You can find the area of composite polygons and the surface area of a solid from its net. This lesson puts both to work on real shapes.
Key idea
Area of a composite figure: split it into rectangles and triangles and add, or start with a big rectangle and subtract the missing piece. Surface area: add the areas of all the faces, which is the area of the net.
Why it works
Area adds up when pieces do not overlap, so any way you cut a figure gives the same total. A solid's faces unfold into a flat net without overlapping, so the surface area is the area of that net.
Worked examples
Worked example: An L-shaped yard
Find the area of the L-shaped yard above.
- Big rectangle: $12 \times 10 = 120$. Missing corner: $4 \times 4 = 16$.
- Area: $120 - 16 = 104$ square meters.
- Check by cutting it into two rectangles: $8 \times 10 = 80$ and $4 \times 6 = 24$, and $80 + 24 = 104$ ✓.
Answer: 104 square meters
Worked example: A box
Find the surface area of a box 8 cm long, 5 cm wide and 4 cm tall.
- Top and bottom: $8 \times 5 = 40$ each. Front and back: $8 \times 4 = 32$ each. Sides: $5 \times 4 = 20$ each.
- Total: $2(40 + 32 + 20) = 2 \times 92 = 184$ square cm.
Answer: 184 square cm
Worked example: A triangular prism
A tent is a triangular prism. Its ends are right triangles with sides 6 ft, 8 ft and 10 ft (legs 6 and 8), and it is 12 ft long. Find its surface area.
- Two triangles: $2 \times \frac{1}{2} \times 6 \times 8 = 48$.
- Three rectangles, each 12 ft long: $(6 + 8 + 10) \times 12 = 288$.
- Total: $48 + 288 = 336$ square ft.
Answer: 336 square ft
Surface area and area are measured in square units (square cm, square ft), because they count unit squares. A quick check: does every face of the net appear exactly once in your sum?