Калькулятор Об'єму Циліндра

Як користуватися цим калькулятором

1. Введіть Radius (r)
2. Введіть Height (h)
3. Натисніть кнопку Розрахувати
4. Прочитайте результат, відображений під калькулятором

Cylinder Volume and Surface Area Formulas

A cylinder is a three-dimensional solid with two parallel circular bases connected by a curved lateral surface. The key measurements are the radius (r) of the circular bases and the height (h) between them.

Volume = πr²h — the base area (πr²) times the height. Intuitively: stacking infinite thin circular discs. Lateral surface area = 2πrh — unrolling the curved surface gives a rectangle of width 2πr (circumference) and height h. Total surface area = 2πr(r+h) = 2πrh + 2πr² — lateral area plus two circular caps.

Example: a cylinder with r=3 and h=10 has volume = π×9×10 = 90π ≈ 282.74 cubic units, lateral surface = 2π×3×10 = 60π ≈ 188.50 square units, total surface = 2π×3×(3+10) = 78π ≈ 245.04 square units.

Types of Cylinders and Related Solids

A right cylinder (the standard type) has its axis perpendicular to its bases. An oblique cylinder has a tilted axis — like a leaning stack of coins. The volume formula πr²h still applies to oblique cylinders (Cavalieri's principle: slices at every height have the same area).

A hollow cylinder (like a pipe) has volume = πh(R²-r²), where R is the outer radius and r is the inner radius. Surface area calculations for hollow cylinders are important in plumbing, pipe insulation, and structural engineering.

The cylinder relates to other solids: a cone with the same base and height has exactly 1/3 the cylinder's volume. A sphere fits exactly inside a cylinder of equal radius and height (h=2r), and has 2/3 the cylinder's volume — a result Archimedes considered his greatest discovery, and had inscribed on his tombstone.

Real-World Applications of Cylinder Calculations

Cylinder volume calculations are essential in manufacturing and daily life. Containers: calculating how much liquid a can, tank, or pipe can hold. A 12 oz soda can has r≈3.2 cm, h≈12.2 cm, volume ≈ 392 cm³ ≈ 392 mL. Engine displacement: the volume swept by pistons in an engine is calculated using cylinder formulas — a 2.0L engine means the pistons sweep 2,000 cm³ total.

Concrete pillars: calculating concrete needed for cylindrical columns. Water towers: estimating storage capacity. Chemical reactors: sizing cylindrical reaction vessels. Pressure vessels: pipes, tanks, and boilers — all modeled as cylinders with specific pressure and material requirements.

The efficiency of cylindrical containers is notable: for a given volume, the optimal cylinder (minimizing surface area, hence material) has height = diameter (h = 2r). This minimizes packaging material. Real-world cans deviate slightly from this optimal ratio for manufacturing and stacking convenience.