What is the difference between volume and surface area?

Volume measures the 3D space inside an object in cubic units like \text{cm}^3 or \text{m}^3. Surface area measures the total area of all faces in square units like \text{cm}^2 or \text{m}^2. For a rectangular prism, volume is V = \ell w h while surface area is SA = 2(\ell w + \ell h + w h).

Does the order matter when multiplying the dimensions?

No. Because multiplication is commutative, \ell \cdot w \cdot h = h \cdot \ell \cdot w = w \cdot h \cdot \ell—any order gives the same volume.

What happens if my dimensions are in different units?

You must convert all dimensions to the same unit before multiplying. For example, convert all to meters or all to centimeters. If you multiply 2 meters, 50 centimeters, and 1 meter directly, the result is meaningless because the units don't align.

How do I find a missing dimension if I know the volume?

Rearrange the formula using division. If you know V, length \ell, and width w, then h = \frac{V}{\ell \cdot w}. Substitute the known values and solve for the missing dimension.

Volume of a Rectangular Prism

Calculates the space inside a rectangular box by multiplying length, width, and height; essential for storage, shipping, and construction problems.

V = \ell w h

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What each symbol means

What Volume of a Rectangular Prism takes

Symbol	Meaning
$V$	The volume or space inside the prism, in cubic units like $\text{cm}^3$ or $\text{m}^3$ ; confusing this with surface area (which uses square units) gives a completely different result.
$\ell$	The length dimension in linear units; if you forget to include it or use its square by mistake, you get area, not volume.
$w$	The width dimension in the same unit as length and height; omitting width gives area, and using different units makes the answer meaningless.
$h$	The height dimension in the same unit as length and width; if height is in different units, the volume is wrong, and if omitted, you only get area.

When to use it

Reach for this formula whenever you need to find how much a rectangular box can hold or what space it occupies.

Level

Usually taught in: Geometry · Appears on: SAT

Worked examples

1. Find volume with whole number dimensions

Problem

A rectangular storage box has length 5 cm, width 4 cm, and height 3 cm. What is its volume?

$V = \ell \cdot w \cdot h = 5 \cdot 4 \cdot 3$
Substitute the given dimensions into the formula: length $\ell = 5$ cm, width $w = 4$ cm, height $h = 3$ cm.
$V = 20 \cdot 3$
Multiply the first two dimensions: $5 \cdot 4 = 20$ cm².
$V = 60 \text{ cm}^{3}$
Multiply by the height to find the final volume.

Answer: $V = 60 \text{ cm}^{3}$

We substituted all three dimensions into the volume formula and multiplied them in order. This is the standard approach when all dimensions are known and measured in the same units.

2. Find volume with decimal dimensions

Problem

A rectangular container has dimensions 2.5 meters by 4 meters by 3.2 meters. Calculate its volume.

$V = \ell \cdot w \cdot h = 2.5 \cdot 4 \cdot 3.2$
Substitute the three dimensions into the volume formula: $\ell = 2.5$ m, $w = 4$ m, $h = 3.2$ m.
$V = 10 \cdot 3.2$
Multiply the first two dimensions: $2.5 \cdot 4 = 10$ m².
$V = 32 \text{ m}^{3}$
Multiply by height: $10 \cdot 3.2 = 32$ m³.

Answer: $V = 32 \text{ m}^{3}$

Decimal measurements work exactly like whole numbers—simply multiply all three dimensions together. The order of multiplication does not matter because multiplication is commutative.

3. Find volume with mixed unit dimensions

Problem

A rectangular fish tank measures 2 feet long, 1.5 feet wide, and 18 inches tall. Calculate its volume in cubic feet.

$h = 18 \text{ in} \div 12 = 1.5 \text{ ft}$
First, convert the height to feet so all measurements use the same unit: $18 \text{ inches} \div 12 \text{ inches per foot} = 1.5$ feet.
$V = \ell \cdot w \cdot h = 2 \cdot 1.5 \cdot 1.5$
Now that all dimensions are in feet, substitute into the formula: $\ell = 2$ ft, $w = 1.5$ ft, $h = 1.5$ ft.
$V = 3 \cdot 1.5$
Multiply the first two dimensions: $2 \cdot 1.5 = 3$ ft².
$V = 4.5 \text{ ft}^{3}$
Multiply by height: $3 \cdot 1.5 = 4.5$ cubic feet.

Answer: $V = 4.5 \text{ ft}^{3}$

This problem requires converting units before multiplying. Always ensure all three dimensions use the same unit before calculating volume; otherwise, your answer and its units become meaningless.

Common mistakes

Where Volume of a Rectangular Prism usually goes wrong

The mistake	Why it is wrong	The fix
A box is 5 cm long and 4 cm wide, so its volume is $V = 5 \cdot 4 = 20$ cm³.	Multiplying only two dimensions gives area, not volume.	Include all three dimensions: $V = 5 \cdot 4 \cdot 3 = 60$ cm³ (where height = 3 cm).
To find the volume of a 5 cm × 4 cm × 3 cm box, I calculate $2(5 \cdot 4 + 5 \cdot 3 + 4 \cdot 3) = 94$ cm².	This is the surface area formula, not the volume formula; they measure different things.	Use the volume formula instead: $V = 5 \cdot 4 \cdot 3 = 60$ cm³.
A box is 1 meter long, 50 centimeters wide, and 0.5 meters tall, so $V = 1 \cdot 50 \cdot 0.5 = 25$ .	Multiplying dimensions in different units produces a meaningless result with incorrect units.	Convert to a single unit first: $V = 1 \text{ m} \cdot 0.5 \text{ m} \cdot 0.5 \text{ m} = 0.25$ m³.

Tips and when to use something else

Always express volume in cubic units—never linear units (cm, m) or square units (cm², m²).
Multiplication is commutative: $\ell \cdot w \cdot h$ gives the same answer regardless of the order you multiply.
For other 3D shapes, use different formulas: Volume of a Cylinder is $\pi r^2 h$ and Volume of a Sphere is $\frac{4}{3}\pi r^3$ .
If you know the volume and two dimensions, find the missing one: $h = \frac{V}{\ell \cdot w}$ .

Related formulas

Area of a RectangleMultiply the length and width of a rectangle to find the total area of flat space it covers, measured in square units.Volume of a CylinderCalculates the volume of a cylinder—the three-dimensional space inside—used for any cylindrical container or object in math and engineering.Volume of a PyramidFind the volume inside a pyramid by multiplying one-third of its base area by its height—the most direct way to measure any pyramid's capacity.

Frequently asked questions

What is the difference between volume and surface area?: Volume measures the 3D space inside an object in cubic units like $\text{cm}^3$ or $\text{m}^3$ . Surface area measures the total area of all faces in square units like $\text{cm}^2$ or $\text{m}^2$ . For a rectangular prism, volume is $V = \ell w h$ while surface area is $SA = 2(\ell w + \ell h + w h)$ .
Does the order matter when multiplying the dimensions?: No. Because multiplication is commutative, $\ell \cdot w \cdot h = h \cdot \ell \cdot w = w \cdot h \cdot \ell$ —any order gives the same volume.
What happens if my dimensions are in different units?: You must convert all dimensions to the same unit before multiplying. For example, convert all to meters or all to centimeters. If you multiply 2 meters, 50 centimeters, and 1 meter directly, the result is meaningless because the units don't align.
How do I find a missing dimension if I know the volume?: Rearrange the formula using division. If you know $V$ , length $\ell$ , and width $w$ , then $h = \frac{V}{\ell \cdot w}$ . Substitute the known values and solve for the missing dimension.

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Reviewed 2026-09-18