MyTrigCalculator.com

Triangle Height Calculator

Find the altitude using area, sides, or angles

Step 1: Choose Your Method

From Area and Base

When you know the triangle's area and base

From Three Sides

When you know all three side lengths

From Base, Side, and Angle (SAS)

When you know base, adjacent side, and angle

Results

Select a method and enter values to calculate

Understanding Triangle Height

The height (or altitude) of a triangle is the perpendicular distance from a vertex to the opposite side (called the base). Every triangle has three different heights, one for each base. Understanding how to calculate triangle height is fundamental for finding area, analyzing geometric properties, and solving real-world problems in engineering and design.

What is Triangle Height?

The height of a triangle is always measured along a line perpendicular (at 90°) to the base. This perpendicular line extends from the opposite vertex down to the base (or the base extended, in the case of obtuse triangles). The key property is that this line forms a right angle with the base.

Since any side can serve as the base, every triangle has three different heights. However, these heights are related to each other through the triangle's area - the area remains constant regardless of which side you choose as the base.

Calculation Methods

Method 1: From Area and Base

This is the most direct method. If you know the triangle's area and have chosen a base, the height is:

height = 2 × Area ÷ base

This formula comes directly from rearranging the triangle area formula: Area = (base × height) / 2. It's the quickest method when area is already known.

Method 2: From Three Sides (Heron's Formula)

When you know all three sides but not the area, first calculate the area using Heron's formula:

s = (a + b + c) / 2

Area = √[s(s-a)(s-b)(s-c)]

Then use the area-base method above to find the height to any chosen base. This is particularly useful when you've measured the triangle's sides directly and want to find the height without measuring angles.

Method 3: From Two Sides and Included Angle (SAS)

If you know a base, an adjacent side, and the angle between them, you can find the height directly using trigonometry:

height = adjacent side × sin(included angle)

This method is common in surveying and engineering applications where angles can be measured with theodolites or other instruments.

Properties of Triangle Height

  • Perpendicularity: The height always forms a 90° angle with the base
  • Three heights per triangle: One for each possible base, all intersecting at the orthocenter
  • Location varies: In acute triangles, all heights are inside; in right triangles, two heights are the legs; in obtuse triangles, two heights fall outside
  • Area relationship: Area = ½ × base × height for any base-height pair
  • Inverse relationship: For a fixed area, longer bases correspond to shorter heights

Special Triangle Heights

Different triangle types have special height properties:

  • Equilateral Triangle: Height = side × √3/2, creating the famous 30-60-90 triangle when bisected
  • Isosceles Triangle: The height from the apex bisects the base and serves as the line of symmetry
  • Right Triangle: The two legs serve as heights for each other; the height to the hypotenuse requires more complex calculation

Real-World Applications

Triangle height calculations are essential in various fields:

  • Architecture and Construction: Determining roof slopes, truss heights, and structural clearances
  • Civil Engineering: Calculating embankment heights, bridge clearances, and slope stabilization
  • Surveying: Measuring inaccessible heights using triangulation methods
  • Computer Graphics: Rendering triangular meshes and calculating surface normals
  • Physics: Analyzing force components, projectile trajectories, and inclined planes
  • Navigation: Computing elevation changes and obstacle heights

Using This Calculator

Our Triangle Height Calculator supports three different input methods to match your available data:

  1. Area + Base: Enter if you already know the area and want to find the height to a specific base
  2. Three Sides: Enter all three sides when you've measured the triangle's perimeter
  3. SAS Method: Enter base, adjacent side, and included angle when working with angle measurements

The calculator automatically determines which method you're using based on your inputs and provides the height along with related properties like area and perimeter. The visual diagram shows the height as a dashed line perpendicular to the base, with a right angle indicator to clarify the perpendicular relationship.

Frequently Asked Questions

How do you find the height of a triangle if you know the area?

Use the formula: height = (2 × area) / base. This comes from rearranging the area formula (Area = base × height / 2). For example, if a triangle has an area of 24 square units and a base of 6 units, the height is (2 × 24) / 6 = 8 units. This is the most direct method when area is known.

Can a triangle have more than one height?

Yes, every triangle has three different heights - one corresponding to each side that can serve as a base. While all three heights are different lengths (unless the triangle is equilateral), they're related through the triangle's constant area. The three heights always meet at a single point called the orthocenter.

What is Heron's formula and how does it relate to triangle height?

Heron's formula calculates a triangle's area using only the three side lengths: Area = √[s(s-a)(s-b)(s-c)], where s is the semiperimeter. Once you have the area from Heron's formula, you can find the height to any base using height = 2 × area / base. This two-step process lets you find height from just the three sides without needing to measure any angles.

Where is the height located in different types of triangles?

The height's location depends on the triangle type. In acute triangles, all three heights fall inside the triangle. In right triangles, two heights are the legs themselves, and the third falls inside. In obtuse triangles, two of the three heights fall outside the triangle, requiring you to extend the base to meet the perpendicular line from the opposite vertex.

How do you calculate height using trigonometry?

If you know a base, an adjacent side, and the angle between them, use: height = adjacent side × sin(included angle). The sine function gives you the ratio of the height to the adjacent side. For example, with a base of 10, an adjacent side of 8, and an included angle of 30°, the height is 8 × sin(30°) = 8 × 0.5 = 4 units.

Why is triangle height always perpendicular to the base?

The height is defined as the perpendicular (90°) distance from a vertex to the opposite side because this gives the shortest distance and creates the proper geometric relationship for area calculation. Any non-perpendicular line from the vertex to the base would be longer and wouldn't correctly represent the true height needed for the area formula Area = ½ × base × height.

Additional Resources