Volume and Surface Area of Solids
Key Formulas
Cube
Volume: V = s³
Surface Area: SA = 6s²
Where s is side length
Cuboid
Volume: V = l × w × h
Surface Area: SA = 2(lw + lh + wh)
Where l is length, w is width, h is height
Sphere
Volume: V = ⁴⁄₃πr³
Surface Area: SA = 4πr²
Where r is radius
Cylinder
Volume: V = πr²h
Surface Area: SA = 2πr² + 2πrh
Where r is radius, h is height
Cone
Volume: V = ⅓πr²h
Surface Area: SA = πr² + πrl
Where r is radius, h is height, l is slant height
Triangular Prism
Volume: V = ½bhl
Surface Area: SA = bh + l(a+b+c)
Where b is base, h is height, l is length
Worked Examples
Example 1: Cylinder
Problem: A cylinder has radius 8 cm and height 12 cm. Calculate its volume and surface area.
Solution:
Volume = πr²h = π × 8² × 12 = π × 64 × 12 ≈ 2412.74 cm³
Surface Area = 2πr² + 2πrh = 2π(64) + 2π(96) = 128π + 192π = 320π ≈ 1005.31 cm²
Example 2: Sphere
Problem: A sphere has volume 80 cm³. Find its radius.
Solution:
V = ⁴⁄₃πr³
80 = ⁴⁄₃πr³
r³ = (80 × 3) ÷ (4π) ≈ 240 ÷ 12.566 ≈ 19.1
r ≈ ∛19.1 ≈ 2.67 cm
Example 3: Cone
Problem: A cone has volume 310 cm³. Its height is twice its radius. Find the slant height.
Solution:
V = ⅓πr²h = ⅓πr²(2r) = ⅔πr³
310 = ⅔πr³
r³ = (310 × 3) ÷ (2π) ≈ 930 ÷ 6.283 ≈ 148.05
r ≈ ∛148.05 ≈ 5.29 cm
h = 2r ≈ 10.58 cm
l = √(r² + h²) = √(28 + 112) = √140 ≈ 11.83 cm
Problem-Solving Strategies
1. Identify the Shape
Determine what type of solid you're working with and recall the appropriate formulas.
2. List Known Values
Write down all given measurements and identify what you need to find.
3. Unit Consistency
Ensure all measurements are in the same units before calculating.
4. Work Step by Step
Break down complex problems into smaller, manageable steps.
Real-World Application: Volume Conversion
A cylindrical tank has radius 50 cm. Water is poured into the tank to a depth of 60 cm. Calculate the volume of water in liters (1 liter = 1000 cm³).
Solution:
1. Volume of water = πr²h = π × 50² × 60
2. = π × 2500 × 60 ≈ 3.1416 × 150000 ≈ 471,240 cm³
3. Convert to liters: 471,240 ÷ 1000 = 471.24 liters
Practice Quiz
Question 1
A cube has a surface area of 384 cm². Find the length of one of its sides.
Question 2
A cylinder has radius 3.6 cm and height 16 cm. Calculate its volume.
Question 3
A cone has radius 4.5 cm and height 10.4 cm. Calculate its volume in terms of π.
Question 4
A sphere has volume 24,430 cm³. Calculate its radius.
Question 5
A cuboid measures 15 cm by 12 cm by 4 cm. Calculate its surface area.
Quiz Answers
- SA = 6s² = 384 → s² = 64 → s = 8 cm
- V = πr²h = π × (3.6)² × 16 ≈ π × 12.96 × 16 ≈ 651.44 cm³
- V = ⅓πr²h = ⅓π × (4.5)² × 10.4 = ⅓π × 20.25 × 10.4 = 70.2π cm³
- V = ⁴⁄₃πr³ = 24,430 → r³ = (24,430 × 3) ÷ (4π) ≈ 18,322.5 → r ≈ ∛18,322.5 ≈ 26.4 cm
- SA = 2(lw + lh + wh) = 2(15×12 + 15×4 + 12×4) = 2(180 + 60 + 48) = 2 × 288 = 576 cm²
Conversion Reference
| Conversion | Relationship | Example |
|---|---|---|
| Volume | 1 liter = 1000 cm³ | 3500 cm³ = 3.5 liters |
| Length | 1 m = 100 cm | 2.5 m = 250 cm |
| Area | 1 m² = 10,000 cm² | 5 m² = 50,000 cm² |
Exam Tips
Show All Steps
Clearly show your working to earn method marks even with calculation errors.
Include Units
Always include correct units in your final answer (cm, cm², cm³, etc.).
Check Reasonableness
Estimate if your answer makes sense in context.
Use Formulas Correctly
Write the formula first, then substitute values.
Common Mistake to Avoid
When calculating surface area, remember to include all faces of the solid. For composite shapes, be careful not to double-count or miss any surfaces.