Surface Area: Prisms, Cylinders, Pyramids
Surface area is just the sum of every face. Unfold the solid into its net, find each face’s area, and add — no face left behind.
1 Key ideas
One strategy for every solid: net it, label every face, add them up.
Rectangular prism & cube
SA = 2(lw + lh + wh) — three pairs of identical faces.
Cube: SA = 6s².
Cylinder
SA = 2πr² + 2πrh — two circles plus the wrapped rectangle.
Use π = 3.14, round to 2 decimals.
Square pyramid
SA = s² + 2sl — base plus four triangles (l = slant height).
Count every face: the most common error is forgetting the bottom (or the second base).
Sketch the net first and number the faces: a prism has 6, a cylinder has 3 pieces, a square pyramid has 5.
Slant height ≠ vertical height. A pyramid’s triangular faces use the slant height
(up the middle of the face). The vertical height is only for volume.
Quick check 1 — Cube with edge 3 cm. Surface area?
Quick check 2 — Rectangular prism 2 × 3 × 4 cm. Surface area?
Quick check 3 — Cylinder r = 3 cm, h = 5 cm. Surface area? (π = 3.14)
Quick check 4 — Square pyramid: base 4 cm, slant height 6 cm. Surface area?
2 Worked examples
Three solids, one method: net, label, add.
Example 1 — rectangular prism
Prism 5 × 2 × 3 cm.
- Face pairs: 5×2 = 10, 5×3 = 15, 2×3 = 6.
- Double each (front/back, top/bottom, left/right): 2(10 + 15 + 6) = 62 cm².
Example 2 — cylinder
Cylinder r = 2 cm, h = 7 cm (π = 3.14).
- Two bases: 2 × 3.14 × 4 = 25.12.
- Lateral: 2 × 3.14 × 2 × 7 = 87.92.
- Total: 25.12 + 87.92 = 113.04 cm².
Example 3 — triangular prism
Right triangular prism: triangle legs 3-4-5, length 7 cm.
- Two triangular bases: 2 × (½ × 3 × 4) = 12.
- Three rectangles: (3 + 4 + 5) × 7 = 84.
- Total: 12 + 84 = 96 cm².
Quick check 5 — A cylinder’s net has 2 circles and 1 rectangle. How many faces does the cylinder have?
Quick check 6 — Triangular prism: 3-4-5 triangle, length 7 cm. Surface area?