13. Surface Area and Nets

Surface area measures the outside covering of a 3D solid.

For a box:
\[ SA=2(lw+lh+wh) \]

For a cube:
\[ SA=6s^2 \]

Example: cube side \(3\). Surface area \(=6(3^2)=54\ \text{square units}\).

Common mistake: giving cubic units for surface area. Surface area covers faces, so it uses square units.

Deepen your understanding
Use this focused guide to turn Surface Area and Nets into a dependable GED® problem-solving skill.

Learning targets

• Interpret a net as every outside face of a three-dimensional solid.
• Find box, cube, and cylinder surface area without including hidden interior regions.
• Adjust a formula for open-top or partially covered objects.

Step-by-step solving routine

1. List the exposed faces or surfaces before calculating any area.
2. Pair congruent faces on a rectangular prism: top/bottom, front/back, and left/right.
3. Calculate each face area, multiply by its count, and add the results.
4. Remove any face that is open, attached to another object, or explicitly not covered.

GED® application
A closed (5\)-by-(3\)-by-(2) box has surface area (2(15+10+6)=62) square units. If the (5\)-by-(3) top is open, subtract (15), leaving (47) square units.

Coach check
Annotate the net with a check mark as each face is counted; this prevents missing a face or including one twice.

Visual study gallery
Study each diagram before the worked examples. Name what is measured, identify the needed dimensions or relationships, and explain which formula or rule the picture supports.

10 worked examples and problem solving
Use these examples as guided practice before attempting the session MCQs. Each problem is self-contained, including any needed table or context.

Example 1
Problem: A cube has side length \(3\). What is its surface area?
Answer: \(54\) square units
Solution: Cube surface area is 6s^\(2\): \(6 \times 3^2 = 54\). Pro tip: surface area uses square units, not cubic units.

Example 2
Problem: A box is \(3\) by \(2\) by \(2\). What is its surface area?
Answer: \(32\) square units
Solution: Surface area of a box is \(2\)(lw+lh+wh): \(2\)(\(6+6+4\)) = \(32\). Pro tip: add the three different face areas, then double.

Example 3
Problem: A cube has side length \(5\). What is its surface area?
Answer: \(150\) square units
Solution: Cube surface area is 6s^\(2\): \(6 \times 5^2 = 150\). Pro tip: surface area uses square units, not cubic units.

Example 4
Problem: A box is \(4\) by \(3\) by \(2\). What is its surface area?
Answer: \(52\) square units
Solution: Surface area of a box is \(2\)(lw+lh+wh): \(2\)(\(12+8+6\)) = \(52\). Pro tip: add the three different face areas, then double.

Example 5
Problem: A cube has side length \(6\). What is its surface area?
Answer: \(216\) square units
Solution: Cube surface area is 6s^\(2\): \(6 \times 6^2 = 216\). Pro tip: surface area uses square units, not cubic units.

Example 6
Problem: A box is \(5\) by \(4\) by \(3\). What is its surface area?
Answer: \(94\) square units
Solution: Surface area of a box is \(2\)(lw+lh+wh): \(2\)(\(20+15+12\)) = \(94\). Pro tip: add the three different face areas, then double.

Example 7
Problem: A cube has side length \(8\). What is its surface area?
Answer: \(384\) square units
Solution: Cube surface area is 6s^\(2\): \(6 \times 8^2 = 384\). Pro tip: surface area uses square units, not cubic units.

Example 8
Problem: A box is \(6\) by \(5\) by \(4\). What is its surface area?
Answer: \(148\) square units
Solution: Surface area of a box is \(2\)(lw+lh+wh): \(2\)(\(30+24+20\)) = \(148\). Pro tip: add the three different face areas, then double.

Example 9
Problem: A cube has side length \(10\). What is its surface area?
Answer: \(600\) square units
Solution: Cube surface area is 6s^\(2\): \(6 \times 10^2 = 600\). Pro tip: surface area uses square units, not cubic units.

Example 10
Problem: A box is \(8\) by \(3\) by \(2\). What is its surface area?
Answer: \(92\) square units
Solution: Surface area of a box is \(2\)(lw+lh+wh): \(2\)(\(24+16+6\)) = \(92\). Pro tip: add the three different face areas, then double.

Additional resources

• Formula card: cube surface area \(SA=6s^2\), box surface area \(SA=2(lw+lh+wh)\).
• Net drill: count faces first so surface area does not become volume.