Base (of a Shape)
Geometry & MeasurementThe 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 bottom side that the shape "sits on." We use the base and the height together to find the area of many shapes.
Example
In a triangle, the base is usually the bottom side. If the bottom side is $8$ cm long, we call that the base. Any side of a triangle can be chosen as the base, and the height is then measured straight up from that side.
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 you measure the height perpendicularly from that side.
Definition
The base of a two-dimensional figure is any side selected as the reference for computing height and area. The corresponding height (altitude) must be perpendicular to the chosen base. For 3-D solids, the base is a face used to determine the volume formula.
Example
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.
Key Insight
The base-height pairing must always be perpendicular. Confusing slant height with perpendicular height is one of the most common errors in area calculations.
Definition
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: $h$ is the perpendicular distance from the opposite vertex (or side) to the line containing $b$. This duality extends to affine transformations, where the product $b \cdot h$ is invariant under shear.
Example
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 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.