How to Calculate the Height of a Triangle

Triangle Height Calculator

Use this calculator to determine the height of a triangle to each of its sides, given the lengths of all three sides.

Calculation Results:

Triangle Area:

Height to Side A:

Height to Side B:

Height to Side C:

Understanding Triangle Height

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

Why is Triangle Height Important?

• Area Calculation: The most common formula for the area of a triangle is Area = 0.5 * base * height. Knowing the height is crucial for this calculation.
• Geometric Properties: Heights are used in various geometric proofs and constructions, such as finding the orthocenter (the intersection point of the three altitudes).
• Engineering and Design: In fields like architecture, engineering, and design, understanding triangle heights is essential for structural stability, calculating volumes, and creating accurate representations.

How to Calculate Triangle Height

There are several methods to calculate the height of a triangle, depending on the information you have available:

Method 1: Given Area and Base

If you already know the area of the triangle and the length of one of its bases, you can easily find the corresponding height using the formula:

Height = (2 * Area) / Base

For example, if a triangle has an area of 12 square units and a base of 6 units, its height to that base would be (2 * 12) / 6 = 4 units.

Method 2: Given All Three Side Lengths (Used in this Calculator)

This is a very common scenario where you only know the lengths of the three sides (let's call them a, b, and c). To find the height, you first need to calculate the triangle's area using Heron's formula, and then apply the area-based height formula.

Steps:

1. Calculate the Semi-Perimeter (s): The semi-perimeter is half the perimeter of the triangle.

   s = (a + b + c) / 2

2. Calculate the Area (A) using Heron's Formula:

   Area = √(s * (s - a) * (s - b) * (s - c))

3. Calculate the Height (h) to each side: Once you have the area, you can find the height corresponding to each side by treating that side as the base.
   • Height to side a (h_a) = (2 * Area) / a
   • Height to side b (h_b) = (2 * Area) / b
   • Height to side c (h_c) = (2 * Area) / c

Method 3: Given Two Sides and an Included Angle

If you know two sides of a triangle and the angle between them (the included angle), you can calculate the area using trigonometry, and then derive the height.

Area = 0.5 * side1 * side2 * sin(included_angle)

Then, use Height = (2 * Area) / Base, where 'Base' would be one of the known sides.

Alternatively, if you want the height to one of the known sides (say, side1), and you have side2 and the angle opposite side1 (let's call it angle A), then Height_to_side1 = side2 * sin(angle_A).

Using the Calculator

Our Triangle Height Calculator simplifies the process for you. Simply enter the lengths of the three sides of your triangle into the respective fields. The calculator will then automatically compute the triangle's area and the height corresponding to each of its sides.

Realistic Examples

Let's look at some practical examples:

Example 1: A Right-Angled Triangle (3-4-5 Triangle)

Consider a right-angled triangle with sides 3, 4, and 5 units.

• Side A = 3
• Side B = 4
• Side C = 5

Calculation:

• Semi-perimeter (s) = (3 + 4 + 5) / 2 = 6
• Area = √(6 * (6 – 3) * (6 – 4) * (6 – 5)) = √(6 * 3 * 2 * 1) = √36 = 6 square units
• Height to Side A (3) = (2 * 6) / 3 = 4 units
• Height to Side B (4) = (2 * 6) / 4 = 3 units
• Height to Side C (5) = (2 * 6) / 5 = 2.4 units

(You can input these values into the calculator to verify.)

Example 2: An Equilateral Triangle

An equilateral triangle has all sides equal. Let's use a side length of 10 units.

• Side A = 10
• Side B = 10
• Side C = 10

Calculation:

• Semi-perimeter (s) = (10 + 10 + 10) / 2 = 15
• Area = √(15 * (15 – 10) * (15 – 10) * (15 – 10)) = √(15 * 5 * 5 * 5) = √1875 ≈ 43.3013 square units
• Height to Side A (10) = (2 * 43.3013) / 10 ≈ 8.6603 units
• Height to Side B (10) = (2 * 43.3013) / 10 ≈ 8.6603 units
• Height to Side C (10) = (2 * 43.3013) / 10 ≈ 8.6603 units

(For an equilateral triangle with side 's', the height is also given by (s * √3) / 2. For s=10, height = (10 * 1.73205) / 2 = 8.66025. Our calculator's result matches.)

Example 3: An Obtuse Triangle

Consider a triangle with sides 7, 10, and 14 units.

• Side A = 7
• Side B = 10
• Side C = 14

Calculation:

• Semi-perimeter (s) = (7 + 10 + 14) / 2 = 15.5
• Area = √(15.5 * (15.5 – 7) * (15.5 – 10) * (15.5 – 14)) = √(15.5 * 8.5 * 5.5 * 1.5) = √1085.0625 ≈ 32.9403 square units
• Height to Side A (7) = (2 * 32.9403) / 7 ≈ 9.4115 units
• Height to Side B (10) = (2 * 32.9403) / 10 ≈ 6.5881 units
• Height to Side C (14) = (2 * 32.9403) / 14 ≈ 4.7058 units

(Input these values to see the results.)

Conclusion

Calculating the height of a triangle is a fundamental skill in geometry with wide-ranging applications. Whether you're working with simple right triangles or more complex scalene or obtuse triangles, understanding the formulas and using tools like this calculator can make the process straightforward and accurate.