Detailed definition
Understanding Altitude
Altitude is a perpendicular segment from a vertex to the opposite side or to the extension of that side. An altitude is a perpendicular segment from a vertex to the opposite side or its extension. Every triangle has three possible altitudes because each side can be treated as a base.
Altitude is central to triangle area because the area formula uses a chosen base together with its corresponding altitude. That means the height must match the selected base, not just any convenient segment.
In obtuse triangles, some altitudes fall outside the triangle because the opposite side must be extended. That makes altitude a good example of why the full geometric definition matters more than the most familiar picture.
Key facts
Important ideas to remember
- An altitude is a perpendicular segment from a vertex to the opposite side or its extension.
- An altitude must meet the opposite side at a right angle.
- A triangle has three altitudes, one from each vertex.
- An altitude can lie outside the triangle in an obtuse case.
Where it is used
Where altitude shows up
- Use altitude when computing triangle area with base times height divided by two.
- Use it in orthocenter problems, since the three altitudes meet there.
- Use it in proofs where perpendicular structure from a vertex matters.
Common mistakes
What to watch out for
- Do not use a slanted side as the altitude unless it is actually perpendicular to the chosen base.
- Do not forget to extend the opposite side when an obtuse triangle requires the altitude outside the figure.
- Do not treat altitude as always vertical; it is perpendicular to the chosen base, whatever the orientation.