# 立体图形体积与表面积计算器

> 计算圆柱、圆锥、球、棱锥、棱柱、圆环体和椭球等14种立体的体积、表面积、斜高及以升表示的容量。

交互版本：https://www.calcopenly.com/zh/geometry/volume-surface-area-calculator
主题：几何计算器

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.

## 输入

- **Solid** (选项：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**
- **长度**
- **宽度**
- **Base edge**
- **半径**
- **Bottom radius (R)**
- **Top radius (r)**
- **高度**
- **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**
- **结果单位** (选项：毫米 (mm), 厘米 (cm), 米 (m), 千米 (km), 英寸 (in), 英尺 (ft), 码 (yd), 英里 (mi))

## 结果

- 体积 — 主要结果
- Total surface area
- Lateral (curved) surface area
- Base area
- Slant height
- Space diagonal
- Capacity (L)

## 公式

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

## 计算示例

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

- Solid: Cylinder
- 半径: 3 cm
- 高度: 10 cm
- 结果单位: 厘米 (cm)
- **体积: 282.743339 cm³**
- **Total surface area: 245.044227 cm²**
- **Lateral (curved) surface area: 188.495559 cm²**
- **Capacity: 0.282743 L**
- 核验来源：Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L

### Box 6 × 4 × 10 m

- Solid: Cuboid (box)
- 长度: 6 m
- 宽度: 4 m
- 高度: 10 m
- 结果单位: 米 (m)
- **体积: 240 m³**
- **Total surface area: 248 m²**
- **Space diagonal: 12.328828 m**
- **Capacity: 240,000 L**
- 核验来源：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
- 半径: 3 cm
- 高度: 4 cm
- 结果单位: 厘米 (cm)
- **Slant height: 5 cm**
- **体积: 37.699112 cm³**
- **Lateral (curved) surface area: 47.12389 cm²**
- **Total surface area: 75.398224 cm²**
- 核验来源：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
- 高度: 4 cm
- 结果单位: 厘米 (cm)
- **Slant height: 4.472136 cm**
- **体积: 205.25072 cm³**
- **Total surface area: 219.211186 cm²**
- 核验来源：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
- 高度: 4 cm
- 结果单位: 厘米 (cm)
- **Slant height: 5 cm**
- **体积: 48 cm³**
- **Lateral (curved) surface area: 60 cm²**
- **Total surface area: 96 cm²**
- 核验来源：Face height √(4² + 3²) = 5; ⅓·36·4; 4 × ½·6·5; + 36

### Hemisphere r = 3 cm

- Solid: Hemisphere
- 半径: 3 cm
- 结果单位: 厘米 (cm)
- **体积: 56.548668 cm³**
- **Total surface area: 84.823002 cm²**
- **Lateral (curved) surface area: 56.548668 cm²**
- 核验来源：Python 3.8 math: 2/3*pi*27, 3*pi*9, 2*pi*9

## 常见问题

### 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.

### “立体图形体积与表面积计算器”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：13。 例如，“Cylinder r = 3 cm, h = 10 cm”根据Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L进行核验。

### 这种方法出自哪里？

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.

## 来源

- 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)