A tail of dust

A passing comet pulled a long, tube-shaped trail of dust behind it – almost perfectly round at the front and back, stretched out evenly in between. “I know that! Our fuel tank at home looked like that – like a circle that had been stretched out lengthways.”

Erklärus: “That is a cylinder. Practically a prism whose base is a circle. Let’s calculate.”

🛢️ What is a cylinder?

A cylinder has two circular, parallel bases, joined by a curved lateral surface.

h r
Liri: “So really like a circle-shaped prism?”
Erklärus: “Exactly. The base is a circle: B = π × r².”
Formula – base area
B = π × r²

📐 The net – the lateral surface as a rectangle

Erklärus: “If you unroll the lateral surface, it becomes a rectangle. Its width is the circumference of the circle C = 2×π×r, its height is h.”
Formula – lateral surface area
L = C × h = 2 × π × r × h

📦 Surface area and volume

Formula – surface area
SA = 2 × B + L = 2πr² + 2πrh
Formula – volume
V = B × h = π × r² × h
r=3 cm, h=10 cm – volume? (to 1 decimal place)

🧮 Liri’s cylinder calculator – interactive

🛢️ Calculate a cylinder
🌌 Cylinders in the universe

Rocket stages and many space-station modules are built in the shape of a cylinder – this shape is easy to seal against pressure and to stack efficiently.

❌ Common mistake

For the lateral surface area the diameter is often used instead of the radius! Remember: L = 2×π×r×h – always calculate with the radius.

📝 Exercise 1
r=2 cm, h=5 cm – base area B? (to 1 decimal place)
📝 Exercise 2
r=4 cm, h=6 cm – lateral surface area L? (to 1 decimal place)
📝 Exercise 3
r=2 cm, h=10 cm – volume V? (to 1 decimal place)

The tank worked out

Liri wrote the measurements of the fuel tank onto her star map – volume included.

Liri: “Now I know exactly how much fuel a tank like that could hold!”
Erklärus: “And another piece of your memory is now captured in numbers.”
“Cylinder: B=πr². L=2πrh. SA=2πr²+2πrh. V=πr²h.”
What you learned today
V = π × r² × h
Area of the circle times height – the cylinder is a “round prism”
🌟

Chapter 52 mastered!

Liri can now work out every cylinder.

Erklärus’s notebook – page 52:
“Cylinder = round prism. B=πr², V=πr²h, L=2πrh.”
← Back to overview 52 – The Cylinder