# Объём и площадь поверхности тел

> Объём, площадь поверхности, наклонная высота и вместимость в литрах для 14 тел, включая цилиндры, конусы, шары, пирамиды, призмы, торы и эллипсоиды.

Интерактивная версия: https://www.calcopenly.com/ru/geometry/volume-surface-area-calculator
Тема: Геометрические калькуляторы

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.

## Входные данные

- **Solid** (варианты: 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**
- **Длина**
- **Ширина**
- **Base edge**
- **Радиус**
- **Bottom radius (R)**
- **Top radius (r)**
- **Высота**
- **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**
- **Единица результатов** (варианты: Миллиметры (mm), Сантиметры (cm), Метры (m), Километры (km), Дюймы (in), Футы (ft), Ярды (yd), Мили (mi))

## Результаты

- Объём — основной результат
- Total surface area
- Lateral (curved) surface area
- Base area
- Slant height
- Space diagonal
- Capacity (L)

## Формула

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

## Примеры с решением

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

- Solid: Cylinder
- Радиус: 3 cm
- Высота: 10 cm
- Единица результатов: Сантиметры (cm)
- **Объём: 282.743339 cm³**
- **Total surface area: 245.044227 cm²**
- **Lateral (curved) surface area: 188.495559 cm²**
- **Capacity: 0.282743 L**
- Источник проверки: Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L

### Box 6 × 4 × 10 m

- Solid: Cuboid (box)
- Длина: 6 m
- Ширина: 4 m
- Высота: 10 m
- Единица результатов: Метры (m)
- **Объём: 240 m³**
- **Total surface area: 248 m²**
- **Space diagonal: 12.328828 m**
- **Capacity: 240,000 L**
- Источник проверки: 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
- Радиус: 3 cm
- Высота: 4 cm
- Единица результатов: Сантиметры (cm)
- **Slant height: 5 cm**
- **Объём: 37.699112 cm³**
- **Lateral (curved) surface area: 47.12389 cm²**
- **Total surface area: 75.398224 cm²**
- Источник проверки: 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
- Высота: 4 cm
- Единица результатов: Сантиметры (cm)
- **Slant height: 4.472136 cm**
- **Объём: 205.25072 cm³**
- **Total surface area: 219.211186 cm²**
- Источник проверки: 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
- Высота: 4 cm
- Единица результатов: Сантиметры (cm)
- **Slant height: 5 cm**
- **Объём: 48 cm³**
- **Lateral (curved) surface area: 60 cm²**
- **Total surface area: 96 cm²**
- Источник проверки: Face height √(4² + 3²) = 5; ⅓·36·4; 4 × ½·6·5; + 36

### Hemisphere r = 3 cm

- Solid: Hemisphere
- Радиус: 3 cm
- Единица результатов: Сантиметры (cm)
- **Объём: 56.548668 cm³**
- **Total surface area: 84.823002 cm²**
- **Lateral (curved) surface area: 56.548668 cm²**
- Источник проверки: Python 3.8 math: 2/3*pi*27, 3*pi*9, 2*pi*9

## Вопросы

### 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.

### Насколько точен «Объём и площадь поверхности тел»?

Точность зависит от введённых данных и допущений метода. Десятичная арифметика использует 50 значащих цифр, но оценки, численные методы и исходные данные могут быть менее точными; округление на экране не устраняет эти ограничения. Решённые примеры, проверенные по независимым источникам: 13. Например, «Cylinder r = 3 cm, h = 10 cm» проверяется по источнику Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L.

### Откуда взята методика?

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.

## Источники

- 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)