Area, surface area & volume
The idea
Three different questions hide inside one box: how much flat space a face covers (area), how much material would wrap the entire outside (surface area), and how much fits inside (volume). You already find areas of rectangles and count unit cubes for volume; this concept assembles those skills for 3D solids. Surface area is just the sum of the areas of every face, and a rectangular box's volume is length × width × height — layers of unit cubes stacked up.
A reliable mental move for surface area is to unfold the solid into a flat net, so no face gets skipped or counted twice; a box has three pairs of matching faces, so you can find three face areas and double the total. The classic confusion is mixing up surface area and volume — the units expose it instantly. Surface area is measured in square units like cm² because it is flat material; volume needs cubic units like cm³ because it fills space. If your units do not match your question, the setup is wrong.
Worked example
A gift box measures 20 cm long, 12 cm wide, and 8 cm tall. How much wrapping paper is needed to cover it exactly, and how much space is inside?
- Find one face from each of the three pairs: top is 20 × 12 = 240 cm², front is 20 × 8 = 160 cm², and side is 12 × 8 = 96 cm².
- Each face has an identical partner on the opposite side of the box, so the surface area doubles the sum: 2 × (240 + 160 + 96) = 2 × 496 = 992 cm².
- Volume stacks layers of unit cubes: the base holds 20 × 12 = 240 cubes per layer, and 8 layers give 240 × 8 = 1920 cm³.
- Check the units against the questions: wrapping paper is flat material, so cm² fits, while interior space is three-dimensional, so cm³ fits — the two answers measure genuinely different things.
Answer. The box needs 992 cm² of wrapping paper and holds 1920 cm³ of space inside.
Check your understanding
- Why does unfolding a solid into a net help you compute surface area without missing a face?
- How can two boxes share the same volume but need different amounts of wrapping paper?
- Why does area use square units while volume uses cubic units — what is each one counting?
- How would you adapt the face-pairing strategy to a solid that is not a rectangular box, like a triangular prism?
Build the foundations first
Area, surface area & volume builds on these concepts. If any feel shaky, start there.