Volume of a Prism
The volume of a prism equals the area of its base multiplied by its height.
Formula
V = B \times h
Definition
To find the volume of any prism (right or oblique), multiply the area of the base by the perpendicular height between the two bases, $V = Bh$; the tricky part is always finding the base area, but the formula itself is the same for every polygon base. This follows from Cavalieri's principle, since every cross-section parallel to the bases is congruent to the base. As an integral, $$V = \int_0^h B \, dz = Bh,$$ which generalizes to any solid with constant cross-section $A$ along an axis, $V = Ah$, making it identical to the cylinder formula; in vector form, for a prism on a parallelogram base with edge vectors $u$, $v$, and height vector $w$, $V = |\det([u,v,w])|$, the 3-D analog of the parallelogram area as a $2 \times 2$ determinant.
Example
A triangular prism with a triangle base of area $15$ cm$^2$ and height $8$ cm has $V = 15 \times 8 = 120$ cm$^3$; a rectangular prism $5 \times 4 \times 6$ has base area $20$ and $V = 20 \times 6 = 120$ cm$^3$. A hexagonal prism with regular hexagon base of side $4$ cm (area $= (3\sqrt{3}/2)(16) = 41.57$ cm$^2$) and height $10$ cm has $V = 41.57 \times 10 = 415.7$ cm$^3$. For a prism on a parallelogram base with vectors $u=(3,1,0)$ and $v=(1,3,0)$ and height vector $w=(0,0,5)$, base area $= |u \times v| = 8$, so $V = 8 \times 5 = 40$, matching $V = \left|\det\begin{bmatrix}3&1&0\\1&3&0\\0&0&5\end{bmatrix}\right| = |5(9-1)| = 40$.
Key Insight
Every prism, no matter what polygon is the base, uses the same formula, base area times height, once you have found the base area. The same formula works for oblique prisms even if the prism leans to one side, so long as the base area and perpendicular height match a right prism, thanks to Cavalieri's principle. The determinant formula for prism volume generalizes directly to the parallelepiped (a prism on a parallelogram base), unifying 2-D and 3-D measurement.