Rectangular Prism
Geometry & MeasurementA rectangular prism is a 3-D solid with six rectangular faces, including a box or cuboid shape.
Formula
V = l \times w \times h; \ SA = 2(lw + lh + wh)
Definition
A rectangular prism (also called a cuboid or box) is a 3-D shape with $6$ rectangular faces, $12$ edges, and $8$ corners. Every face is a rectangle.
Example
A shoebox, a brick, and a book are all rectangular prisms. If a box is $10$ cm long, $5$ cm wide, and $4$ cm tall, its volume is $10 \times 5 \times 4 = 200$ cm$^3$.
Key Insight
A cube is a special rectangular prism where all three dimensions are equal. All six faces of a cube are squares. Every cube is a rectangular prism, but not every rectangular prism is a cube.
Definition
A rectangular prism (cuboid) has $6$ rectangular faces in $3$ pairs of parallel, congruent rectangles. Volume $V = lwh$; Surface area $SA = 2(lw + lh + wh)$. The space diagonal (longest interior segment) has length $\sqrt{l^2 + w^2 + h^2}$.
Example
A shipping box $30$ cm $\times$ $20$ cm $\times$ $15$ cm: $V = 9000$ cm$^3 = 9$ liters. $SA = 2(600 + 450 + 300) = 2700$ cm$^2$. Space diagonal $= \sqrt{900 + 400 + 225} = \sqrt{1525} = 39.1$ cm.
Key Insight
The space diagonal formula $\sqrt{l^2 + w^2 + h^2}$ is a 3-D extension of the Pythagorean theorem. First apply the theorem in the base to get $\sqrt{l^2 + w^2}$, then again vertically to get $\sqrt{l^2 + w^2 + h^2}$.
Definition
A rectangular prism is the Cartesian product $[0,l] \times [0,w] \times [0,h]$ in $\mathbb{R}^3$, an axis-aligned box. Its volume $lwh$ is the determinant of the diagonal matrix $\text{diag}(l,w,h)$, connecting the box to linear algebra. In 3-D packing problems, rectangular prisms tile $\mathbb{R}^3$ by translation, unlike most other polyhedra.
Example
The surface area to volume ratio $SA/V = 2(lw+lh+wh)/(lwh) = 2(1/h + 1/w + 1/l)$. For a cube ($l=w=h=s$), $SA/V = 6/s$, which decreases as $s$ increases. This is why large animals stay warmer relative to their mass than small animals.
Key Insight
The AM-GM inequality shows that for fixed volume $V = lwh$, the surface area $2(lw+lh+wh)$ is minimized when $l=w=h$ (the cube). This is the 3-D analog of the rectangle isoperimetric result and explains why cubes are efficient packaging shapes.