Rectangular prism
A rectangular prism is a three-dimensional solid bounded by six rectangular faces. Opposite faces are congruent and parallel, adjacent faces meet at right angles, and the solid is completely determined by three perpendicular dimensions: length l, width w, and height h. In geometry, the rectangular prism is also called a cuboid, and a cube appears as the special case in which l = w = h.
Geometric structure
The rectangular prism has 6 faces, 12 edges, and 8 vertices. The three face pairs have areas lw, lh, and wh. Because each face area occurs twice, both the surface area and many measurement problems reduce to combinations of these three products.
Volume of a rectangular prism
The volume measures the amount of three-dimensional space enclosed by the solid. Since the base area is lw and the height is h, the product of base area and height gives the volume:
V = lwh
This formula remains valid regardless of which face is regarded as the base, because multiplication is commutative:
lwh = wlh = hlw.
Surface area of a rectangular prism
The surface area is the sum of the areas of all six faces. Two faces have area lw, two have area lh, and two have area wh. Therefore:
S = 2lw + 2lh + 2wh
S = 2(lw + lh + wh)
This expression shows immediately that surface area depends on pairwise products of the dimensions rather than on the triple product that appears in volume.
Diagonal of a rectangular prism
The space diagonal joins two opposite vertices of the solid. Its length follows from the Pythagorean theorem applied twice. The diagonal of the base rectangle is
dbase = √(l2 + w2).
Treating that base diagonal and the height h as perpendicular sides of a right triangle gives the full space diagonal:
d2 = dbase2 + h2
d2 = (l2 + w2) + h2
d = √(l2 + w2 + h2).
Face diagonals
A rectangular prism has three distinct face-diagonal formulas, one for each type of rectangular face:
| Face | Dimensions | Face diagonal |
|---|---|---|
| Top and bottom faces | l and w | √(l2 + w2) |
| Front and back faces | l and h | √(l2 + h2) |
| Left and right faces | w and h | √(w2 + h2) |
Worked example
Consider a rectangular prism with length 8, width 5, and height 3.
V = lwh = 8 · 5 · 3 = 120
S = 2(lw + lh + wh) = 2(8 · 5 + 8 · 3 + 5 · 3) = 2(40 + 24 + 15) = 158
d = √(l2 + w2 + h2) = √(82 + 52 + 32) = √(64 + 25 + 9) = √98 = 7√2.
The prism therefore has volume 120, surface area 158, and space diagonal 7√2.
Special case: cube
If all three dimensions are equal, so that l = w = h = a, the rectangular prism becomes a cube. The general formulas simplify to
V = a3, S = 6a2, d = a√3.
The cube is therefore the most symmetric rectangular prism, but the rectangular prism formulas remain the broader framework.
Comparison of major formulas
| Quantity | Formula | Interpretation |
|---|---|---|
| Volume | V = lwh | Product of the three perpendicular dimensions |
| Surface area | S = 2(lw + lh + wh) | Sum of the areas of all six faces |
| Space diagonal | d = √(l2 + w2 + h2) | Distance between opposite vertices |
Common misconceptions
Volume and surface area measure different geometric quantities. Volume describes enclosed space and is measured in cubic units, while surface area describes exposed boundary and is measured in square units.
The formula l + w + h is not a meaningful total size measure for a rectangular prism in standard geometry. Perimeter belongs to two-dimensional figures, whereas surface area and volume are the central measurements for three-dimensional solids.
The space diagonal is not obtained by adding edge lengths. It follows from the Euclidean distance formula in three dimensions and therefore includes squares and a square root.
A rectangular prism is one of the most fundamental solids in geometry because it connects area, volume, distance, and orthogonality in a single model. Its formulas, V = lwh, S = 2(lw + lh + wh), and d = √(l2 + w2 + h2), form the standard foundation for a wide range of geometric and applied measurement problems.