# Altitude of a Triangle: Formulas & Properties

## TL;DR

The altitude of a triangle is the perpendicular segment from a vertex straight down to the line containing the opposite side, and its length is the height used in the area formula. This article covers the definition, the formulas for scalene, isosceles, equilateral, and right triangles, the orthocentre, six worked examples, and the mistakes students make most.

## What Is the Altitude of a Triangle?

The **altitude of a triangle** is a line segment drawn from a vertex perpendicular to the line containing the opposite side. The point where it meets that side is the **foot** of the altitude, and the length of the segment from vertex to foot is the triangle's **height**.

Every triangle has **three altitudes**, one from each vertex, because any of the three sides can be treated as the base. The altitude always meets its base at a right angle (90°). Depending on the triangle, an altitude can fall **inside** the triangle, lie **along a side**, or fall **outside** it, which is a point we come back to for obtuse triangles.

## Properties of the Altitude of a Triangle

The perpendicular-from-a-vertex definition forces a clear set of properties:

- **Three altitudes per triangle**, one to each side.
- **Each altitude is perpendicular to its base**, meeting it at exactly 90°.
- **The three altitudes meet at one point**, the orthocentre.
- **An altitude need not bisect the base** — this is the key difference from a median, which always does.
- **Position depends on the triangle's type.** In an acute triangle all three altitudes are inside; in a right triangle two of them are the legs themselves; in an obtuse triangle two altitudes fall outside the triangle.

## The General Altitude Formula

Because an altitude is the height in the area formula, the two are tied together directly. The area of a triangle is half the base times the height:

Area = \( \frac{1}{2} \times \text{base} \times \text{height} \)  
Rearranging for the height gives the general formula:

\( h = \frac{2 \times \text{Area}}{\text{base}} \)  
So if you know a triangle's area and the side you want the altitude to, one division gives the altitude.

## Altitude Formulas by Triangle Type

Different triangle types give the altitude a cleaner closed form. Each formula below is just the general formula with the area worked out for that shape.

| Triangle type         | Altitude formula                                 | What the variables mean                                                  |
|-----------------------|--------------------------------------------------|-------------------------------------------------------------------------|
| Scalene (any)        | \( h = \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b} \) | a,b,c are the sides; s=\( \frac{a+b+c}{2} \) is the semi-perimeter; altitude is to side b. |
| Isosceles             | \( h = \sqrt{a^2 - \frac{b^2}{4}} \)                  | a is each equal side; b is the base; altitude is from the apex        |
| Equilateral          | \( h = \frac{a \sqrt{3}}{2} \)                     | a is the side length                                                   |
| Right (to a leg)    | the other leg                                   | the two legs are altitudes of each other                               |
| Right (to hypotenuse)| \( h = \sqrt{xy} \)                            | x,y are the two segments the foot makes on the hypotenuse              |

## What Is the Orthocentre?

A natural question once you have three altitudes: do they meet anywhere special? They do. The three altitudes of any triangle always pass through a single common point called the **orthocentre** (often written H). Where that point sits tells you about the triangle:

- **Acute triangle** — the orthocentre lies **inside** the triangle.
- **Right triangle** — the orthocentre sits **exactly at the right-angle vertex.**
- **Obtuse triangle** — the orthocentre falls **outside** the triangle.

## Altitude vs Median: What Is the Difference?

Both run from a vertex to the opposite side, but they are built on different rules. An **altitude** is defined by an angle (it must be perpendicular); a **median** is defined by a point (it must hit the midpoint).

| Feature               | Altitude                           | Median                                   |
|-----------------------|------------------------------------|------------------------------------------|
| Goes from a vertex to | the opposite side, at 90°         | the **midpoint** of the opposite side    |
| Always perpendicular?  | Yes                                | No, not usually                          |
| Always bisects the base? | No                             | Yes, by definition                       |
| Stays inside the triangle? | No (outside for obtuse)      | Yes, always                             |
| Three of them meet at | the **orthocentre**                | the **centroid**                         |

## Examples of Altitude of a Triangle

### Example 1: A triangle has an area of 24 cm² and a base of 6 cm. Find the altitude to that base

Use the general formula:
\( h = \frac{2A}{b} = \frac{2 \times 24}{6} = 8 \text{ cm} \)

Final answer: 8 cm.

### Example 2: Find the altitude to the base of an isosceles triangle with equal sides 10 cm and base 12 cm

Done correctly with the isosceles formula:
\( h = \sqrt{10^2 - \frac{12^2}{4}} = 8 \text{ cm} \)

### Example 3: Find the altitude of an equilateral triangle with side 6 cm

\( h = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \approx 5.20 \text{ cm} \)

### Example 4: A scalene triangle has sides 7 cm, 8 cm, and 9 cm. Find the altitude to the 8 cm side

First, the semi-perimeter and area:
\( s = \frac{7 + 8 + 9}{2} = 12 \)  
\( A = \sqrt{12(12-7)(12-8)(12-9)} \approx 26.83\text{ cm}^2 \)

Then the altitude to the 8 cm base:
\( h = \frac{2A}{b} \approx 6.71 \text{ cm} \)

### Example 5: In a right triangle, the altitude from the right angle meets the hypotenuse and splits it into segments of 4 cm and 9 cm. Find the altitude

\( h = \sqrt{4 \times 9} = 6 \text{ cm} \)

### Example 6: A triangle has sides 5 cm, 12 cm, and 13 cm. Find the altitude to the 13 cm side

\( h = \frac{2 \times 30}{13} \approx 4.62 \text{ cm} \)

## Why the Altitude of a Triangle Matters

The altitude connects area, the Pythagorean theorem, and triangle centres, making it essential for understanding geometry.

## Where Students Trip Up on Altitudes

### Mistake 1: Using a slant side as the height

### Mistake 2: Confusing the altitude with the median

### Mistake 3: Expecting every altitude to stay inside the triangle

## Key Takeaways

- The **altitude of a triangle** is the perpendicular segment from a vertex to the opposite side.
- The general formula is \( h = \frac{2A}{b} \);
- Every triangle has three altitudes that meet at the orthocentre.

## Practice These Problems to Solidify Your Understanding

1. A triangle has area 45 cm² and base 9 cm.
2. Find the altitude of an equilateral triangle with side 10 cm.
3. An isosceles triangle has equal sides 13 cm and base 10 cm.