3-D Figures (Solids) • Topic 4 of 5

Surface Area

Surface area is the total area of all the outer faces, measured in square units — think of it as the amount of wrapping paper or paint needed to cover the solid. A box has three pairs of matching faces, so its surface area is 2(lw + lh + wh). A cube’s six identical square faces give the tidy 6s². A cylinder’s surface is the two circular ends (2πr²) plus the unrolled curved side (2πrh), which factor together as 2πr(r + h). The common GRE trap is confusing surface area with volume — one is square units of covering, the other is cubic units of filling — so read whether the question asks about covering the outside or filling the inside. Reverse questions on a cube divide the surface area by 6 and take a square root to find the edge.

An unfolded net of a box showing its six rectangular facesSurface area (net)topfrontbottombackleftrightSA = 2(lw + lh + wh)

✅ Solved examples

1. A box measures 3 by 4 by 5. Find its surface area.
SA = 2(lw + lh + wh) = 2(3×4 + 3×5 + 4×5) = 2(12 + 15 + 20) = 2 × 47 = 94.
2. A cube has an edge of 5. Find its surface area.
SA = 6s² = 6 × 25 = 150.
3. A cube has a surface area of 96. Find the length of an edge.
6s² = 96 → s² = 16 → s = 4.
4. A cylinder has a radius of 3 and a height of 4. Find its total surface area.
SA = 2πr(r + h) = 2π × 3 × (3 + 4) = 6π × 7 = 42π.

✏️ Practice — try these, take hints as needed

1. A box measures 2 by 3 by 4. Find its surface area.
SA = 2(lw + lh + wh).
2(6 + 8 + 12).
2 × 26.
52
2. A cube has an edge of 3. Find its surface area.
SA = 6s².
6 × 9.
Multiply.
54
3. A cube has a surface area of 150. Find the length of an edge.
6s² = 150.
s² = 25.
Take the square root.
5
4. A cylinder has a radius of 2 and a height of 5. Find its total surface area.
SA = 2πr(r + h).
2π × 2 × (2 + 5).
4π × 7.
28π
5. A cube has an edge of 10. Find its surface area.
SA = 6s².
6 × 100.
Multiply.
600

📝 Topic test — 8 questions

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