Surface Area of 3D Shapes

Surface Area Concept

Surface area: Total area of all outer surfaces of a 3D shape

Measured in square units: cm², m², in², ft²

Think: How much paper/paint needed to cover the shape

Surface Area of Cylinders

Cylinder has:

• 2 circular bases (top and bottom)
• 1 rectangular side (curved surface)

Formula: SA = 2πr² + 2πrh

Where:

• 2πr² = area of both circles
• 2πrh = area of curved surface

Alternative: SA = 2πr(r + h)

Example 1: Basic Cylinder

Radius = 4 cm, Height = 10 cm

Calculate:

SA = 2πr² + 2πrh
SA = 2π(4)² + 2π(4)(10)
SA = 2π(16) + 2π(40)
SA = 32π + 80π
SA = 112π cm²
SA ≈ 351.9 cm²

Answer: 112π cm²

Example 2: Factored Form

Radius = 5 m, Height = 8 m

Using SA = 2πr(r + h):

SA = 2π(5)(5 + 8)
SA = 2π(5)(13)
SA = 130π m²
SA ≈ 408.4 m²

Answer: 130π m²

Example 3: Lateral Surface Area Only

Lateral SA (no top/bottom): LSA = 2πrh

Radius = 3 in, Height = 7 in

Calculate:

LSA = 2π(3)(7)
LSA = 42π in²
LSA ≈ 131.9 in²

Use: Open cylinder like a pipe

Surface Area of Cones

Cone has:

• 1 circular base
• 1 curved surface (lateral area)

Formula: SA = πr² + πrℓ

Where:

• πr² = area of base
• πrℓ = lateral surface area
• ℓ = slant height (from edge of base to apex)

Alternative: SA = πr(r + ℓ)

Finding slant height: ℓ² = r² + h² (Pythagorean theorem)

Example 1: Given Slant Height

Radius = 5 cm, Slant height = 13 cm

Calculate:

SA = πr² + πrℓ
SA = π(5)² + π(5)(13)
SA = 25π + 65π
SA = 90π cm²
SA ≈ 282.7 cm²

Answer: 90π cm²

Example 2: Find Slant Height First

Radius = 6 m, Height = 8 m

Find ℓ:

ℓ² = 6² + 8²
ℓ² = 36 + 64
ℓ² = 100
ℓ = 10 m

Calculate SA:

SA = π(6)(6 + 10)
SA = π(6)(16)
SA = 96π m²

Answer: 96π m²

Example 3: Lateral Surface Only

LSA of cone: πrℓ

Radius = 4 ft, Slant height = 10 ft

Calculate:

LSA = π(4)(10) = 40π ft²

Surface Area of Spheres

Sphere: Perfectly round, all surface is curved

Formula: SA = 4πr²

Where r = radius

Note: No base or top, just continuous surface

Example 1: Basic Sphere

Radius = 7 cm

Calculate:

SA = 4πr²
SA = 4π(7)²
SA = 4π(49)
SA = 196π cm²
SA ≈ 615.8 cm²

Answer: 196π cm²

Example 2: Given Diameter

Diameter = 12 m

Radius = 6 m

Calculate:

SA = 4π(6)²
SA = 4π(36)
SA = 144π m²
SA ≈ 452.4 m²

Answer: 144π m²

Example 3: Solve for Radius

SA = 100π ft², Find r

Use formula:

100π = 4πr²
100 = 4r²
r² = 25
r = 5 ft

Answer: r = 5 ft

Hemisphere Surface Area

Hemisphere: Half of a sphere

Has:

• Curved surface (half of sphere)
• Flat circular base

Formula: SA = 2πr² + πr² = 3πr²

Where:

• 2πr² = curved half-sphere
• πr² = flat circular base

Example: Hemisphere

Radius = 6 cm

Calculate:

SA = 3πr²
SA = 3π(6)²
SA = 3π(36)
SA = 108π cm²

Answer: 108π cm²

Summary of Formulas

Cylinder: SA = 2πr² + 2πrh or 2πr(r + h)

Cone: SA = πr² + πrℓ or πr(r + ℓ)

• Remember: ℓ² = r² + h²

Sphere: SA = 4πr²

Hemisphere: SA = 3πr²

Composite Shapes

Add and subtract surface areas

Example 1: Cylinder with Hemisphere Top

Cylinder: r = 4 m, h = 10 m Hemisphere: r = 4 m

Cylinder lateral: 2π(4)(10) = 80π Cylinder bottom: π(4)² = 16π Hemisphere curved: 2π(4)² = 32π

Total:

SA = 80π + 16π + 32π = 128π m²

Note: Don't count cylinder top (covered by hemisphere)

Example 2: Cone on Cylinder

Both have radius 3 cm Cylinder height 8 cm, Cone slant height 5 cm

Cylinder lateral: 2π(3)(8) = 48π Cylinder bottom: π(3)² = 9π Cone lateral: π(3)(5) = 15π

Total:

SA = 48π + 9π + 15π = 72π cm²

Real-World Applications

Painting: Calculate paint needed for curved surfaces

Packaging: Material needed for containers

Construction: Siding for silos, domes

Manufacturing: Sheet metal for tanks

Example 1: Paint a Cylindrical Tank

Tank: diameter 6 m, height 15 m Paint covers 10 m² per liter

Radius = 3 m

Surface area (no bottom, underground):

SA = πr² + 2πrh  (top and sides only)
SA = π(3)² + 2π(3)(15)
SA = 9π + 90π
SA = 99π m²
SA ≈ 311 m²

Paint needed:

311 ÷ 10 = 31.1 liters

Answer: About 31 liters

Example 2: Soccer Ball

Approximate as sphere, diameter 22 cm

Radius = 11 cm

Surface area:

SA = 4π(11)²
SA = 4π(121)
SA = 484π cm²
SA ≈ 1520.5 cm²

Comparing Volume and Surface Area

Different units!

• Volume: cubic units (cm³, m³)
• Surface Area: square units (cm², m²)

Example: Same Dimensions, Different Measures

Sphere with radius 3 cm:

Volume:

V = (4/3)π(3)³ = 36π cm³

Surface Area:

SA = 4π(3)² = 36π cm²

Same number, different units and meanings!

Practice