Cylinders · Ages 13–14
Cylinders at work
Rise = volume ÷ πr²; hollow π(R² − r²)h.
How to do it
Method
Rise = volume ÷ πr²; hollow π(R² − r²)h
r = 5, 100π cm³ sunk: up 100π ÷ 25π = 4 cm
Another way
One cylinder minus another
πR²h − πr²h = π(R² − r²)h
In full
Capacity and volume: 1 mL = 1 cm³, 1 L = 1000 cm³, 1 m³ = 1000 L. Water in a cylindrical container: its volume = base area × depth = πr² × depth. Water rising: a thing sunk right under has the volume πr² × the rise, so the rise = volume ÷ (πr²). A hollow cylinder (a pipe, a ring, a cup with walls): volume = (outer circle − inner circle) × length = π(R² − r²)h; for a cup with a base, the inside height is the outside less the base's thickness. A rod taller than the water stood in it isn't under the water, so its volume doesn't count: the water keeps its volume but fills only the ring round the rod, new depth = water's volume ÷ π(R² − r²). Water poured from one cylinder into another keeps its volume: the new depth = volume ÷ the new base area. Another way: see a hollow cylinder as a big one less a small one, πR²h − πr²h — the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A cylindrical container of base radius 2 cm has some water in it. 12π cm³ more is poured in, none spilling. How far does the water rise?
- A. 3 cm
- B. 12π ÷ (2π × 2) cm
- C. 12π ÷ 2π cm
- D. 12 cm
Choose the right answer
Show the answer and working
Answer: A. 3 cm
The water poured in is a cylinder on the container's base: rise = 12π ÷ (π × 22) = 12 ÷ 4 = 3 cm.
Example 2
A cylindrical tank has a base radius of 10 cm, with water 3 cm deep. A solid cylindrical rod of base radius 5 cm and height 59 cm is stood upright on the tank's floor; it rises above the water, and none spills. How deep is the water now, in cm?
Show the answer and working
Answer: 4 cm
The water keeps its volume, 100π × 3 = 300π, but now fills only the ring round the rod: π(102 − 52) = 75π. New depth = 300π ÷ 75π = 4 cm. The rod isn't under the water: its whole volume doesn't count.
Example 3
1800 mL of water is poured into an empty cylindrical cup of base radius 7 cm. Using your calculator's π key, how deep is the water, in cm, to 3 significant figures?
Show the answer and working
Answer: 11.7 cm
1800 mL = 1800 cm³; the base is π × 72 = 49π cm², so the depth = 1800 ÷ 49π ≈ 11.7 cm (3 significant figures).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A cylindrical container of base radius 2 cm has some water in it. 20π cm³ more is poured in, none spilling. How far does the water rise?
- A. 20π ÷ 2π cm
- B. 20π ÷ (2π × 2) cm
- C. 5 cm
- D. 20 cm
Choose the right answer
Answer:
2.The cylindrical glass in the picture has a base radius of 6 cm, with water 6 cm deep. In terms of π, how many cm³ of water does it hold?
Answer:
3.The hollow cylinder in the picture (a length of pipe) has an outer radius of 5 cm and an inner radius of 3 cm, and is 23 cm long. What is the volume of the pipe itself, in terms of π?
Answer:
4.The cylindrical container in the picture has a base radius of 5 cm and water 7 cm deep. A stone is dropped in and sinks right under, and the water rises 1 cm. What is the stone's volume, in terms of π?
Answer:
5.1500 mL of water is poured into an empty cylindrical cup of base radius 7 cm. Using your calculator's π key, how deep is the water, in cm, to 3 significant figures?
Answer:
All the answers
- 1. C. 5 cm
- 2. 216 π cm³
- 3. 368 π cm³
- 4. 25 π cm³
- 5. 9.74 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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