Ice cream, a cone, a cone
Take a cylinder and cut it to a point — you have a cone. An ice-cream cone, a traffic cone, the tip of a pencil. A circle at the bottom, a vertex at the top.
Good news: if you can do a cylinder, a cone is almost free. The volume of a cone is a third of the volume of a cylinder with the same base and height: V = (π · r² · v) / 3. The same story as with a pyramid — the point eats space and leaves only a third.
The surface has two parts: a circular base (π · r²) and the lateral surface — that wrapped “hat”. When you cut the lateral surface and unroll it, it is not a rectangle like with a cylinder, but a sector of a circle. Its area is π · r · s, where s is the slant height of the cone — the distance from the edge of the base to the point, measured along the surface.
Again we have two lengths: height v (perpendicular inside) and slant height s (slanted along the lateral surface). Height belongs in volume, slant height in surface. The same logic as with a pyramid, just round.
We calculate with π ≈ 3.14 and with a comma in decimal numbers. Let’s go.
Tereza and Filip buy cones for the school ice-cream party. On the box it says: diameter 6 cm, depth 12 cm. “How much ice cream fits in one?” Filip wonders. Tereza counts: “Radius 3 cm. A cylinder would have 3.14 times 9 times 12, about 339 millilitres. A cone is a third — about 113 ml.” Filip whistles: “So a scoop on top is not a luxury, but a necessity. Less fits into the point than it looks.”
Check what the question gives: diameter, or radius? Always divide the diameter by two first. Half the mistakes with a cone and a cylinder are a forgotten halving of the diameter.
Volume and surface of a cone
Volume:
Step 1: Radius of the base r (from the diameter divided by two). Step 2: Area of the base: Sp = π · r² ≈ 3.14 · r². Step 3: Times height v, divided by three: V = (π · r² · v) / 3.
Example: r = 3 cm, v = 10 cm. Sp = 3.14 · 9 = 28.26 cm². V = (28.26 · 10)/3 = 282.6/3 = 94.2 cm³.
Surface:
Step 1: Base: π · r². Step 2: Lateral surface: Spl = π · r · s, where s is the slant height (the slanted length along the surface to the point). Step 3: Add: S = π · r² + π · r · s = π · r · (r + s).
Example: r = 3 cm, s = 5 cm. S = 3.14 · 3 · (3 + 5) = 3.14 · 24 = 75.36 cm².
Height and slant height are friends through Pythagoras: slant height s is the hypotenuse of a right-angled triangle with legs r and v. So s² = r² + v². For r = 3 and v = 4: s² = 9 + 16 = 25, s = 5. When the question gives only the height and wants the surface, you calculate the slant height — the triple 3, 4, 5 waits for you suspiciously often.
Example 1: the volume of an ice-cream cone
Question: A cone has diameter 6 cm and height 12 cm. How many cm³ of ice cream does it hold?
Method: Diameter 6 cm → radius r = 3 cm.
Base: 3.14 · 3² = 3.14 · 9 = 28.26 cm².
Volume: V = (28.26 · 12)/3 = 339.12/3 = 113.04 cm³.
Sense check: that is about 113 ml — a small cup. Believable for a cone.
Answer: The cone holds about 113 cm³ of ice cream.
Example 2: the surface of a party hat
Question: A paper hat (a cone without a base — a head does not need a bottom) has radius 7 cm and slant height 20 cm. How much paper for one hat?
Method: Without a base, the lateral surface is enough: Spl = π · r · s.
Spl = 3.14 · 7 · 20 = 439.6 cm².
Answer: About 440 cm² of paper is needed for a hat.
Notice the reading of the question: “without a base” crosses out the term π · r². With a cone always think whether the solid has a bottom (you do not close an ice-cream cone from below, a traffic cone neither).
Example 3: find the slant height with Pythagoras
Question: A cone has radius 3 cm and height 4 cm. Calculate the volume and the whole surface.
Method: Volume from the height: V = (3.14 · 9 · 4)/3 = 113.04/3 = 37.68 cm³.
For the surface I need slant height s: s² = r² + v² = 9 + 16 = 25, so s = 5 cm.
Surface: S = π · r · (r + s) = 3.14 · 3 · 8 = 75.36 cm².
Sense check: the slant height (5) came out longer than the height (4) — a slanted path is longer, it matches.
Answer: V = 37.68 cm³, S = 75.36 cm².
Height into volume, slant height into surface. The same trap as with a pyramid, it just has another name: v goes perpendicular inside, s slanted along the lateral surface and it is always longer. When in the question you see only one of them and you need the other, Pythagoras joins them: s² = r² + v².
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.
Practise the volume of a cylinder — a cone is exactly a third of it. Calculate with π ≈ 3.14, write decimal numbers with a comma.
On paper
A sketch with r, v and s marked for every problem. Calculate with π ≈ 3.14.
-
A cone: r = 5 cm, v = 6 cm. Calculate the volume and check that it is a third of a cylinder with the same sizes.
-
A traffic cone without a base: r = 15 cm, s = 50 cm. How many cm² of material for the lateral surface?
-
A cone: r = 6 cm, v = 8 cm. Find the slant height with Pythagoras and calculate the whole surface.
-
An ice-cream cone: diameter 8 cm, height 15 cm. How many ml does it hold? (1 cm³ = 1 ml. Do not forget to divide the diameter by two.)
Now you 💪
- You can derive the volume of a cone from a cylinder (a third) and you do not have to remember the formula in isolation.
- You tell apart height v (volume) and slant height s (surface) and you can join them with Pythagoras.
- From a diameter you make a radius before you reach for a formula.
Done when: You have calculated the volume and the surface of a cone, the slant height found with Pythagoras and one volume checked against a cylinder.
What to take from this lesson
- Volume of a cone: V = (π · r² · v) / 3 — a third of a cylinder.
- Surface: base π · r² plus lateral surface π · r · s.
- Height v goes inside (volume), slant height s along the lateral surface (surface).
- Slant height and height are joined by Pythagoras: s² = r² + v².
- Divide the diameter by two first. Always.
© 2026 Ing. Martin Polak / AlgoRhino · 콘텐츠 이용 약관