Rectangular Prism Calculator
Calculate the volume, surface area, and space diagonal of a rectangular prism (box) from its length, width, and height. Ideal for shipping box sizing, room volume, and material estimation.
About this calculator
A rectangular prism (cuboid) has three perpendicular dimensions: length l, width w, and height h. Its volume is V = l × w × h, giving the total space enclosed. The total surface area covers six rectangular faces: A = 2(lw + lh + wh). The space diagonal — the longest straight line fitting inside — is d = √(l² + w² + h²), derived from applying the Pythagorean theorem twice. A cube is the special case where l = w = h, giving V = l³ and A = 6l². These formulas are foundational in architecture, logistics, packaging design, and any context involving three-dimensional space.
How to use
Suppose a box is 4 m long, 3 m wide, and 2 m high. Volume: V = 4 × 3 × 2 = 24 m³. Surface area: A = 2(4×3 + 4×2 + 3×2) = 2(12 + 8 + 6) = 2 × 26 = 52 m². Space diagonal: d = √(4² + 3² + 2²) = √(16 + 9 + 4) = √29 ≈ 5.39 m. Enter 4, 3, and 2 in the Length, Width, and Height fields respectively and select which quantity to calculate.
Frequently asked questions
How do you calculate the surface area of a rectangular prism?
The surface area formula is A = 2(lw + lh + wh), where l, w, and h are the three dimensions. It accounts for all six faces: two l×w faces (top and bottom), two l×h faces (front and back), and two w×h faces (left and right). For a box 5 × 4 × 3 units: A = 2(20 + 15 + 12) = 2 × 47 = 94 square units. This is especially useful for calculating material needed to wrap or paint a box.
What is the space diagonal of a rectangular prism and how is it calculated?
The space diagonal is the straight-line distance between two opposite corners of the prism, passing through its interior. It is calculated as d = √(l² + w² + h²), applying the 3D version of the Pythagorean theorem. For a 3 × 4 × 12 box: d = √(9 + 16 + 144) = √169 = 13 units. This measurement is important when determining whether a long object (like a rod or tube) can fit inside a box.
How is a rectangular prism different from a cube, and do the same formulas apply?
A cube is a special rectangular prism where all three dimensions are equal (l = w = h). The same formulas apply: volume = l³ and surface area = 6l². For a general rectangular prism the three dimensions can differ independently, giving more flexibility. In practice, cubes maximize volume relative to surface area among all rectangular prisms with a fixed perimeter sum, which is why cube-shaped containers are material-efficient. This calculator handles both cases — just enter equal values for a cube.