A parallelepiped is a polyhedron with six parallelogram faces arranged as three pairs of parallel, congruent opposites. It has 8 vertices and 12 edges. A rectangular parallelepiped, or rectangular prism, is the special case in which all faces are rectangles and the three adjacent dimensions are perpendicular.
Perspective does not change the solid's measurements
A flat drawing suggests depth with slanted segments, but their apparent angles are not measured angles of the solid. For a rectangular prism, identify length a, width b, and height c. Parallel edges have equal lengths.
Change the rectangular prism's dimensions
Move the controls. The solid keeps its proportions and updates area, volume, and diagonal without overlapping labels.
Surface area: count each pair of faces once
The three distinct face areas are ab, ac, and bc, and each occurs twice. A closed prism therefore has surface area A = 2(ab + ac + bc). For an open box, subtract the missing face. Area is measured in square units.
Example with dimensions 8 cm, 5 cm, and 3 cm
The distinct face areas are 8 × 5 = 40 cm², 8 × 3 = 24 cm², and 5 × 3 = 15 cm². Counting both of each gives A = 2(40 + 24 + 15) = 158 cm². This describes an ideal closed exterior without tabs or material thickness.
Volume and capacity
The rectangular-prism volume is V = abc. In the example, V = 8 × 5 × 3 = 120 cm³. Internal measurements can describe capacity; external measurements overstate usable space when walls have thickness. Convert all dimensions to one unit before multiplying.
The space diagonal uses the Pythagorean theorem twice
A face diagonal is √(a² + b²). Together with height c it forms another right triangle, so the space diagonal is d = √(a² + b² + c²). For 8, 5, and 3, d = √98 = 7√2 ≈ 9.90 cm.
Do not confuse a face diagonal with a space diagonal. The former stays on one face; the latter joins opposite vertices through the interior. An oblique parallelepiped also has six parallelogram faces, but its area and diagonal depend on angles, so the rectangular formulas cannot be inferred from three lengths alone.
Scale as a reasonableness check
If all dimensions are multiplied by k, surface area is multiplied by k² and volume by k³. Doubling a similar block quadruples its surface and multiplies its volume by 8. This check helps distinguish area formulas from volume formulas and reveals incorrect units.
