Triangle Height Calculator

Determine the altitude of a triangle from known properties.

Select a method and enter the known values to calculate the triangle's height.

Practical Examples

Explore common scenarios for calculating triangle height.

Classic Right Triangle

Three Sides (Heron's Formula)

Calculating the heights of a 3-4-5 right triangle.

Side A: 3

Side B: 4

Side C: 5

Given Area and Base

Base and Area

Finding the height when the area is 24 and the base is 8.

Area: 24

Base: 8

SAS (Side-Angle-Side)

Two Sides and Angle

Calculating height with two sides (10, 12) and the angle (60°) between them.

Side A: 10

Side B: 12

Angle: 60

Isosceles Triangle

Three Sides (Heron's Formula)

Finding the heights of an isosceles triangle with sides 7, 7, and 10.

Side A: 7

Side B: 7

Side C: 10

Other Titles
Understanding Triangle Height: A Comprehensive Guide
An in-depth look at the concepts, formulas, and applications related to the altitude of a triangle.

What is the Height (Altitude) of a Triangle?

  • Defining the Altitude
  • The Orthocenter
  • Heights in Different Triangle Types
The height of a triangle, also known as its altitude, is the perpendicular distance from a vertex to the line containing the opposite side. Every triangle has three altitudes, one for each vertex. This measurement is fundamental in geometry, particularly for calculating a triangle's area.
Key Properties
An altitude forms a right (90-degree) angle with the base. The three altitudes of a triangle are concurrent, meaning they intersect at a single point called the orthocenter. The location of the orthocenter depends on the type of triangle: inside for an acute triangle, on the vertex of the right angle for a right triangle, and outside for an obtuse triangle.

Step-by-Step Guide to Using the Triangle Height Calculator

  • Selecting Your Calculation Method
  • Entering Your Data
  • Interpreting the Results
Our calculator simplifies finding the height. First, choose a calculation method from the dropdown menu based on the information you have.
Input Methods
Using 3 Sides: If you know the lengths of all three sides (a, b, c), select this method. It uses Heron's formula to first find the area, then calculates each height.
Using Base and Area: The most direct method. If you already know the area and the length of one base, this will give you the height relative to that base.
Using 2 Sides and an Angle: If you have the lengths of two sides and the measure of the angle between them (Side-Angle-Side or SAS), this method uses trigonometry to find the height.
After entering your data into the appropriate fields, click 'Calculate Height'. The tool will display the result, and for the 3-side method, it will provide all three corresponding heights.

Mathematical Formulas for Triangle Height

  • Formula 1: Using Area
  • Formula 2: Using Heron's Formula (3 Sides)
  • Formula 3: Using Trigonometry
1. Height from Area and Base
This is the most fundamental height formula. If 'A' is the area and 'b' is the length of the base, the height 'h' relative to that base is: h = (2 * Area) / base
2. Height from Three Sides (Heron's Formula)
When all three sides (a, b, c) are known, we first calculate the semi-perimeter 's': s = (a + b + c) / 2
Then, we find the area using Heron's formula: Area = sqrt(s (s - a) (s - b) * (s - c))
Finally, we can find the height relative to any side, for example, side 'a': hₐ = (2 * Area) / a
3. Height from Two Sides and an Included Angle
If you know sides 'a' and 'b' and the included angle 'C', the height relative to base 'b' (hₑ) is: hₑ = a * sin(C)

Real-World Applications

  • Architecture and Engineering
  • Land Surveying
  • Physics and Navigation
Calculating triangle height is not just an academic exercise; it has numerous practical applications.
Construction and Design
Architects and engineers constantly use these calculations to determine the stability of structures, the slope of roofs, and the forces acting on triangular components like trusses and braces. The height is critical for ensuring structural integrity.
Surveying and Cartography
Land surveyors use triangulation to determine distances and elevations. Calculating the height of a triangle formed by three points helps in creating accurate topographical maps and establishing property boundaries.

Common Questions and Misconceptions

  • Is the Altitude Always Inside the Triangle?
  • Are Median and Altitude the Same?
  • Does a Triangle Have Only One Height?
Altitude vs. Median
A common point of confusion is the difference between an altitude and a median. An altitude is a perpendicular line from a vertex to the opposite side. A median is a line from a vertex to the midpoint of the opposite side. They are only the same in an equilateral triangle or for the unique side of an isosceles triangle.
Location of the Altitude
No, the altitude is not always inside the triangle. In an obtuse triangle, two of the altitudes lie outside the triangle, extending from their vertices to the line containing the opposite side.