Base (of a Shape)
The base of a shape is the side or face used as the reference bottom for measuring height and computing area or volume.
Definition
The base of a shape is the side it "sits on," used together with the height to find area; the corresponding height (altitude) must always be perpendicular to whichever side is chosen as the base, whether that side happens to be at the bottom of the figure or not. For 3-D solids, the base is a face used to determine the volume formula. In the generalized area formula $A = (1/2) \cdot \text{base} \cdot \text{height}$ for triangles (or $\text{base} \cdot \text{height}$ for parallelograms), the base $b$ and altitude $h$ are dual quantities, with $h$ the perpendicular distance from the opposite vertex (or side) to the line containing $b$, a duality that extends to affine transformations where the product $b \cdot h$ is invariant under shear.
Example
In a triangle, any side can be chosen as the base, with the height measured straight up (perpendicular) from that side. A parallelogram with sides $10$ cm and $6$ cm: if the $10$ cm side is the base and the perpendicular height is $4$ cm, the area is $10 \times 4 = 40$ cm$^2$, not $10 \times 6$; choosing the slant side as the base would require a different height. For a triangle with vertices at $(0,0)$, $(6,0)$, and $(2,5)$, choosing the side from $(0,0)$ to $(6,0)$ as base gives $b = 6$ and $h = 5$ (the y-coordinate of the opposite vertex), so $A = (1/2)(6)(5) = 15$.
Key Insight
The base does not have to be at the bottom of the figure; any side can be called the base as long as the height is measured perpendicularly from it. Confusing slant height with perpendicular height is one of the most common errors in area calculations. The invariance of base times height under shear transformations (which preserve area) explains why any parallelogram with the same base and height as a rectangle has the same area, even though their shapes differ.