Volume & Surface Area: Surface Area of a Box

Learn how to calculate the surface area of boxes, cubes, composite shapes, and shapes with cutouts. Practice exercises with step-by-step solutions.

Practice

📖 Learning Guide

The surface area of a 3D shape is the total area of all its outer faces. Imagine covering the entire shape with stickers without overlaps. The total area of all the stickers needed to cover the shape is its surface area.

💡 Stickers for the Top, Front, and Right faces hovering outside the box.

1. Surface Area of a Box

A box (rectangular prism) has 6 rectangular faces. These faces come in 3 matching pairs (front/back, top/bottom, left/right). We find the surface area by adding the areas of all 6 faces:

Surface Area of a Box

A = 2(l \times w + l \times h + w \times h)

2. Cube as a Special Case

A cube is a special case of a box where all 6 faces are identical squares of side length

s

. The area of each square face is

s^2

, so the total surface area is:

Surface Area of a Cube

A = 6s^2

3. Surface Area of Composite Shapes

When two boxes are joined to form a composite shape, they share an overlapping contact area. Because this overlapping area is inside the shape, it is no longer part of the outer surface.

Therefore, the surface area of the composite shape is the sum of the individual surface areas minus twice the overlapping area (since the overlap covers a face on both boxes):

Surface Area of Composite Shapes

A_{\text{total}} = A_1 + A_2 - 2 \times A_{\text{overlap}}

When boxes share a face, subtract twice the overlapping area from the sum of their individual surface areas.

4. Surface Area with a Cutout

When a rectangular channel is cut out of a box from above through its full width:
• The top face loses an area of

w_{\text{inner}} \times l_{\text{outer}}

, but this is exactly replaced by the bottom of the cut channel.
• The front and back faces each lose a notch of size

w_{\text{inner}} \times h_{\text{inner}}

, reducing the surface area by

2 \times (w_{\text{inner}} \times h_{\text{inner}})

.
• Inside the channel, two new side wall faces are exposed, each of area

l_{\text{outer}} \times h_{\text{inner}}

, increasing the surface area by

2 \times (l_{\text{outer}} \times h_{\text{inner}})

.

So, the remaining surface area is:

Surface Area with a Cutout

\begin{aligned} A_{\text{remaining}} &= A_{\text{outer}} - 2(w_{\text{inner}} \times h_{\text{inner}}) \\ &\quad + 2(l_{\text{outer}} \times h_{\text{inner}}) \end{aligned}

The surface area of a box with a channel cutout is the outer surface area minus twice the removed front/back notch area plus twice the new inner wall area.

Learning Topics

Frequently Asked Questions

What is volume?

Volume is the amount of three-dimensional space occupied by an object. It is measured in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).

How do you calculate the volume of a box?

The volume of a box (rectangular prism) is calculated by multiplying its length, width, and height: V = l × w × h.

What is the surface area of a shape?

Surface area is the total area of all the exterior faces of a 3D shape. It represents how many square units are needed to completely cover the outside of the shape without any overlaps.