Triangle Height Calculator

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. For a deeper exploration of these concepts, you can learn about triangle altitude properties at Wolfram MathWorld.

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. You can also review altitude formulas for different triangles for additional calculation methods.

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:

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:

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:

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

How to Find the Height of a Triangle Without the Area

One of the most common questions students and DIYers ask: can I find the height if I don't know the area? The answer is yes — and you have two reliable paths.

If you know all three sides, use Heron's formula. It computes area from the three side lengths alone, after which dividing by twice the chosen base gives you the height. This is the standard approach when you've measured the triangle physically with a tape measure.

If you know two sides and the angle between them, you can skip the area entirely. The height drops out of basic right-triangle trigonometry: h = side × sin(angle). This is the fastest method and is widely used in surveying and structural layout, where angles can be measured directly with a transit or smartphone level.

Height Formulas for Equilateral, Isosceles, and Right Triangles

For special triangles, you don't need a general method — there are direct shortcut formulas you should know.

Equilateral Triangle Height: h = a√3/2

For an equilateral triangle with side length a, the height to any side is:

This formula comes from the Pythagorean theorem: drop a perpendicular from the apex to the base. It bisects the base into two segments of length a/2 and creates a 30-60-90 right triangle whose hypotenuse is a and short leg is a/2. By Pythagoras: h² + (a/2)² = a², so h = √(a² − a²/4) = (a/2)√3. Try our equilateral triangle calculator for a complete property breakdown.

Isosceles Triangle Height

For an isosceles triangle with two equal sides a and base b, the height from the apex to the base is:

This altitude is special — it's also the median, the angle bisector of the apex angle, and the axis of symmetry, all in one line. See the isosceles triangle calculator for the full set of formulas.

Right Triangle Height: h = ab/c

In a right triangle, two of the three altitudes are simply the two legs (since each leg is already perpendicular to the other). The interesting one is the altitude from the right-angle vertex to the hypotenuse:

This drops out of the area formula two ways: Area = ½ab (using the legs) equals Area = ½ch (using the hypotenuse as base). Setting them equal gives h = ab/c. Use our right triangle calculator to solve any right triangle from two known values.

Step-by-Step Worked Examples

Example 1: Height from area and base

A triangular garden bed has an area of 18 m² and a base of 6 m. Find the height to that base.

Verify above: switch the calculator to "From Area and Base" and enter 18 and 6.

Example 2: Height from three sides (Heron's)

A triangular sail has sides 5, 12, and 13 ft. Find the height to the 13 ft side.

(Note: this is actually a 5-12-13 right triangle, so its area equals ½ × 5 × 12 = 30 — Heron's confirms the answer.)

Example 3: Height from SAS (two sides + angle)

A surveyor measures two sides of a plot — 40 m and 60 m — meeting at an angle of 35°. Find the height of the triangle (using the 60 m side as base).

Example 4: Right triangle altitude to the hypotenuse

A right triangle has legs 6 and 8. Find the altitude from the right angle to the hypotenuse.

The two legs themselves are also altitudes — but to each other, not to the hypotenuse.

Where Does the Height Fall in an Obtuse Triangle?

This is the single most common confusion in triangle altitude problems. In an acute triangle, all three heights fall inside the triangle — clean and intuitive. In a right triangle, the two legs are heights (perpendicular to each other already), and the third altitude (from the right angle to the hypotenuse) falls inside.

But in an obtuse triangle, two of the three altitudes fall outside the triangle. To draw them, you must extend the base as a dashed line beyond the triangle, and drop a perpendicular from the opposite vertex to that extended line. The calculation is identical — the perpendicular distance is still the height — it just doesn't visually land inside the shape.

This is why our calculator and diagrams clearly mark the right-angle indicator: it shows where the perpendicular meets the base (or its extension), removing any ambiguity about which line is the true altitude.

What Is the Orthocenter of a Triangle?

The three altitudes of any triangle always meet at a single point called the orthocenter. This is one of the four classical "centers" of a triangle, alongside the centroid (intersection of medians), incenter (intersection of angle bisectors), and circumcenter (intersection of perpendicular bisectors).

The orthocenter's location depends on the triangle's shape:

• Acute triangles: orthocenter sits inside the triangle
• Right triangles: orthocenter coincides with the right-angle vertex
• Obtuse triangles: orthocenter falls outside the triangle

Quick-Reference Formula Table

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.