🎂 Surface Area

Grade 8 · Geometry · FractionRush

Surface Area

Surface area is the total area of all faces of a 3D shape — like unfolding a box and measuring every piece of cardboard.

This topic covers nets, formulae, and practical problems for cuboids, prisms, cylinders, pyramids, cones, and spheres.

Cuboid
SA = 2(lw + lh + wh)
Cylinder
SA = 2πr² + 2πrh
Sphere
SA = 4πr²
Cone
SA = πr² + πrl
Prism
SA = 2×base + perimeter×length
Pyramid
SA = base + ½×P×slant h

📖 Learn

What is Surface Area?

Surface area is the sum of the areas of all faces (flat or curved) of a 3D solid.

A useful strategy is to think about the net — the flat shape you get when you unfold the solid. Surface area = total area of the net.

💡 Units: if lengths are in cm, surface area is in cm². Always check units!

Cuboid and Prisms

Cuboid: SA = 2(lw + lh + wh)
Prism: SA = 2 × (base area) + perimeter of base × length

For any prism: the total surface area is two bases plus all rectangular side faces. The side faces together form a rectangle with width = perimeter of the base and height = length of the prism.

Triangular prism example: base is a triangle with sides a, b, c and area A. Length = L.
SA = 2A + (a + b + c) × L

💡 For a triangular prism with right-angle triangle base: area = ½ × base × height. The three sides are found using Pythagoras if needed.

Cylinder

Curved surface area (CSA): 2πrh
Total surface area: 2πr² + 2πrh = 2πr(r + h)

Think of unwrapping the curved surface — it becomes a rectangle with width 2πr (the circumference) and height h.

Add the two circular ends (each = πr²) to get the total SA.

If the question asks for open top (like a cup), subtract one πr²: SA = πr² + 2πrh

Pyramid and Cone

Square pyramid: SA = l² + 2ls (where l = base side, s = slant height)
General pyramid: SA = base area + ½ × perimeter × slant height
Cone: SA = πr² + πrl (where l = slant height)

The slant height l is the length along the sloping face, not the vertical height h.

To find slant height from vertical height: l² = h² + r² (Pythagoras)

💡 Curved surface area of a cone = πrl. Add πr² for the base if it's a closed cone.

Sphere and Hemisphere

Sphere: SA = 4πr²
Hemisphere (half sphere): curved SA = 2πr²; total SA = 3πr² (add the flat circle)

The sphere formula SA = 4πr² is equivalent to the area of exactly 4 great circles.

For a closed hemisphere (like a bowl with a lid): SA = 2πr² + πr² = 3πr²

For a bowl (open hemisphere — curved only): CSA = 2πr²

✏️ Worked Examples

Example 1: Surface area of a cuboid

Find the surface area of a cuboid with length 8 cm, width 5 cm, height 3 cm.

SA = 2(lw + lh + wh)
= 2(8×5 + 8×3 + 5×3)
= 2(40 + 24 + 15)
= 2 × 79 = 158 cm²

Example 2: Surface area of a triangular prism

A triangular prism has a right-angled triangle base with legs 3 cm and 4 cm. The prism length is 10 cm.

Hypotenuse = √(3² + 4²) = √25 = 5 cm
Base area = ½ × 3 × 4 = 6 cm²
Perimeter of triangle = 3 + 4 + 5 = 12 cm
SA = 2 × 6 + 12 × 10 = 12 + 120 = 132 cm²

Example 3: Surface area of a cylinder

Find the total surface area of a cylinder with radius 4 cm and height 9 cm. (Give answer to 1 d.p.)

SA = 2πr² + 2πrh
= 2π(4)² + 2π(4)(9)
= 2π×16 + 72π
= 32π + 72π = 104π
= 326.7 cm²

Example 4: Surface area of a cone

A cone has base radius 5 cm and vertical height 12 cm. Find the total surface area.

Slant height: l = √(r² + h²) = √(25 + 144) = √169 = 13 cm
SA = πr² + πrl = π(5)² + π(5)(13)
= 25π + 65π = 90π
= 282.7 cm²

🎨 Visualizer

Surface Area Calculator

Enter dimensions above.

Net Explorer

See how a cylinder unfolds into a net — two circles and a rectangle.

Slant Height Finder

Find the slant height of a cone or pyramid from its vertical height and base radius/half-side.

Ex 1 — Cuboids & Cubes

Ex 2 — Prisms

Ex 3 — Cylinders

Ex 4 — Cones & Pyramids

Ex 5 — Spheres & Mixed

⭐ Practice — 20 Questions

🔥 Challenge — 8 Questions