How this instrument works
A rectangular prism, more plainly called a box, is described by three independent dimensions: length, width, and height. Its volume is simply the product of all three, V = l × w × h — the amount of space enclosed inside. Its total surface area sums the areas of all six faces, but since a box has three PAIRS of identical opposite faces (top/bottom, front/back, left/right), the formula only needs to compute three distinct face areas and double them: A = 2(lw + lh + wh).
This is the fully general version a cube's simpler formulas are a special case of — set l = w = h in both formulas here and they collapse exactly to a cube's V = s³ and A = 6s², since all three of a cube's dimensions are, by definition, the same length.
Volume and surface area respond very differently to changing just one dimension. Stretching only the length (leaving width and height fixed) increases volume in direct proportion to that stretch, but increases surface area more slowly, since only the two faces that include the length dimension (front/back and top/bottom, in most labelings) grow at all — the third pair (left/right) stays completely unchanged.
- Enter the box's length into the Length field.
- Enter its width into the Width field.
- Enter its height into the Height field.
- Read Volume and Surface area: the sheet computes both simultaneously from the three dimensions.
Worked example — a 5×4×3 box
A rectangular prism has length 5, width 4, and height 3. Its volume is 5 × 4 × 3 = 60 cubic units, and its total surface area is 2 × (5×4 + 5×3 + 4×3) = 2 × (20+15+12) = 2 × 47 = 94 square units — one figure for the space enclosed, another for the material needed to wrap or coat the whole exterior.
A longer, flatter box measuring 10 by 6 by 2 has volume 10×6×2 = 120 and surface area 2×(60+20+12) = 184 — a larger volume than the golden example, but reached through very different proportions, with the flattened height keeping two of the three face-pairs comparatively small.
Questions
What is the formula for the volume of a rectangular prism?
V = l × w × h — simply multiply the length, width, and height together. This is the same 'cross-section area times length' logic every prism shares, here with a rectangular cross-section (length × width) extruded through the height.
What is the formula for the surface area of a rectangular prism?
A = 2(lw + lh + wh) — add the areas of the three distinct face shapes (formed by each pair of dimensions) and double the total, since each of those three shapes appears twice as opposite faces on the box.
How is a rectangular prism different from a cube?
A cube is the special case of a rectangular prism where all three dimensions happen to be equal. The general rectangular-prism formulas here reduce exactly to a cube's simpler s³ and 6s² formulas once length, width, and height are all set to the same value.
Does stretching one dimension affect volume and surface area equally?
No — stretching just one dimension (say, the length) increases volume in direct proportion to that stretch, since volume is a straight product of all three. Surface area increases too, but more slowly, since only two of the three face-pairs include that dimension at all.
What if one dimension is zero?
The volume becomes exactly zero — the box has flattened into a two-dimensional rectangle with no depth. Surface area, however, doesn't necessarily vanish, since the flattened shape can still have area on its remaining front and back faces.