How to Solve: Volume of a Cylinder

How to Solve: Volume of a Cylinder

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Ever wondered how much soda fits in a can, or how much water your pool holds? Mastering the volume of a cylinder unlocks real-world problems—and easy marks on your geometry exam!

What You Need To Know First

Before tackling cylinder volume, you must understand:
1. Area of a circle – The formula A = πr² (where r is the radius).
2. Units of measurement – Volume is always in cubic units (e.g., cm³, m³).
3. Height vs. radius – Height (h) is the distance between the two circular bases; radius (r) is half the diameter.

If you’re shaky on any of these, pause and review them first!

Key Vocabulary

Formulas To Know

1. Volume of a Cylinder

Formula: V = πr²h

Variables: - V = Volume (in cubic units, e.g., cm³, m³) - r = Radius of the circular base (in linear units, e.g., cm, m) - h = Height of the cylinder (in linear units)

Memorise this? ✅ YES! (Not always given on exam sheets.)

2. Diameter to Radius Conversion

Formula: r = d/2

When to use: If the problem gives the diameter instead of the radius, convert it first!

Step-by-Step Method

Follow these steps exactly for every cylinder volume problem.

1. Identify the given values.
2. Write down r (radius) and h (height).

3. If given diameter (d), convert to radius: r = d/2.

4. Write the formula.

5. V = πr²h

6. Substitute the values into the formula.

7. Replace r and h with the numbers from Step 1.

8. Calculate r² first.

9. Square the radius (r × r).

10. Multiply by h and π.

11. π × r² × h

12. Simplify and add units.

13. Round π to 3.14 (or 22/7 if specified).

14. Volume is always in cubic units (e.g., cm³, m³).

15. Check your answer.

16. Does it make sense? (A small can shouldn’t hold 1000 cm³!)

Worked Example Using the Steps

Problem: Find the volume of a cylinder with radius 5 cm and height 10 cm.

1. Given:
2. r = 5 cm

3. h = 10 cm

4. Formula:

5. V = πr²h

6. Substitute:

7. V = π × (5)² × 10

8. Calculate r²:

9. 5² = 25

10. V = π × 25 × 10

11. Multiply:

12. 25 × 10 = 250

13. V = 250π

14. Simplify (use π ≈ 3.14):

15. V ≈ 250 × 3.14 = 785 cm³

16. Check:

17. A cylinder with radius 5 cm and height 10 cm should hold ~785 cm³. ✅

Final Answer: 785 cm³

Worked Examples

Example 1 – Basic

Problem: A cylinder has a radius of 3 m and a height of 7 m. Find its volume.

1. Given:
2. r = 3 m

3. h = 7 m

4. Formula:

5. V = πr²h

6. Substitute:

7. V = π × (3)² × 7

8. Calculate r²:

9. 3² = 9

10. V = π × 9 × 7

11. Multiply:

12. 9 × 7 = 63

13. V = 63π

14. Simplify (π ≈ 3.14):

15. V ≈ 63 × 3.14 = 197.82 m³

16. Check:

17. A cylinder this size should hold ~200 m³. ✅

Final Answer: 198 m³ (rounded to 3 significant figures)

What we did and why: - We followed the steps in order to avoid mistakes. - We squared the radius before multiplying by height and π.

Example 2 – Medium (Diameter Given)

Problem: A can has a diameter of 8 cm and a height of 12 cm. What is its volume?

1. Given:
2. d = 8 cm → r = d/2 = 4 cm

3. h = 12 cm

4. Formula:

5. V = πr²h

6. Substitute:

7. V = π × (4)² × 12

8. Calculate r²:

9. 4² = 16

10. V = π × 16 × 12

11. Multiply:

12. 16 × 12 = 192

13. V = 192π

14. Simplify (π ≈ 3.14):

15. V ≈ 192 × 3.14 = 602.88 cm³

16. Check:

17. A can with diameter 8 cm and height 12 cm should hold ~600 cm³. ✅

Final Answer: 603 cm³ (rounded to 3 significant figures)

What we did and why: - We converted diameter to radius first (a common mistake is forgetting this!). - We kept π as 192π until the final step for accuracy.

Example 3 – Exam Style (Disguised Problem)

Problem: A cylindrical water tank has a circumference of 31.4 m and a height of 5 m. What is its volume? (Use π ≈ 3.14)

1. Given:
2. Circumference (C) = 31.4 m

3. h = 5 m

4. Find radius first (using circumference formula):

5. C = 2πr
6. 31.4 = 2 × 3.14 × r
7. 31.4 = 6.28r

8. r = 31.4 / 6.28 = 5 m

9. Now use volume formula:

10. V = πr²h

11. V = 3.14 × (5)² × 5

12. Calculate r²:

13. 5² = 25

14. V = 3.14 × 25 × 5

15. Multiply:

16. 25 × 5 = 125

17. V = 3.14 × 125 = 392.5 m³

18. Check:

19. A tank with circumference 31.4 m and height 5 m should hold ~400 m³. ✅

Final Answer: 393 m³ (rounded to 3 significant figures)

What we did and why: - We used circumference to find radius first (a sneaky exam trick!). - We kept units consistent (meters throughout).

Common Mistakes

Exam Traps

1-Minute Recap

(Speak naturally, as if to a student the night before the exam.)

"Okay, listen up—this is your 60-second cheat sheet for cylinder volume. First, memorise the formula: V = πr²h. That’s π times radius squared times height. If they give you diameter, cut it in half to get the radius. Plug in the numbers, square the radius first, then multiply by height and π. Always check your units—volume is in cubic units, like cm³ or m³. Watch out for tricks: if they give you circumference, use C = 2πr to find the radius first. And if the answer seems way too big or small, recheck your steps. You’ve got this—now go ace that exam!