How this instrument works
A cuboid is a box with three edge lengths free to differ — length, width, and height — unlike a cube, where all three are forced equal. Volume follows directly from stacking: one layer covering the l × w floor has area lw, and h identical layers stacked on top give V = lwh cubic units, the three-dimensional cousin of a rectangle's length-times-width.
Surface area works face by face. A cuboid has six rectangular faces, but only three distinct shapes among them, because each face is mirrored by an identical twin on the opposite side: the floor and ceiling each measure l × w, the front and back each measure w × h, and the two sides each measure h × l. Add the three unique areas and double the sum — SA = 2(lw + wh + hl) — rather than list all six separately.
The space diagonal needs the Pythagorean theorem run twice, not once. First treat the floor as a flat rectangle: its corner-to-corner diagonal is √(l² + w²), an ordinary 2D distance. Then tip that flat diagonal up to meet the height, forming a second right triangle whose hypotenuse — the true interior diagonal — is d = √(l² + w² + h²). Set h to 0 and the formula collapses back to the flat-rectangle case, since a cuboid with no height is really just a rectangle.
- Enter the box's edge lengths in Length, Width, and Height, using the same unit for all three.
- Read Volume for how much space the cuboid encloses — length times width times height.
- Read Surface area for the combined area of all six rectangular faces.
- Read Space diagonal for the straight interior line from one corner to the farthest opposite corner.
- Relabel which edge you call Length versus Width or Height freely — all three results stay the same.
Worked example — a 3 × 4 × 5 crate
Take a crate measuring Length l = 3, Width w = 4, and Height h = 5 units. Volume is the straightforward product: V = 3 × 4 × 5 = 60 cubic units. Surface area sums the three distinct face shapes — lw = 12, wh = 20, and hl = 15 square units — then doubles the total for the matching opposite faces: SA = 2(12 + 20 + 15) = 2 × 47 = 94 square units.
The space diagonal turns up a tidy accident of this particular box. The floor's diagonal is √(3² + 4²) = √25 = 5 — the familiar 3-4-5 triangle — and that value happens to equal the height, 5. So the second right triangle has two matching legs, both 5, giving d = √(5² + 5²) = √50 = 5√2 = 7.0710678118654755 units, a 45-45-90 triangle hiding inside an otherwise ordinary box.
Questions
What is the formula for the volume of a cuboid?
Volume is the product of the three edges: V = l × w × h, where l, w, and h are the length, width, and height in matching units. It extends a rectangle's area formula, length × width, one dimension further — stack h layers of an l × w floor and the total space enclosed is lwh cubic units.
How do you find the surface area of a cuboid?
Add the areas of the three distinct faces — lw, wh, and hl — then double that sum, because each face has an identical twin on the opposite side of the box: SA = 2(lw + wh + hl). A 3×4×5 box, for instance, gives 2(12 + 20 + 15) = 94 square units.
What does the space diagonal of a cuboid measure?
It is the straight line from one corner, through the box's interior, to the corner farthest from it — the longest line that fits inside the shape. It is found with d = √(l² + w² + h²), the Pythagorean theorem applied first to the floor and then again to the height.
How is a cuboid different from a cube?
A cube is the special case of a cuboid where all three edges match: l = w = h. A cuboid — a shoebox, a shipping container, a brick — is usually made of three different lengths, but the same three formulas for volume, surface area, and diagonal hold for both shapes without any change.
Why calculate volume, surface area, and diagonal together?
All three answers come from the same three inputs, so entering Length, Width, and Height once covers how much the box holds, how much material would wrap it, and how long a rod would need to be to span it corner to corner — three separate practical questions answered from a single measurement pass.
Does swapping which edge is called length, width, or height change the results?
No. Every formula here — V = lwh, SA = 2(lw + wh + hl), and d = √(l² + w² + h²) — treats l, w, and h symmetrically, so relabeling the edges only changes which field you typed a number into, never the volume, surface area, or diagonal that comes back.