Height / Altitude
The 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 (or altitude) of a shape is the straight-up distance from the base to the opposite vertex or side, measured at a right angle ($90$ degrees) to the base, never along a slanted side. The altitude of a polygon is formally a line segment drawn perpendicular to the base from the opposite vertex; every triangle has three altitudes, one for each possible base, and for an obtuse triangle the altitude from an acute vertex falls outside the triangle. The altitude from vertex $A$ to side $a$ can be computed as $h_a = 2 \cdot \text{Area} / a$, and 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
Picture a triangle like a tilted mountain: the base is the bottom side, and the height is the straight vertical line from the bottom to the tip, forming a right-angle corner where it meets the base. In an obtuse triangle with the longest side as base, the foot of the perpendicular from the top vertex may lie outside the triangle, yet the area is still $(1/2) \cdot \text{base} \cdot \text{altitude}$. For a triangle with area $30$ and base $10$, the altitude to that base is $h = 2(30)/10 = 6$, letting you find a missing altitude from just the area and base, without coordinates.
Key Insight
If you tilt a triangle, the height line may even fall outside the triangle itself; height is always measured straight up, never along a slant. The three altitudes of any triangle are concurrent, meeting at a point called the orthocenter, which lies inside acute triangles, outside obtuse triangles, and at the right-angle vertex of right triangles. In any triangle, the product of a side and its corresponding altitude equals twice the area, $a h_a = b h_b = c h_c = 2K$, an elegant symmetry showing the three altitudes carry equal geometric weight.