Rectangular prism — volume and surface area
A rectangular prism (also called a cuboid) is a 3-D box: six rectangular faces, twelve edges, eight corners. Every problem about a box boils down to one of two calculations — how much space it holds (volume) or how much material covers its outside (surface area).
Core formulas
Volume: $V = l \times w \times h$
Surface area: $SA = 2(lw + lh + wh)$
where $l$ = length, $w$ = width, $h$ = height.
Labeling the dimensions
Assign any orientation you like — which side is length, width, or height is a convention choice. What matters is that you use all three distinct dimensions and stay consistent within a single problem.
Six rectangular faces
The surface area formula counts every face exactly once. A rectangular prism has three pairs of identical opposite faces:
- Front and back: each has area $l \times h$.
- Left and right (sides): each has area $w \times h$.
- Top and bottom: each has area $l \times w$.
Adding all six: $2lh + 2wh + 2lw = 2(lw + lh + wh)$.
Volume — worked example
A box is 8 units long, 5 units wide, and 3 units tall. Find the volume.
Formula: $V = l \times w \times h$
Substitute: $V = 8 \times 5 \times 3$
Multiply: $V = 40 \times 3 = 120$
Volume = 120 cubic units
Tip: the order of multiplication never matters — $8 \times 5 \times 3 = 5 \times 8 \times 3 = 3 \times 8 \times 5$.
Surface area — worked example
The same box: length 8, width 5, height 3. Find the total surface area.
Formula: $SA = 2(lw + lh + wh)$
Pairs:
$lw = 8 \times 5 = 40$
$lh = 8 \times 3 = 24$
$wh = 5 \times 3 = 15$
Sum the pairs: $40 + 24 + 15 = 79$
Double: $SA = 2 \times 79 = 158$
Surface area = 158 square units
Finding a missing dimension from volume
Rearrange $V = lwh$ to isolate whichever dimension is unknown:
Find l
$l = \dfrac{V}{w \times h}$
Find w
$w = \dfrac{V}{l \times h}$
Find h
$h = \dfrac{V}{l \times w}$
Try a sample problem
Each sample is either a volume or surface area question. Click Refresh example to switch between types.