Cylindrical Sector Volume

What is a Cylindrical Sector?

A cylindrical sector (or sector of a cylinder) is a portion of a right circular cylinder bounded by its lateral surface, two planes passing through the cylinder axis forming a dihedral angle θ, and two circular sectors in the bases. In cross-section, a cylindrical sector has the shape of a circular sector — like a "slice of pie".

If you need to calculate the volume of a portion of a cylinder cut by a plane parallel to the axis, please use the cylindrical segment volume calculator — that is a different shape.

Parameters of a Cylindrical Sector

• r — radius of the cylinder;
• h — height (length) of the cylinder;
• θ — central angle of the sector (in degrees, 0 < θ ≤ 360). In the formulas below, θ is used in radians.

Formulas for the Volume of a Cylindrical Sector

The volume of a cylindrical sector is calculated as the area of the circular sector at the base multiplied by the height of the cylinder:

$$V=S_{\mathrm{sector}}·h=\frac{1}{2}·r^2·\theta ·h$$

where:

• S_sector — area of the circular sector of the base: $S_{\mathrm{sector}}=\frac{1}{2}{r}^2\theta$;
• r — radius of the cylinder;
• h — height of the cylinder;
• θ — central angle of the sector in radians.

Special Cases

• θ = π (180°) — half-cylinder: the sector occupies exactly half of the cylinder, and the volume equals half the volume of the full cylinder: $V=\frac{1}{2}\pi r^{2}h$.
• θ = 2π (360°) — full cylinder: the sector becomes a full cylinder, and the volume equals: $V=\pi r^{2}h$.
• θ < π — minor sector (less than half of the cylinder).
• θ > π — major sector (more than half of the cylinder).

Applications of Cylindrical Sector Volume Calculation

• Ventilation and heating. Calculating the volume of sector duct components is essential for selecting appropriate fans and determining the air exchange rate in a room.
• Liquid tanks and flumes. Estimating fluid volume in sector-shaped collectors, horizontal tanks, or channels with a sector cross-section during dosed supply.
• Mechanical engineering and pumps. Working chambers of centrifugal and rotary pumps are often divided into cylindrical sectors. The volume of each sector determines the pump's displacement per revolution.
• Architecture and construction. Sector dome elements, columns, niches, or circular bay windows in buildings occupy a certain usable space, the volume of which must be accounted for in HVAC calculations.
• Food industry and dispensers. Dosing equipment for bulk materials and liquids works by filling a cylindrical sector-shaped measuring cup. Volume calculation helps calibrate product portions.
• Hydrodynamics and rotary machines. In rotary vane compressors and vacuum pumps, the working volume between the moving rotor vanes represents a variable cylindrical sector. Calculating this volume is essential for determining the theoretical capacity and compression ratio of the compressor.

How to Use the Calculator?

1. Enter the cylinder radius r.
2. Enter the cylinder height h.
3. Enter the central angle θ in degrees (if the angle is in radians — convert to degrees: θ° = θ_rad × 180/π) or the arc length L.
4. Specify the desired precision of the result.
5. Click "Calculate" — the volume of the cylindrical sector will appear instantly.

Frequently Asked Questions (FAQ)

How does a cylindrical sector differ from a cylindrical segment?

A cylindrical sector is formed by planes passing directly through its axis, creating a circular sector in cross-section. A cylindrical segment is cut by a plane parallel to the axis, creating a region bounded by a chord and an arc, rather than two radii. To calculate the segment volume, please use the cylindrical segment volume calculator.

How does the arc length affect the volume?

The arc length L is directly proportional to the central angle θ (θ = L/r). Accordingly, as the arc length increases, the volume of the cylindrical sector increases linearly.

Why is the volume of a half-cylinder equal to half the total volume?

Since the angle of a half-cylinder is 180° or π radians, which is exactly half of a full circle (2π radians or 360°), its base area equals half of the circle's area, and consequently, its volume is exactly half that of the entire cylinder.