Triangle Altitudes and Height-to-Base Formulas
An altitude (also called the height) of a triangle is a line segment drawn from a vertex
perpendicular to the opposite side. If the triangle is obtuse, the altitude may meet the extension of the opposite side
rather than the side segment itself. The point where the altitude hits the opposite line is called the foot of the altitude.
Every (non-degenerate) triangle has three altitudes: one from each vertex.
The most useful relationship connecting altitude, base, and area is:
\[
A=\frac{1}{2}bh
\quad\Longleftrightarrow\quad
h=\frac{2A}{b}.
\]
Here \(b\) is a chosen base length and \(h\) is the altitude to that base. This formula works for any triangle type (acute,
right, or obtuse) as long as \(A>0\) and \(b>0\). In practice, if you know the triangle’s area and a base length, you can compute
the matching altitude immediately using \(h=\dfrac{2A}{b}\). Conversely, if you know base and height, you can compute the area with
\(A=\dfrac12 bh\).
Using Three Sides (Heron’s Formula)
Sometimes you do not know the area directly but you do know all three side lengths \(a\), \(b\), and \(c\). In that case, you can
compute the area using Heron’s formula:
\[
s=\frac{a+b+c}{2},\qquad
A=\sqrt{s(s-a)(s-b)(s-c)}.
\]
This requires the triangle inequality \(a+b>c\), \(a+c>b\), and \(b+c>a\). Once \(A\) is known, the three altitudes follow from:
\[
h_a=\frac{2A}{a},\qquad
h_b=\frac{2A}{b},\qquad
h_c=\frac{2A}{c},
\]
where \(h_a\) is the altitude to side \(a\) (the side opposite vertex \(A\)), and similarly for \(h_b\) and \(h_c\).
Orthocenter and Triangle Types
The three altitudes are concurrent: they intersect at a single point called the orthocenter. Its position depends on
the triangle type:
- Acute triangle: the orthocenter lies inside the triangle.
- Right triangle: the orthocenter is exactly the right-angle vertex (two legs are already altitudes).
- Obtuse triangle: the orthocenter lies outside the triangle (because two altitudes meet the extensions of sides).
Geometric Interpretation
Altitudes connect area to “base times perpendicular height,” which is why they appear in many geometry problems (area, similarity,
trigonometry, and coordinate geometry). On a coordinate plane, the foot of an altitude is a projection of a vertex onto the
opposite side’s line. This tool visualizes that perpendicular drop and helps you see when the foot falls on the side segment (acute)
versus on an extension (obtuse). Try changing the inputs to watch how the orthocenter shifts, and use the “All altitudes” view to see
the concurrency directly.
