relative to side a (h_a): h_a = (2 * A) / a
   • Height relative to side b (h_b): h_b = (2 * A) / b
   • Height relative to side c (h_c): h_c = (2 * A) / c

Example: Consider a triangle with sides a = 5, b = 6, and c = 7 units.

• Semi-perimeter s = (5 + 6 + 7) / 2 = 18 / 2 = 9 units.
• Area A = sqrt(9 * (9 - 5) * (9 - 6) * (9 - 7)) = sqrt(9 * 4 * 3 * 2) = sqrt(216) ≈ 14.6969 square units.
• Height for side a: h_a = (2 * 14.6969) / 5 ≈ 5.8788 units.
• Height for side b: h_b = (2 * 14.6969) / 6 ≈ 4.8990 units.
• Height for side c: h_c = (2 * 14.6969) / 7 ≈ 4.1991 units.

3. Using Trigonometry (for specific triangles)

If you know a side length and an angle, you can use trigonometric functions (sine, cosine, tangent) to find the height, especially in right-angled triangles or when an angle and an adjacent side are known.

For example, in a right-angled triangle, if one leg is the base, the other leg is the height. If you know the hypotenuse and an angle, height = hypotenuse * sin(angle).

Triangle Inequality Theorem

It's important to remember that not any three lengths can form a triangle. For three segments to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the Triangle Inequality Theorem:

• a + b > c
• a + c > b
• b + c > a

Our calculator automatically checks this condition to ensure you are working with a valid triangle.

Use the calculator above to quickly determine the heights of your triangle based on its side lengths!