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Altitude of a Triangle: Definition, Height & Orthocenter

Altitude of a Triangle
Height of a Triangle

The distance between a vertex of a triangle and the opposite side. Formally, the shortest line segment between a vertex of a triangle and the (possibly extended) opposite side. Altitude also refers to the length of this segment.

Note: The three altitudes of a triangle are concurrent, intersecting at the orthocenter.

Triangle with dashed perpendicular altitude from the top vertex to the base.

Key Formula

A=12 b hA = \frac{1}{2} \, b \, h
Where:
  • AA = Area of the triangle
  • bb = Length of the base (the side to which the altitude is drawn)
  • hh = Altitude (height) to that base, measured perpendicular to it

Worked Example

Problem: A triangle has vertices A(0, 0), B(8, 0), and C(3, 6). Find the length of the altitude from vertex C to side AB.
Step 1: Identify the base. Side AB lies along the x-axis from (0, 0) to (8, 0), so it is a horizontal segment.
Step 2: Since the altitude from C must be perpendicular to AB, and AB is horizontal, the altitude is a vertical drop from C straight down to the x-axis.
Step 3: The foot of the altitude is the point directly below C on the x-axis, which is (3, 0). The altitude length is the difference in y-coordinates.
h=6−0=6h = 6 - 0 = 6
Step 4: Verify using the area formula. The base AB has length 8, so the area should be:
A=12(8)(6)=24A = \frac{1}{2}(8)(6) = 24
Answer: The altitude from C to side AB is 6 units long, and the triangle's area is 24 square units.

Another Example

Problem: In an obtuse triangle, the sides have lengths a = 13, b = 14, and c = 15. Find the altitude from the vertex opposite the side of length 14.
Step 1: First, compute the area using Heron's formula. The semi-perimeter is:
s=13+14+152=21s = \frac{13 + 14 + 15}{2} = 21
Step 2: Apply Heron's formula:
A=s(s−a)(s−b)(s−c)=21⋅8⋅7⋅6=7056=84A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84
Step 3: Use the area formula solved for the altitude, with the base b = 14:
h=2Ab=2(84)14=12h = \frac{2A}{b} = \frac{2(84)}{14} = 12
Answer: The altitude to the side of length 14 is 12 units.

Frequently Asked Questions

Can an altitude of a triangle fall outside the triangle?
Yes. In an obtuse triangle, the altitudes drawn from the two acute-angled vertices land on the extensions of the opposite sides, not on the sides themselves. The foot of such an altitude lies outside the triangle. This is why the formal definition mentions the 'possibly extended' opposite side.
How many altitudes does a triangle have?
Every triangle has exactly three altitudes, one from each vertex. All three altitudes (or their extensions) meet at a single point called the orthocenter. In an acute triangle the orthocenter is inside the triangle; in a right triangle it sits at the right-angle vertex; and in an obtuse triangle it lies outside the triangle.

Altitude vs. Median

An altitude drops perpendicularly from a vertex to the opposite side, while a median connects a vertex to the midpoint of the opposite side. A median always lies entirely inside the triangle; an altitude may not. The three altitudes meet at the orthocenter, whereas the three medians meet at the centroid. A median bisects the opposite side; an altitude generally does not (unless the triangle is isosceles with that vertex as the apex).

Why It Matters

The altitude is essential for computing a triangle's area: every area formula relies on a base-height pair. Altitudes also define the orthocenter, one of the four classical triangle centers studied in geometry. In real-world problems — from architecture to surveying — finding the perpendicular height of a triangular shape is one of the most common measurement tasks.

Common Mistakes

Mistake: Drawing the altitude to the midpoint of the opposite side instead of dropping it perpendicularly.
Correction: The altitude must be perpendicular to the opposite side. The segment to the midpoint is the median, which is a different concept. Always ensure a 90° angle at the foot of the altitude.
Mistake: Assuming the foot of every altitude lies on the triangle's side.
Correction: In an obtuse triangle, two of the three altitude feet lie on the extensions of the sides, outside the triangle. You may need to extend the base line to locate the foot.

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