Practice Maths

Surface Area

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Key Terms

Surface area
The total area of all faces and curved surfaces of a 3D solid; measured in square units.
Net
A flat 2D diagram that can be folded to form a 3D solid; useful for visualising and calculating surface area.
Cylinder SA
SA = 2πr² + 2πrh (two circular ends plus the curved rectangular surface).
Cone SA
SA = πr² + πrl where l = slant height = √(r² + h²).
Sphere SA
SA = 4πr².
Composite solid
Split into recognisable parts; add exposed surface areas and subtract any hidden (internal) faces.

The surface area of a 3D solid is the total area of all its outer faces. Think of it as the area of material needed to wrap or cover the solid completely.

Surface Area Formulas

SolidFormulaNotes
Rectangular prismSA = 2(lw + lh + wh)l = length, w = width, h = height
Cylinder (closed)SA = 2πr² + 2πrhTwo circular ends + curved surface
Cylinder CSACSA = 2πrhCurved surface area only
Square pyramidSA = l² + 2lsl = base side length, s = slant height
Cone (total)TSA = πr² + πrll = slant height
Cone slant heightl = √(r² + h²)h = perpendicular height
SphereSA = 4πr²—
Rect. Prism l h w Cylinder r h Pyramid s l Cone r h l Sphere r

Worked Example 1 — Surface area of a closed cylinder

Find the total surface area of a cylinder with r = 4 cm and h = 7 cm.

SA = 2πr² + 2πrh

= 2 × π × 16 + 2 × π × 4 × 7

= 32π + 56π

= 88π

≈ 276.46 cm²

Worked Example 2 — Surface area of a cone

Find the TSA of a cone with r = 5 cm and h = 12 cm.

Step 1: Find slant height.

l = √(r² + h²) = √(25 + 144) = √169 = 13 cm

Step 2: Find TSA.

TSA = πr² + πrl = π × 25 + π × 5 × 13 = 25π + 65π = 90π ≈ 282.74 cm²

Hot Tip: When a solid is “open” (e.g., a tin can with no lid), omit that face from the formula. Read each question carefully to determine which faces are included. For cones and pyramids, always find the slant height first using Pythagoras’ theorem if only the perpendicular height is given.

Full Lesson

Surface area is calculated by finding the area of every face of a 3D solid and adding them together. A helpful way to think about this is to imagine unfolding the solid into a flat net — the surface area equals the total area of that net.

Nets and Face Identification

Rectangular prism: Has 6 rectangular faces arranged in 3 pairs of identical rectangles. The formula SA = 2(lw + lh + wh) captures all three pairs in one expression.

Cylinder: When unfolded, a cylinder has two circular ends (area = πr² each) and one curved surface that becomes a rectangle. The curved surface has width = h and length = circumference = 2πr, giving CSA = 2πrh.

Pyramid: Has one square base (l²) and four triangular faces. Each triangle has base l and slant height s, so area = ½ls. Four faces = 4 × ½ls = 2ls.

Cone: Has one circular base (πr²) and a curved surface. The curved surface “unrolls” into a sector. Its area = πrl where l is the slant height.

Sphere: SA = 4πr². This is exactly 4 times the area of a great circle cross-section — a remarkable result.

Guide Example — Rectangular prism

Find the SA of a box 10 cm × 6 cm × 4 cm.

Face 1 (top/bottom): 10 × 6 = 60 cm² (two of these = 120 cm²)

Face 2 (front/back): 10 × 4 = 40 cm² (two of these = 80 cm²)

Face 3 (sides): 6 × 4 = 24 cm² (two of these = 48 cm²)

SA = 120 + 80 + 48 = 248 cm²

Using formula: SA = 2(lw + lh + wh) = 2(60 + 40 + 24) = 2 × 124 = 248 cm² ✓

Guide Example — Composite solid

A cylinder (r = 3, h = 5) sits on top of a rectangular platform. Find the external SA of the cylinder alone (including the bottom circle).

TSA = 2πr² + 2πrh = 2π(9) + 2π(3)(5) = 18π + 30π = 48π ≈ 150.80 cm²

If the cylinder’s base is glued to a surface, the bottom circle is internal and not counted: SA = πr² + 2πrh = 9π + 30π = 39π ≈ 122.52 cm²

Tip: For composite solids, identify which faces are internal (hidden or joined) and which are external (exposed). Only count external faces.

Mastery Practice

See Answers ➔

  1. Q1 — Surface area of rectangular prisms

    Fluency

    (a) Find the surface area of a rectangular prism with dimensions 5 cm × 3 cm × 4 cm.

    (b) Find the surface area of a rectangular prism with dimensions 12 cm × 8 cm × 6 cm.

  2. Q2 — Surface area of cylinders

    Fluency

    (a) Find the total surface area of a cylinder with r = 5 cm and h = 10 cm.

    (b) Find the total surface area of a cylinder with r = 3.5 cm and h = 8 cm.

  3. Q3 — Surface area of spheres

    Fluency

    (a) Find the surface area of a sphere with r = 7 cm.

    (b) Find the surface area of a sphere with d = 15 m.

  4. Q4 — Open cylinder (no lid)

    Understanding

    An open cylinder (base only, no top) has r = 4 cm and h = 9 cm. Find the total surface area.

  5. Q5 — Square pyramid

    Understanding

    A square pyramid has base side length 8 m and slant height 10 m. Find the total surface area.

  6. Q6 — Cone with given perpendicular height

    Understanding

    A cone has r = 6 cm and perpendicular height h = 8 cm.

    (a) Find the slant height.

    (b) Find the total surface area.

  7. Q7 — Hemisphere

    Understanding

    A solid hemisphere (half-sphere) has r = 5 cm.

    (a) Find the curved surface area.

    (b) Find the total surface area, including the flat circular base.

  8. Q8 — Tin can manufacturing

    Understanding

    A tin can (cylinder with no lid) has r = 4 cm and h = 12 cm.

    (a) Find the CSA (material for the label).

    (b) Find the area of the base.

    (c) How many tins can be made from 10 m² of tin? (Hint: convert units.)

  9. Q9 — Composite solid: cylinder with hemisphere on top

    Problem Solving

    A composite solid consists of a cylinder (r = 5 cm, h = 10 cm) with a hemisphere (r = 5 cm) on top. The flat face of the hemisphere sits on top of the cylinder (this join is internal). The base of the cylinder is external.

    Find the total external surface area.

  10. Q10 — Painting a conical roof on a silo

    Problem Solving

    A grain silo has a cylindrical body (r = 4 m, h = 6 m) topped by a conical roof (r = 4 m, h = 3 m). Only the outer curved surfaces are to be painted (not the base of the cylinder or any internal surfaces).

    (a) Find the slant height of the cone.

    (b) Find the CSA of the cone.

    (c) Find the CSA of the cylinder.

    (d) Find the total area to be painted.

    (e) Paint covers 8 m² per litre. How many litres are needed (round up)?