Height / Altitude
Geometry & MeasurementThe height or altitude of a shape is the perpendicular distance from a base to the opposite vertex or parallel side.
Formula
h \text{ is perpendicular to the base}
Definition
The height of a shape is the straight-up distance from the base to the top, measured at a right angle ($90$ degrees) to the base. It is sometimes called the altitude.
Example
A triangle that looks like a tilted mountain: the base is the bottom side, and the height is the straight vertical line from the bottom to the tip of the triangle. The height forms a little square corner where it meets the base.
Key Insight
Height is always measured straight up, never along a slanted side. If you tilt a triangle, the height line may even fall outside the triangle itself.
Definition
The altitude (height) of a polygon is a line segment drawn perpendicular to the base from the opposite vertex or side. For an obtuse triangle, the altitude from an acute vertex falls outside the triangle. Every triangle has three altitudes, one for each possible base.
Example
In an obtuse triangle with the longest side as base, drop a perpendicular from the top vertex to the line containing the base. The foot of that perpendicular may lie outside the triangle. The area is still $(1/2) \cdot \text{base} \cdot \text{altitude}$.
Key Insight
The three altitudes of any triangle are concurrent, meeting at a point called the orthocenter. For acute triangles the orthocenter is inside; for obtuse triangles it is outside; for right triangles it is at the right-angle vertex.
Definition
The altitude from vertex $A$ of a triangle to side $a$ is $h_a = 2 \cdot \text{Area} / a$. In coordinate geometry, the altitude from point $P$ to line $ax + by + c = 0$ has length $|ax_0 + by_0 + c| / \sqrt{a^2 + b^2}$. In 3-D, the altitude of a solid is the perpendicular distance between its two bases.
Example
For a triangle with area $30$ and base $10$, the altitude to that base is $h = 2(30)/10 = 6$. This formula allows finding a missing altitude from just the area and base, without coordinates.
Key Insight
In any triangle, the product of a side and its corresponding altitude equals twice the area: $a h_a = b h_b = c h_c = 2K$. This elegant symmetry means the three altitudes carry equal geometric weight.