A box is made of flat faces. Unfold it and you get a flat pattern called a net. Each face is a rectangle or a triangle whose area you can find.
Key idea
The surface area of a solid is the total area of all its faces, which is the area of its net. For a box: $SA = 2\ell w + 2\ell h + 2wh$.
Why it works
Folding and unfolding moves faces around but does not stretch or shrink any of them, so the total area stays the same. Laying the faces flat lets you see every one, including the bottom and the back, which are hidden when the box sits on a table. Opposite faces of a box match, which is why each area appears twice in the formula.
Worked example: A box
Find the surface area of a box 4 cm long, 3 cm wide and 2 cm tall.
- Top and bottom: $2 \times (4 \times 3) = 24$.
- Front and back: $2 \times (4 \times 2) = 16$.
- Left and right: $2 \times (3 \times 2) = 12$.
- Total: $24 + 16 + 12 = 52$.
Answer: 52 square centimeters
Worked example: A cube
Find the surface area of a cube with 5-inch edges.
- A cube has 6 square faces, each $5 \times 5 = 25$.
- $6 \times 25 = 150$.
Answer: 150 square inches
Worked example: A square pyramid
A pyramid has a square base 6 m on a side, and each triangular face has height 5 m. Find its surface area.
- Base: $6 \times 6 = 36$.
- Each triangle: $\frac{1}{2} \times 6 \times 5 = 15$. Four of them: 60.
- Total: $36 + 60 = 96$.
Answer: 96 square meters
Common mistake
Missing the hidden faces (like the bottom of a box) or counting a face twice. Quick check: count the faces before adding. A box has 6, a square pyramid has 5 (1 square and 4 triangles), and a triangular prism has 5 (2 triangles and 3 rectangles).