Cylindrical Sector Calculator
Calculator and formulas for calculating a cylindrical sector
Cylindrical Sector Calculator
The Cylindrical Sector
The cylindrical sector is a sector-shaped cylinder section with a defined central angle φ.
Cylindrical Sector Properties
The Cylindrical Sector: Sector-shaped section from a circular cylinder
Cylindrical Sector Visualization
Cylindrical Sector
Sector-shaped cylinder section
Parameters
r: Radius of the cylinder
h: Height of the cylindrical sector
φ: Central angle (in degrees)
AB: Sector-shaped base area
Sector-shaped cylinder section.
Defined by central angle φ.
What is a Cylindrical Sector?
The cylindrical sector is an important geometric body:
- Definition: Sector-shaped section from a circular cylinder
- Central angle φ: Determines the size of the sector section
- Properties: Circular sector as base and top surface
- Side surfaces: Rectangular cutting surfaces at the sides
- Applications: Mechanical engineering, architecture, piping technology
- Mathematics: Combination of sector and cylinder geometry
Geometric Properties of the Cylindrical Sector
The cylindrical sector shows characteristic geometric properties:
Basic Parameters
- Radius r: Radius of the circular base
- Height h: Height of the cylindrical sector
- Central angle φ: Angle of the sector section (in degrees)
- Base area AB: Sector-shaped base and top surface
Special Properties
- Sector-shaped cross-section: Circular sector as characteristic base
- Rectangular side surfaces: Flat cutting surfaces at the radii
- Curved lateral surface: Cylindrical surface section
- Angle-dependent sizes: All areas proportional to φ/360°
Mathematical Relationships of the Cylindrical Sector
The cylindrical sector follows elegant mathematical laws:
Volume Formula
Volume proportional to central angle. Elegant sector relationship.
Base Area Formula
Sector-shaped base area as fraction of full circle area.
Applications of the Cylindrical Sector
Cylindrical sectors find applications in various fields:
Mechanical Engineering & Automotive
- Gear segments
- Clutch sectors
- Turbine wheel sections
- Valve housing parts
Architecture & Construction
- Column segments
- Vault sections
- Stair railing elements
- Roof constructions
Piping Technology
- Pipe bend sectors
- Branch elements
- Valve segments
- Flange components
Process Engineering
- Reactor sectors
- Container segments
- Mixing devices
- Heat exchanger parts
Formulas for the Cylindrical Sector
Volume (V)
Volume proportional to central angle
Base Area (AB)
Sector-shaped base and top surface
Side Area (AS)
Rectangular cutting surface at the radii
Lateral Area (L)
Curved cylinder surface
Total Surface Area (S)
Sum of all partial areas: two base areas, two side areas and lateral area
Additional Parameters
\[b = \frac{2\pi r \phi}{360°}\]
\[\phi_{rad} = \frac{\phi \cdot \pi}{180°}\]
\[\frac{\phi}{360°} = \text{Fraction}\]
\[\text{Radius } r\]
The cylindrical sector connects 2D sector geometry with 3D cylinder form
Calculation Example for a Cylindrical Sector
Given
Find: All parameters of the cylindrical sector
1. Calculate Base Area
For r = 5 cm, φ = 90°:
\[A_B = \frac{r^2 \pi \phi}{360°}\] \[A_B = \frac{5^2 \cdot \pi \cdot 90°}{360°}\] \[A_B = \frac{25\pi \cdot 90°}{360°} = \frac{25\pi}{4}\] \[A_B ≈ 19.63 \text{ cm}^2\]The base area is approximately 19.63 cm²
2. Calculate Side Area
Rectangular cutting surfaces:
\[A_S = r \cdot h\] \[A_S = 5 \cdot 6 = 30 \text{ cm}^2\]Two side surfaces:
\[2 \cdot A_S = 2 \cdot 30 = 60 \text{ cm}^2\]The side surfaces total 60 cm²
3. Calculate Lateral Area
Curved cylinder surface:
\[L = \frac{2\pi r h \phi}{360°}\] \[L = \frac{2\pi \cdot 5 \cdot 6 \cdot 90°}{360°}\] \[L = \frac{60\pi \cdot 90°}{360°} = 15\pi\] \[L ≈ 47.12 \text{ cm}^2\]The lateral area is approximately 47.12 cm²
4. Calculate Volume
Volume of the cylindrical sector:
\[V = \frac{\pi r^2 h \phi}{360°}\] \[V = \frac{\pi \cdot 25 \cdot 6 \cdot 90°}{360°}\] \[V = \frac{150\pi \cdot 90°}{360°} = \frac{150\pi}{4}\] \[V ≈ 117.81 \text{ cm}^3\]The volume is approximately 117.81 cm³
5. Calculate Total Surface Area
Sum of all partial areas:
\[S = 2A_B + 2A_S + L\] \[S = 2 \cdot 19.63 + 60 + 47.12\] \[S = 39.26 + 60 + 47.12 = 146.38 \text{ cm}^2\]The total surface area is approximately 146.38 cm²
6. Geometric Analysis
90°/360° = 1/4 = 25% of full cylinder
V_full = πr²h = π·25·6 ≈ 471.24 cm³
117.81/471.24 = 25.0% (exact!)
The sector corresponds exactly to 25% of the full cylinder - perfect sector geometry!
7. Summary
The cylindrical sector with 90° - a perfect quarter of the full cylinder
8. Practical Example: Gear Segment
V ≈ 117.81 cm³
≈ 0.12 liters
S ≈ 146.38 cm²
Coating area
90° sector
Quarter symmetry
A gear segment with 90° sector for optimal force transmission
9. Other Angle Examples
V ≈ 58.90 cm³
12.5% of full cylinder
V ≈ 157.08 cm³
33.3% of full cylinder
V ≈ 235.62 cm³
Half cylinder
Different sector sizes for various applications
The Cylindrical Sector: Elegance of 3D Sector Geometry
The cylindrical sector is a fascinating geometric body that combines the elegance of two-dimensional sector geometry with the practicality of three-dimensional cylinder forms. As a sector-shaped section from a circular cylinder, it unites the mathematical clarity of proportional angular relationships with technical applicability in mechanical engineering, architecture, and process engineering. This unique combination of sectorial precision and cylindrical functionality makes it an indispensable element in modern construction technology.
The Geometry of Sector Form
The cylindrical sector shows the perfection of three-dimensional sector geometry:
- Central angle φ: Determines all size relationships proportional to φ/360°
- Sector-shaped base areas: AB = (πr²φ)/360° as characteristic base surface
- Rectangular side surfaces: AS = r·h as flat cutting surfaces at the radii
- Curved lateral surface: L = (2πrhφ)/360° as cylinder section
- Volume proportionality: V = (πr²hφ)/360° directly proportional to angle
- Surface symmetry: S = 2AB + 2AS + L with all components
- Angle invariance: All sizes scale linearly with φ
Versatile Applications
Mechanical Engineering Precision
Gear segments and clutch sectors use precise angular geometry for optimal force transmission and uniform load distribution in rotating systems.
Architectural Innovation
Column segments and vault sections use the sector form for structurally stable and aesthetically pleasing structures with characteristic radial symmetry.
Piping Efficiency
Pipe bend sectors and branch elements use sector geometry for flow-optimized connections with minimal pressure losses and uniform flow distribution.
Process Engineering Perfection
Reactor sectors and container segments use proportional size relationships for precise process control and optimized heat and mass transfer.
Summary
The cylindrical sector embodies the perfect synthesis of geometric proportionality and practical functionality. Its form, characterized by the central angle φ and described by the elegant proportional formulas V = (πr²hφ)/360° and AB = (πr²φ)/360°, shows the mathematical beauty of linear scaling. The total surface area S = 2AB + 2AS + L as the sum of all partial areas demonstrates how complex three-dimensional forms are systematically constructed. From precise gear segments to elegant column sections to flow-optimized pipe bend sectors - the cylindrical sector proves that the simplest mathematical relationships often lead to the most functional technical solutions. It is living proof that sector geometry possesses not only two-dimensional elegance but also enables three-dimensional applications of highest precision, showing how angular proportionality and cylinder form merge in perfect mathematical harmony.