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what is altitude of a triangle

The altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side (or the line containing that side), and it is also called the height of the triangle.

Quick Scoop: Core Idea

  • An altitude starts at one vertex of the triangle.
  • It meets the opposite side (the “base”) at a right angle, forming a 90° angle.
  • Every triangle has three possible altitudes, one from each vertex.
  • All three altitudes intersect at a special point called the orthocenter.

A simple way to picture it: imagine your triangle is a tent, and you drop a straight pole from the very top vertex down to the ground so that it stands perfectly upright; that pole is the altitude.

Why altitudes matter

  • The altitude is what appears as the “height” in the area formula Area=12×base×height\text{Area}=\tfrac{1}{2}\times \text{base}\times \text{height}Area=21​×base×height.
  • If you know the area and the base, you can find the altitude with
    Altitude=2×Areabase\text{Altitude}=\dfrac{2\times \text{Area}}{\text{base}}Altitude=base2×Area​.

Example:
If a triangle has area 24 and a chosen base of length 8, then its altitude to that base is 2×248=6\dfrac{2\times 24}{8}=682×24​=6.

Where the altitude lies (by triangle type)

  • Acute triangle: All angles are less than 90°, so all three altitudes lie inside the triangle.
  • Right triangle: Two sides themselves are altitudes (the legs), and a third altitude can be drawn from the right-angle vertex to the hypotenuse.
  • Obtuse triangle: The altitude from an obtuse vertex falls outside the triangle, so you extend the opposite side and drop a perpendicular to that extended line.

One short comparison table

Below is a compact comparison of how altitudes behave in common triangle types.

html

<table>
  <tr>
    <th>Triangle type</th>
    <th>Where altitudes lie</th>
    <th>Extra note</th>
  </tr>
  <tr>
    <td>Acute</td>
    <td>All three altitudes inside the triangle.[web:3][web:8]</td>
    <td>Orthocenter is inside.[web:3][web:8]</td>
  </tr>
  <tr>
    <td>Right</td>
    <td>Two altitudes are the legs; the third goes to the hypotenuse.[web:1][web:3]</td>
    <td>Orthocenter is at the right-angle vertex.[web:3]</td>
  </tr>
  <tr>
    <td>Obtuse</td>
    <td>Some altitudes fall outside the triangle.[web:3][web:7][web:8]</td>
    <td>Orthocenter lies outside.[web:3][web:8]</td>
  </tr>
</table>

Tiny “story” to remember it

Imagine shining a flashlight straight down from each vertex of the triangle so the light beam hits the opposite side at a perfect right angle. Each beam is an altitude, and the three beams would all cross at one special point in space: the orthocenter.

TL;DR: The altitude of a triangle is the perpendicular height from a vertex to the opposite side, used to measure area and meeting with the other altitudes at the orthocenter.

Information gathered from public forums or data available on the internet and portrayed here.