Volume of a Prism
Geometry & MeasurementThe 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, multiply the area of the base (the shape at the end) by the height (how long the prism is). $V = B \times h$.
Example
A triangular prism with a triangle base of area $15$ cm$^2$ and height $8$ cm: $V = 15 \times 8 = 120$ cm$^3$. A rectangular prism $5 \times 4 \times 6$: base area $= 20$, $V = 20 \times 6 = 120$ cm$^3$.
Key Insight
Every prism, no matter what polygon is the base, uses the same formula: base area times height. The tricky part is always finding the base area. Once you have that, just multiply by the height.
Definition
For any prism (right or oblique), $V = Bh$, where $B$ is the area of the polygonal base and $h$ is the perpendicular height between the two bases. This follows from Cavalieri's principle: every cross-section parallel to the bases is congruent to the base, so volume = base area $\times$ height.
Example
Hexagonal prism: regular hexagon base with side $4$ cm (area $= (3\sqrt{3}/2)(16) = 41.57$ cm$^2$), height $10$ cm. $V = 41.57 \times 10 = 415.7$ cm$^3$. The base area formula for regular polygons is $(ns^2)/(4\tan(\pi/n))$.
Key Insight
The same formula works for oblique prisms. Even if the prism leans to one side, if the base area and perpendicular height are the same as a right prism, the volume is the same. This is Cavalieri's principle in action.
Definition
The volume of a prism is the integral of the constant cross-sectional area $B$ over the height interval $[0, h]$: $$V = \int_0^h B \, dz = Bh.$$ For oblique prisms, $h$ is still the perpendicular distance between base planes. The formula generalizes: for any solid with constant cross-section $A$ along axis $z$, $V = Ah$, making it identical to the cylinder formula.
Example
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 $= |u \times v| = |(0,0,9-1)| = 8$. $V = 8 \times 5 = 40$. Equivalently, $V = |\det([u,v,w])| = \left|\det\begin{bmatrix}3&1&0\\1&3&0\\0&0&5\end{bmatrix}\right| = |5(9-1)| = 40$.
Key Insight
The determinant formula for prism volume directly generalizes to the parallelepiped (a prism on a parallelogram base): $V = |\det(A)|$ where $A$ is the $3 \times 3$ matrix of edge vectors. This is the 3-D analog of the parallelogram area as a $2 \times 2$ determinant, unifying 2-D and 3-D measurement.