CalcCalcolatrice di Volume del Cono
Ultimo aggiornamento: 2026-05-09
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| Raggio | Altezza | |
|---|---|---|
| Caso basico | 1.2 | 4.0 |
| Caso tipico | 2.1 | 7.0 |
| Caso medio | 3.0 | 10.0 |
| Caso avanzado | 4.5 | 15.0 |
| Caso extremo | 7.5 | 25.0 |
Cone Volume & Surface Area Calculator
A right circular cone is a geometric solid with a circular base and a vertex that rises perpendicularly above the center of the base. This calculator computes both the volume and total surface area from the radius and height.
Cone formulas
For a cone with radius r, height h and slant height l:
- Volume: V = (1/3) × π × r² × h
- Total surface area: SA = π × r × (r + l)
- Slant height: l = √(r² + h²)
The slant height is found using the Pythagorean theorem, as it forms a right triangle with the radius and the height.
Example 1: cone with integer dimensions
Problem: A cone has radius r = 6 cm and height h = 8 cm.
- Slant height:
- l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
- Volume:
- V = (1/3) × π × 6² × 8 = (1/3) × π × 36 × 8 ≈ 301.59 cm³.
- Surface area:
- SA = π × 6 × (6 + 10) = π × 6 × 16 ≈ 301.59 cm².
Answer: V ≈ 301.59 cm³, SA ≈ 301.59 cm², l = 10 cm.
Example 2: cone with decimal measurements
Problem: A cone has radius r = 3.5 m and height h = 5.2 m.
- Slant height:
- l = √(3.5² + 5.2²) = √(12.25 + 27.04) = √39.29 ≈ 6.27 m.
- Volume:
- V = (1/3) × π × 3.5² × 5.2 ≈ 66.73 m³.
- Surface area:
- SA = π × 3.5 × (3.5 + 6.27) ≈ 107.43 m².
Answer: V ≈ 66.73 m³, SA ≈ 107.43 m², l ≈ 6.27 m.
Usi comuni
- Computing the capacity of funnels, traffic cones and conical containers.
- Estimating materials for conical roofs, tents and umbrellas.
- Determining volumes of sand, gravel or bulk material piles.
- Solving solid geometry problems in mathematics courses.
- Designing decorative and architectural conical elements.
- Calculating the surface area of ice cream cones and food packaging.
Common mistakes when working with cones
- Using the vertical height instead of the slant height in the surface area formula.
- Forgetting the 1/3 factor in the volume formula.
- Confusing lateral area with total surface area (which includes the base).
- Not computing the slant height when only radius and height are known.
Consiglio professionale
Remember that a cone's volume is exactly one-third of a cylinder's volume with the same base and height. This relationship can help you mentally verify your calculations.
The slant height is the distance from the vertex to any point on the edge of the base. It is computed as l = √(r² + h²) using the Pythagorean theorem.
A cone occupies exactly one-third of the volume of a cylinder with the same base and height. This can be proven through integration or Cavalieri's principle.
The volume is the same if the perpendicular height is maintained, but the surface area changes. These formulas are for right circular cones.
The lateral area is LA = π × r × l. It does not include the circular base area. Total surface area adds π × r².