Surface Area of Triangular Prisms and Cylinders

Same idea as a box: unfold it flat, then add up every piece.

▶ Practice this concept

🧱 See It

Surface area never changes its meaning: it's the total area of every face, once you unfold the solid flat into its net. A box gave three pairs of rectangles. These two shapes just give different flat pieces.

A triangular prism (think of a Toblerone bar or a tent) unfolds into 2 triangles (the two ends) plus 3 rectangles (the three long sides). A cylinder (a soup can) unfolds into 2 circles (the top and bottom lids) plus 1 rectangle — the label, peeled off and laid flat.

✏️ Draw It

A triangular prism — two triangular ends, three rectangles wrapping between them:

2 triangular ends + 3 rectangular sides = 5 faces in the net.

A cylinder — the real can, then that same can unrolled flat:

r h

The real can: radius r across the lid, height h up the side.

top bottom width = circumference h

Unrolled: the rectangle's width is the circle's circumference (π × d) — the label wraps exactly once around the lid.

🔢 Write It

Triangular prism — two triangle ends plus the three rectangles:

SA = 2 × (triangle area) + (triangle perimeter) × (length)

The three rectangles always share the same length (the prism's length); their widths are the three sides of the triangle. Adding the three widths first gives the triangle's perimeter, so perimeter × length covers all three rectangles at once.

Cylinder — two circle lids plus the one wrapped rectangle:

SA = 2 × (π r²) + (2 π r) × h

The rectangle's width is the circumference 2πr, NOT the diameter — the single most common cylinder-surface-area mistake. The label has to reach all the way around, not just across.

💡 Worked Examples

Example 1 — Triangular prism. The triangular end has base 6 cm, height 4 cm, and its three sides are 5 cm, 5 cm, and 6 cm. The prism is 10 cm long.

  1. One triangle's area: ½ × 6 × 4 = 12. Two ends: 2 × 12 = 24
  2. Triangle's perimeter: 5 + 5 + 6 = 16
  3. All three rectangles: 16 × 10 = 160
  4. Add: 24 + 160 = 184
  5. Answer: Surface Area = 184 cm²

Example 2 — Cylinder. Radius 3 cm, height 10 cm (use π ≈ 3.14).

  1. Two circle lids: 2 × π × 3² = 2 × π × 9 = 18π
  2. Wrapped rectangle: (2 × π × 3) × 10 = 60π
  3. Add: 18π + 60π = 78π
  4. As a number: 78 × 3.14 ≈ 245
  5. Answer: Surface Area ≈ 245 cm²

🧠 Quick Reference

Triangular prism: 2 triangles + 3 rectangles → 2 × triangle area + perimeter × length. Cylinder: 2 circles + 1 rectangle → 2πr² + 2πr × h. Unfold it into the net first, and the "which pieces?" question answers itself.
👪 For Parents & Tutors

BC Curriculum (Level 3 / Grade 8, Geometry & Measurement): "Surface area and volume of regular solids, including triangular and other right prisms and cylinders."

Watch for: the diameter-vs-circumference slip on the cylinder — kids reach for the width they can "see" across the lid (the diameter) instead of the distance the label actually travels (the circumference, 2πr). Physically peeling the label off a can and measuring how long it is makes this concrete in a way a formula can't. Also watch for counting only ONE circle on a cylinder or forgetting a triangular end on the prism — unfolding into the net and literally counting the pieces (2 + 3, or 2 + 1) is the guard.

Comes up again: this extends Surface Area of Prisms (rectangular boxes) to the two curved/triangular solids the curriculum names, and pairs with Volume of Triangular Prisms and Level 2's Volume of Cylinders — same solids, outside vs. inside.