Volume and surface area of a cuboid

The cuboid is the box: a shape with a height $h$, a length $l$ and a width $w$. Its volume can be found by multiplying the height, the length and the width together, and its surface area by adding up the area of all its sides.

When do I use this?

Whenever a task asks how much fits inside a box shape, or how much material it takes to cover one. Volume is what is inside, surface area is what is around the outside. The two ideas are explained in 3D shapes: volume and surface area.

The volume of a cuboid

The volume of a cuboid can be found by multiplying the height, the length and the width with each other:

$$V = h \times l \times w$$

Here $V$ is the volume. We can see it as finding the area of the bottom of the box, which is $l \times w$, and then multiplying by how tall the box is.

Take a box that is $5$ cm long, $2$ cm wide and $3$ cm high. Then

$$V = 3 \times 5 \times 2 = 30 \text{ cm}^3$$

So $30$ cubic centimetres fit inside the box. The unit is cm³ because we multiplied three lengths together.

The surface area of a cuboid

To find the surface area $S$ of the box, we use the formula

$$S = 2 \times (w \times h + l \times w + l \times h)$$

In this formula we're actually just adding up the area of all the sides. A box has six sides, and they come in three pairs: the front and the back ($w \times h$ each), the bottom and the top ($l \times w$ each), and the two ends ($l \times h$ each). That is where the $2$ in front comes from.

For the same box, with $l = 5$, $w = 2$ and $h = 3$:

$$S = 2 \times (2 \times 3 + 5 \times 2 + 5 \times 3) = 2 \times (6 + 10 + 15) = 2 \times 31 = 62 \text{ cm}^2$$

So we'd need $62$ square centimetres of wrapping paper to cover the box. The unit is cm² this time, because a surface area is an area.

The two answers side by side

For the $5$ by $2$ by $3$ box:

• Volume: $V = 30$ cm³, what fits inside.
• Surface area: $S = 62$ cm², what is around the outside.

Same box, two different questions, two different units. Keeping the unit right is a good way to check you answered the question that was asked.

Common mistakes

• Mixing up volume and surface area. Volume is how much fits inside the box, $h \times l \times w$. Surface area is the area of all the sides added together. Read the question again if you're unsure which one it wants.
• Giving the answer in the wrong unit. Volume is in cubic units (cm³), surface area in square units (cm²). If your unit doesn't match, you have probably calculated wrong somewhere.
• Forgetting the $2$ in the surface area formula. The sides come in pairs, so the bracket only counts three of the six sides. The $2$ counts the other three.

Related

The overview of all the shapes and what volume and surface area mean is in 3D shapes: volume and surface area. The cylinder works the same way, the area of the bottom times the height, in volume and surface area of a cylinder. The area of the rectangle at the bottom is explained in area of a square and a rectangle.