# Volume et aire de surface des solides

> Volume, aire de surface, hauteur oblique et capacité en litres de 14 solides : cylindres, cônes, sphères, pyramides, prismes, tores et ellipsoïdes.

Version interactive : https://www.calcopenly.com/fr/geometry/volume-surface-area-calculator
Sujet : Calculatrices de géométrie

Choose one of 14 solids and enter its dimensions to get the volume and total surface area, plus the curved (lateral) surface, base area, slant height and space diagonal where the solid has them. The volume is also given as a capacity in litres, using 1 L = 1,000 cm³. The formulas are the standard ones, for example V = πr²h for a cylinder, V = ⅓πh(R² + Rr + r²) for a frustum and V = 2π²Rr² for a torus.

The default cylinder, 3 cm in radius and 10 cm tall, holds 282.743 cm³ (0.283 L) and has 245.044 cm² of surface, of which 188.496 cm² is the curved side. Volume sizes tanks, containers and concrete pours; surface area sizes paint, insulation and sheet material.

Every result is exact except the surface of a general ellipsoid, which uses Knud Thomsen's approximation, within about 1.06%. The 3D model is drawn to the proportions entered; drag it to rotate.

## Données

- **Solid** (options : Cube, Cuboid (box), Sphere, Hemisphere, Cylinder, Cone, Frustum of a cone, Square pyramid, Triangular prism (equilateral base), Hexagonal prism (regular base), Torus, Ellipsoid, Capsule, Regular tetrahedron)
- **Edge length**
- **Longueur**
- **Largeur**
- **Base edge**
- **Rayon**
- **Bottom radius (R)**
- **Top radius (r)**
- **Hauteur**
- **Length of the straight part**: Distance between the two hemispherical caps
- **Major radius (R)**: From the centre of the hole to the centre of the tube
- **Tube radius (r)**
- **Semi-axis a**
- **Semi-axis b**
- **Semi-axis c**
- **Unité des résultats** (options : Millimètres (mm), Centimètres (cm), Mètres (m), Kilomètres (km), Pouces (in), Pieds (ft), Yards (yd), Miles (mi))

## Résultats

- Volume — résultat principal
- Total surface area
- Lateral (curved) surface area
- Base area
- Slant height
- Space diagonal
- Capacity (L)

## Formule

$$
\begin{gathered} V_\text{sphere} = \tfrac{4}{3}\pi r^3,\quad V_\text{cone} = \tfrac{1}{3}\pi r^2 h \\ V_\text{frustum} = \tfrac{1}{3}\pi h(R^2 + Rr + r^2),\quad V_\text{torus} = 2\pi^2 R r^2 \end{gathered}
$$

## Exemples détaillés

### Cylinder r = 3 cm, h = 10 cm

- Solid: Cylinder
- Rayon: 3 cm
- Hauteur: 10 cm
- Unité des résultats: Centimètres (cm)
- **Volume: 282.743339 cm³**
- **Total surface area: 245.044227 cm²**
- **Lateral (curved) surface area: 188.495559 cm²**
- **Capacity: 0.282743 L**
- Source de vérification : Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L

### Box 6 × 4 × 10 m

- Solid: Cuboid (box)
- Longueur: 6 m
- Largeur: 4 m
- Hauteur: 10 m
- Unité des résultats: Mètres (m)
- **Volume: 240 m³**
- **Total surface area: 248 m²**
- **Space diagonal: 12.328828 m**
- **Capacity: 240,000 L**
- Source de vérification : 6·4·10; 2(24 + 60 + 40); Python 3.8 math.sqrt(152); 1 m³ = 1000 L

### Cone r = 3, h = 4 (slant 5)

- Solid: Cone
- Rayon: 3 cm
- Hauteur: 4 cm
- Unité des résultats: Centimètres (cm)
- **Slant height: 5 cm**
- **Volume: 37.699112 cm³**
- **Lateral (curved) surface area: 47.12389 cm²**
- **Total surface area: 75.398224 cm²**
- Source de vérification : 3-4-5 slant; Python 3.8 math: 12*pi, 15*pi, 24*pi

### Frustum R = 5, r = 3, h = 4

- Solid: Frustum of a cone
- Bottom radius (R): 5 cm
- Top radius (r): 3 cm
- Hauteur: 4 cm
- Unité des résultats: Centimètres (cm)
- **Slant height: 4.472136 cm**
- **Volume: 205.25072 cm³**
- **Total surface area: 219.211186 cm²**
- Source de vérification : Python 3.8 math: sqrt(20), pi*4*(25+15+9)/3, pi*8*sqrt(20) + 34*pi

### Square pyramid a = 6, h = 4

- Solid: Square pyramid
- Base edge: 6 cm
- Hauteur: 4 cm
- Unité des résultats: Centimètres (cm)
- **Slant height: 5 cm**
- **Volume: 48 cm³**
- **Lateral (curved) surface area: 60 cm²**
- **Total surface area: 96 cm²**
- Source de vérification : Face height √(4² + 3²) = 5; ⅓·36·4; 4 × ½·6·5; + 36

### Hemisphere r = 3 cm

- Solid: Hemisphere
- Rayon: 3 cm
- Unité des résultats: Centimètres (cm)
- **Volume: 56.548668 cm³**
- **Total surface area: 84.823002 cm²**
- **Lateral (curved) surface area: 56.548668 cm²**
- Source de vérification : Python 3.8 math: 2/3*pi*27, 3*pi*9, 2*pi*9

## Questions

### How do you calculate the volume of a cylinder?

Multiply the area of the circular end by the height: V = πr²h. A radius of 3 cm and a height of 10 cm give 90π ≈ 282.74 cm³, or 0.283 L. Use the radius, not the diameter: a tank 1 m across and 1.5 m tall has r = 0.5 m and holds π × 0.25 × 1.5 ≈ 1.178 m³, which is 1,178 L.

### How do you find the surface area of a cylinder?

Add the curved side, 2πrh, to the two circular ends, 2πr²: S = 2πr(r + h). For r = 3 cm and h = 10 cm the curved side is 60π ≈ 188.50 cm² and the total is 78π ≈ 245.04 cm². An open-topped tank has only one end, so subtract πr², here 28.27 cm², to get 216.77 cm².

### How many litres are in a cubic metre?

1,000. A litre is exactly one cubic decimetre, 1,000 cm³, a definition fixed by the General Conference on Weights and Measures (CGPM) in 1964. Divide cubic centimetres by 1,000 to get litres. In US units, one gallon is exactly 231 in³, or 3.785411784 L, and one cubic foot is 28.316846592 L.

### What is the formula for the volume of a sphere?

V = ⁴⁄₃πr³, which is two thirds of the cylinder that just encloses the sphere, as Archimedes showed. A sphere of radius 3 cm holds 36π ≈ 113.10 cm³, and a ball 1 m across holds 0.5236 m³, or 523.6 L. Volume grows with the cube of the radius, so doubling the diameter multiplies the volume by 8.

### How do you calculate the volume of a cone or a pyramid?

Take one third of the base area times the height: V = ⅓Bh. A cone of radius 3 and height 4 has V = ⅓π × 9 × 4 = 12π ≈ 37.70, and a square pyramid with a 6 × 6 base and height 4 has V = ⅓ × 36 × 4 = 48. A cone holds exactly one third of the cylinder with the same base and height.

### Quelle est la précision de « Volume et aire de surface des solides » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 13. Par exemple, « Cylinder r = 3 cm, h = 10 cm » est vérifié à l’aide de Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L.

### D’où vient cette méthode ?

Zwillinger, D. (ed.) CRC Standard Mathematical Tables and Formulas, 33rd ed., §4.6 (solids); Weisstein, E. W. “Torus”, “Spherical Cap”, “Ellipsoid” — MathWorld; Thomsen, K. — ellipsoid surface approximation, as quoted in “Ellipsoid § Surface area”, Wikipedia.

## Sources

- Zwillinger, D. (ed.) CRC Standard Mathematical Tables and Formulas, 33rd ed., §4.6 (solids)
- [Weisstein, E. W. “Torus”, “Spherical Cap”, “Ellipsoid” — MathWorld](https://mathworld.wolfram.com/Ellipsoid.html)
- [Thomsen, K. — ellipsoid surface approximation, as quoted in “Ellipsoid § Surface area”, Wikipedia](https://en.wikipedia.org/wiki/Ellipsoid#Surface_area)