Cylinder: surface area and volume · Lập trình với Martin

A tin, a mug and a barrel are cylinders. How to calculate how much fits into a cylinder and how much sheet metal is needed for it.

A tin, a mug, a barrel

Take a circle and pull it up — you get a cylinder. A tin, a mug, a glass, a barrel, a pipe, a lemonade can. The most common solid in your kitchen.

A cylinder has two bases (circles at the bottom and the top) and a curved surface (the wall all around). Two numbers describe it: the base radius r and the height v.

Two questions, two formulae:

How much fits inside? Volume: V = π · r² · v. It is the area of the circular base times the height — like a column of water: the base and how many floors on top of each other.

How much material for the surface? Surface area: S = 2 · π · r² + 2 · π · r · v. Two bases plus the unrolled curved surface.

It looks complicated, but everything in those formulae you already can: area of a circle from last lesson, circumference from the one before. A cylinder is just a circle that got promoted.

Sofie and dad are jam-making and they have run out of jars. In the shop there are two sizes: a narrow tall one (radius 4 cm, height 12 cm) and a wide short one (radius 6 cm, height 6 cm). “The tall one looks bigger,” says Sofie. Dad: “Calculate it.” Tall: 3.14 · 16 · 12 = 602.88 cm³. Wide: 3.14 · 36 · 6 = 678.24 cm³. Sofie blinks: the short and wide one holds more. “The radius is squared in the formula, so the width of the cylinder counts for more than the height.” They buy the wide one.

💡

Into volume and surface area belongs the RADIUS of the base. But glasses and mugs are measured across the mouth — that is the diameter. Before you substitute, halve.

Volume and surface area step by step

Volume V = π · r² · v:

Calculate the area of the base: Sp = π · r² (last lesson).
Multiply by the height: V = Sp · v.
Unit: cm³ or m³ — three dimensions, a three in the unit.

Example: r = 3 cm, v = 10 cm. Base: 3.14 · 9 = 28.26 cm². Volume: 28.26 · 10 = 282.6 cm³.

For liquids: 1 litre = 1000 cm³, so this cylinder holds about 0.28 litres — a normal mug.

Surface area S = 2πr² + 2πr · v:

Cut the cylinder open: two circular bases (each πr², hence 2πr²) and the curved surface. The curved surface unrolled on the table is a rectangle: width = circumference of the base 2πr, height = v. From that 2πr · v.

Example with the same cylinder: bases 2 · 28.26 = 56.52 cm². Curved surface: 2 · 3.14 · 3 · 10 = 188.4 cm². Total 244.92 cm².

When which: how much FITS (water, jam, petrol) = volume. How much MATERIAL (sheet metal, a label, paint) = surface area. A label around a tin is only the curved surface — without the bases.

Example 1: how much water in a barrel

A rain barrel has radius 30 cm and height 90 cm. How many litres does it hold?

Base: Sp = 3.14 · 30² = 3.14 · 900 = 2826 cm².

Volume: V = 2826 · 90 = 254,340 cm³.

Litres: 254,340 : 1000 = 254 litres.

Estimate as a check: a box 60 × 60 × 90 cm would have 324 litres; a cylinder inscribed in it is smaller — 254 fits.

Example 2: a label on a tin

A tin has diameter 8 cm and height 11 cm. How big a paper does the label around need?

The label is only the curved surface: a rectangle with width equal to the circumference and height of the tin.

Circumference: o = π · d = 3.14 · 8 = 25.12 cm.

Label: 25.12 · 11 = 276.32 cm² — paper roughly 25 × 11 cm.

The bases are not counted: the lid and the bottom have no label. Always think which parts of the surface the problem really wants.

Example 3: the surface of a whole can

A lemonade can: r = 3.3 cm, v = 11.5 cm. How much sheet metal is needed for it?

Bases: 2 · 3.14 · 3.3² = 2 · 3.14 · 10.89 = 68.39 cm².

Curved surface: 2 · 3.14 · 3.3 · 11.5 = 238.33 cm².

Total: 68.39 + 238.33 = 306.72 cm² — about half a sheet of A4.

Calculate in parts and write mid-results. One long calculation in a calculator = invisible mistakes.

⚠️

Square first, then multiply: with V = π · r² · v you square ONLY the radius, not the height. And do not swap the formulae: volume has r² and the unit cm³, surface area is made of bases and a curved surface and has cm². The unit always gives you away.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Calculate the volume of a cylinder: π · r² · v, with π ≈ 3.14. Write decimals with a point.

Practise the questions

On paper

Measure a real cylinder. The kitchen is full of test problems.

Take a mug: measure the diameter of the mouth and the height. Calculate the volume in cm³ and convert to litres. Then pour water in with a measuring jug and compare.
Calculate the volume of a cylinder with radius 5 cm and height 20 cm. Then a cylinder with radius 10 cm and height 5 cm. Which is bigger and why?
A tin has diameter 10 cm and height 12 cm. Calculate the area of the label (only the curved surface).
A rainwater pipe has radius 6 cm and length 3 m. Calculate the volume (watch the units: convert metres to centimetres).

Now you 💪

I calculate volume as area of the base times height: V = πr²v.
I build surface area from two bases and the curved surface.
I know 1 litre = 1000 cm³, and I watch the units.

✅

Done when: You have a mug measured and checked with water, volumes calculated, a label and a pipe, and a run of five correct in practice.

What to take from this lesson

Volume of a cylinder: V = π · r² · v — base times height.
Surface area: S = 2πr² + 2πrv — bases plus curved surface.
The curved surface unrolled is a rectangle: circumference of the base times height.
Fits = volume (cm³), material = surface area (cm²).
1 litre = 1000 cm³. A wide cylinder beats a tall one.