Rectangular prism volume and surface area
Measure the space inside and total exterior area of a rectangular prism.
Rectangular prism volume and surface area is one of 8 solids formulas in the geometry section of this library, and it is used at middle school level.
Why rectangular prism volume and surface area works
Volume counts unit cubes in a block that is so many cubes long, so many wide and so many tall, so the three edge lengths multiply. The outside comes in three matching pairs of faces, front with back, left with right, top with bottom, which is why each product is counted twice.
What each symbol means
$\ell,w,h$ are length, width, and perpendicular height.
Rectangular prism volume and surface area: when it holds
All dimensions must be nonnegative and use the same unit.
When it stops applying
Every corner must be a right angle. Skew the box into a leaning parallelepiped and the edges keep their lengths while the space inside shrinks, so multiplying the three edges no longer measures the contents.
Rectangular prism volume and surface area: a worked example
For $\ell=3,w=4,h=5$, $V=60$ and $S=94$.
The mistake to avoid
What people do: Adding the three different face areas once each and stopping.
Why it goes wrong: That covers only half the box: for a 3 by 4 by 5 box it gives 47 when the full outside is 94 square units.
Do this instead: Double the sum of the three products, because every face has an identical partner on the opposite side.
Rectangular prism volume and surface area: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\ell,w,h$ are length, width, and perpendicular height.
- Check the conditions before substituting. All dimensions must be nonnegative and use the same unit.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Solids
- Level
- Middle school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where rectangular prism volume and surface area comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about rectangular prism volume and surface area
Does it matter which edge I call the height?
No. The three lengths are simply multiplied together, so however you set the box down the volume comes out the same.
How much wrapping paper does a 3 by 4 by 5 box need?
At least 94 square units, which is its total outside, plus some extra for the overlap at the seams.
Do equal volumes force equal surface areas?
No. A 1 by 1 by 64 stick and a 4 by 4 by 4 cube both hold 64 cubic units, yet their outsides measure 258 and 96 square units.
How do I find a missing edge from the volume?
Divide the volume by the product of the two edges you know, since the three lengths multiply together to make it.