Cylindrical Segment Volume

Definition of a Cylindrical Segment

A cylindrical segment is a three-dimensional solid formed by cutting a right circular cylinder with a plane parallel to its longitudinal axis. Computing the capacity of such a solid is a fundamental problem in spatial geometry with many engineering applications.

Formula for Computing Capacity

The volumetric calculation is based on the cross-sectional area of the circular segment and the overall length (height) of the cylindrical body. The total volume is the product of the base segment area and the cylinder height:

$$V=S_{\mathrm{seg}}·H$$

The area of the base circular segment is determined by the radius and the central angle:

$$S_{\mathrm{seg}}=\frac{R^2}{2}·(\theta -\sin \theta )$$

The central angle θ (in radians) is calculated using the following relationship:

$$\theta =2·\arccos \frac{R-h}{R}$$

Geometric Parameters for Calculation

• R — radius of the circular base;
• H — height (length) of the cylinder;
• h — segment height (fluid depth or distance from the chord to the top of the arc);
• c — chord length (the distance between the intersection points of the cutting plane and the base, = 2√(2Rh − h²)).

Geometric Cases of Filling

• Minor segment (h < R): the cutting plane lies above the cylinder axis, and the solid occupies less than half of the total space.
• Half-cylinder (h = R): the cutting plane passes exactly through the cylinder axis. The volume is: V = πR²H / 2.
• Major segment (h > R): the cutting plane lies below the cylinder axis, and the solid occupies more than half of the total space.
• Full cylinder (h = 2R): the limiting case where the volume equals the full capacity of the cylinder: V = πR²H.

Practical Applications of Capacity Calculation

Tank Farms & Oil Depots. Determining the exact volume of fuel, oil, or liquefied gas in horizontal storage tanks using measuring rods (dipsticks).

Gravity Flow Piping Design. Calculating the actual fill level of non-pressurized pipelines, sewer mains, and storm drains to optimize flow and prevent overflow.

Food & Brewing Industries. Monitoring raw materials in horizontal fermentation, pasteurization, or storage vessels for milk, juices, and other liquids.

Logistics & Cargo Weight Assessment. Estimating the weight of liquid cargo in rail tankers or road tankers based on the fill level and fluid density.

Mechanical Counterweights. In cranes and lift systems, cylindrical segment weights are used for balancing, where the target mass determines the required volume.

Hydrological Monitoring. Assessing water volume in semi-circular open channels and flumes at varying water levels.

How to Use the Calculator?

1. Choose the input method: by segment height h or chord length c.
2. Enter the cylinder radius R, length H, and the chosen value (h or c).
3. Specify the number of decimal places for rounding the result.
4. Click the "Calculate" button to obtain the result instantly.

Frequently Asked Questions about Segment Capacity (FAQ)

How do I find the liquid volume in a horizontal tank filled more than halfway?

If the fill height h exceeds the radius R, calculate the volume of the empty space at the top (using the empty height 2R - h as the segment height) and subtract it from the total volume of the cylinder.

How do I convert the calculated volume into liters or gallons?

The calculator outputs the result in cubic units of the input parameters. If you input parameters in meters, the result is in cubic meters (m³). Multiply by 1000 to convert to liters (1 m³ = 1000 liters). To convert cubic meters to US gallons, multiply by 264.172.

Can I calculate the volume of a tilted horizontal cylinder?

No, if the cylinder is tilted, the liquid surface is not parallel to the cylinder's longitudinal axis, creating a complex shape with a variable cross-section. This calculator assumes a perfectly level cylinder; tilted tanks require calibration charts or complex integration.