Cylinders

Skills and methods in this topic, easy to hard

1. 1Volume of a cylinderV = πr²h; halve a diameter first

   Methodr = 3, h = 10: 9π × 10 = 90π cm³

   Another way: The π key at the end90π ≈ 283 (3 sig. fig.)

2. 2Surface area of a cylinderCurved 2πrh; total 2πr² + 2πrh

   Methodr = 3, h = 10: 60π + 18π = 78π cm²

   Another way: Take out the factor2πr(r + h) = 6π × 13 = 78π

3. 3Cylinders at workRise = volume ÷ πr²; hollow π(R² − r²)h

   Methodr = 5, 100π cm³ sunk: up 100π ÷ 25π = 4 cm

   Another way: One cylinder minus anotherπR²h − πr²h = π(R² − r²)h

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Volume of a cylinder →

   The cylinder in the picture has a base radius of 7 cm and a height of 2 cm. What is its volume, in terms of π?

   ▶Show the answer and working

   Answer: 98 π cm³

   Base area π × 7² = 49π cm², volume = base area × height = 49π × 2 = 98π cm³.

2. Example 2Surface area of a cylinder →

   The cylinder in the picture has a base radius of 8 cm and a height of 2 cm. What is its total surface area, in terms of π?

   ▶Show the answer and working

   Answer: 160 π cm²

   Curved 2π × 8 × 2 = 32π, the two ends 2 × π × 8² = 128π, 160π cm² in all.

3. Example 3Cylinders at work →

   A cylindrical container of base radius 4 cm has some water in it. 112π cm³ more is poured in, none spilling. How far does the water rise?

   • A. 7 cm
   • B. 112 cm
   • C. 112π ÷ (2π × 4) cm
   • D. 112π ÷ 4π cm

   Choose the right answer

   ▶Show the answer and working

   Answer: A. 7 cm

   The water poured in is a cylinder on the container's base: rise = 112π ÷ (π × 4²) = 112 ÷ 16 = 7 cm.