Calculate the total and individual face surface areas, volume, space diagonal, and edge length of a rectangular prism (box). Supports open-top boxes with presets for common objects.
A rectangular prism — also called a cuboid or simply a box — is one of the most common three-dimensional shapes you encounter in daily life. From shoe boxes and shipping cartons to entire rooms, everything with six rectangular faces is a rectangular prism. Understanding its surface area is essential for packaging design, material estimation, painting, wrapping, and thermal analysis.
The total surface area of a closed rectangular prism with length l, width w, and height h is SA = 2(lw + lh + wh). Each pair of opposite faces contributes one term: the top and bottom (l × w each), the front and back (l × h each), and the two sides (w × h each). If the box has an open top — like a fish tank, a tray, or a planter — you subtract one l × w face, giving SA = lw + 2lh + 2wh.
Beyond surface area, this calculator also computes the volume (l × w × h), the space diagonal √(l² + w² + h²) — the longest straight line that fits inside the box — the three face diagonals, the total edge length (sum of all 12 edges), and the surface-area-to-volume ratio, which indicates how efficiently the shape encloses space. A cube (l = w = h) minimizes that ratio among all rectangular prisms of equal volume.
Practical applications range from estimating paint for a room (treat the room as an open-top box minus window/door area) to computing heat loss through a container's walls, to choosing the right sized mailing box. Presets for common objects — shoe box, shipping container, room — let you explore real scenarios instantly.
Use this calculator when you need more than a single surface-area number. Packaging teams can separate lid, base, and side material to estimate cardboard or plastic usage. Interior projects can model a room as an open-top rectangular prism to estimate wall area before subtracting doors and windows. Students can also see how changing one dimension affects not just total surface area, but the space diagonal, face diagonals, and the surface-area-to-volume ratio.
The face-by-face breakdown is especially helpful when one dimension dominates the shape. A shallow storage bin, for example, spends most of its material on the base, while a tall shipping box spends more on the side panels. That makes this tool useful for comparing designs, checking hand calculations, and choosing dimensions that balance capacity against material use.
Total SA (closed) = 2(lw + lh + wh) Total SA (open top) = lw + 2lh + 2wh Top/Bottom pair = 2 × l × w Front/Back pair = 2 × l × h Left/Right pair = 2 × w × h Volume = l × w × h Space diagonal = √(l² + w² + h²) Edge length sum = 4(l + w + h)
Result: Total SA = 340 cm², Volume = 400 cm³, Space diagonal ≈ 13.75 cm
For a 10 × 5 × 8 cm box: Top/Bottom = 2(10 × 5) = 100 cm². Front/Back = 2(10 × 8) = 160 cm². Left/Right = 2(5 × 8) = 80 cm². Total SA = 100 + 160 + 80 = 340 cm². Volume = 10 × 5 × 8 = 400 cm³. Space diagonal = √(100 + 25 + 64) = √189 ≈ 13.75 cm.
A closed rectangular prism has six faces, so its total surface area includes both the top and bottom. Open-top containers behave differently because the lid is missing. That distinction matters in practical work: a planter, tray, fish tank, or room interior should not be treated like a sealed box. This calculator handles both cases, letting you see exactly how much area belongs to the base pair, front and back, and left and right faces.
Two boxes can hold similar volumes while using very different amounts of material. Long, flat boxes tend to devote more area to the top and bottom panels, while tall narrow boxes shift more area into the side walls. The breakdown bars and face-diagonal table help you see where the area is concentrated, which is useful for packaging design, insulation planning, and estimating wrap, paint, or panel material.
The most common error is forgetting that opposite faces come in pairs. Another is mixing up surface area with volume: surface area uses square units, while volume uses cubic units. For open-top problems, people often leave the missing lid in the formula by accident. It also helps to label length, width, and height clearly before substituting values so you do not assign the wrong dimensions to a face pair.
SA = 2(lw + lh + wh), where l is the length, w is the width, and h is the height. This sums the areas of all six rectangular faces.
They are the same thing. "Rectangular prism" emphasizes the prism structure, while "cuboid" emphasizes the box shape. Both refer to a 3D shape with six rectangular faces.
For an open-top box, SA = lw + 2lh + 2wh. You include only one base (the bottom) instead of two.
The space diagonal is the longest line segment inside the box, running from one corner to the opposite corner: d = √(l² + w² + h²). Use this as a practical reminder before finalizing the result.
Surface area matters more for material costs (cardboard, wrapping paper), heat transfer (insulation), and painting. Volume matters for capacity (how much fits inside).
For a given volume, a cube always has the minimum surface area among rectangular prisms. This is why nature favors cube-like proportions for minimizing surface exposure.