Volume & Surface Area
3D shapes, prisms, cylinders and composite solids
Explanation
Rectangular Prism (Cuboid)
Volume
V = l × w × h
l = length w = width h = height
Surface Area
SA = 2(lw + lh + wh)
Add each pair of opposite faces × 2
Worked Example: A box is 8 cm long, 5 cm wide, 3 cm tall.
V = 8 × 5 × 3 = 120 cm³
SA = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2 × 79 = 158 cm²
V = 8 × 5 × 3 = 120 cm³
SA = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2 × 79 = 158 cm²
Cube
Volume
V = s³
s = side length (all sides equal)
Surface Area
SA = 6s²
6 identical square faces
Worked Example: A cube has side length 4 cm.
V = 4³ = 64 cm³
SA = 6 × 4² = 6 × 16 = 96 cm²
V = 4³ = 64 cm³
SA = 6 × 4² = 6 × 16 = 96 cm²
Cylinder
Volume
πr²h
r = radius h = height
Surface Area
SA = 2πr² + 2πrh
2 circular ends + curved surface
Worked Example: Cylinder with radius 5 cm, height 10 cm.
V = π × 5² × 10 = 250π ≈ 785.4 cm³
SA = 2π(5)² + 2π(5)(10) = 50π + 100π = 150π ≈ 471.2 cm²
V = π × 5² × 10 = 250π ≈ 785.4 cm³
SA = 2π(5)² + 2π(5)(10) = 50π + 100π = 150π ≈ 471.2 cm²
Triangular Prism
Volume
V = ½ × b × h × l
b = base of triangle h = height of triangle l = length of prism
Surface Area
SA = 2(½bh) + (b + s&sub1; + s&sub2;) × l
2 triangular ends + 3 rectangular faces
s₁, s₂ = other two sides of triangle
s₁, s₂ = other two sides of triangle
Worked Example: Triangular prism with base 6 cm, triangle height 4 cm, length 10 cm. (right-angled triangle, third side = 5 cm)
V = ½ × 6 × 4 × 10 = 120 cm³
SA = 2(½ × 6 × 4) + (6 + 4 + 5) × 10 = 24 + 150 = 174 cm²
V = ½ × 6 × 4 × 10 = 120 cm³
SA = 2(½ × 6 × 4) + (6 + 4 + 5) × 10 = 24 + 150 = 174 cm²
Square Pyramid
Volume
V = ⅓ × b² × h
b = base side length h = perpendicular height
Surface Area
SA = b² + 2bl
b² = square base l = slant height of triangular face
Worked Example: Square pyramid, base 6 cm, height 4 cm, slant height 5 cm.
V = ⅓ × 6² × 4 = ⅓ × 36 × 4 = 48 cm³
SA = 6² + 2 × 6 × 5 = 36 + 60 = 96 cm²
V = ⅓ × 6² × 4 = ⅓ × 36 × 4 = 48 cm³
SA = 6² + 2 × 6 × 5 = 36 + 60 = 96 cm²
Key Tips
Units to Remember
Volume: cm³, m³
1 cm³ = 1 mL • 1000 cm³ = 1 L
Volume: cm³, m³
1 cm³ = 1 mL • 1000 cm³ = 1 L
Tips
• Surface area is in square units (cm²)
• For composite solids, split into parts
• Don't count shared/touching faces
• Surface area is in square units (cm²)
• For composite solids, split into parts
• Don't count shared/touching faces
Practice Questions
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