Understanding the Calculation of the Volume of a Cistern

Calculating the volume of a cistern is essential for efficient water storage management. It involves determining the exact capacity based on shape and dimensions.

This article explores detailed formulas, common values, and real-world applications for precise cistern volume calculations. You will gain expert-level insights and practical examples.

• Calculate the volume of a cylindrical cistern with a radius of 2 meters and height of 5 meters.
• Determine the volume of a rectangular cistern measuring 3m by 4m by 2m.
• Find the volume of a spherical cistern with a diameter of 6 meters.
• Compute the volume of a horizontal cylindrical cistern with radius 1.5m and length 6m partially filled to 1m depth.

Common Dimensions and Volumes of Cisterns: Reference Tables

Below are extensive tables listing common cistern shapes, their dimensions, and corresponding volumes. These values serve as quick references for engineers and technicians.

Fundamental Formulas for Calculating Cistern Volume

Calculating the volume of a cistern depends on its geometric shape. Below are the primary formulas used, with detailed explanations of each variable and typical value ranges.

1. Cylindrical Cistern (Vertical)

The volume V of a vertical cylindrical cistern is calculated by:

V = π × r2 × h

• V: Volume of the cistern (cubic meters, m³)
• r: Radius of the base (meters, m)
• h: Height of the cistern (meters, m)
• π: Pi, approximately 3.1416

Typical values: Radius usually ranges from 0.5 m to 5 m; height varies from 1 m to 10 m depending on design.

2. Cylindrical Cistern (Horizontal)

For a horizontal cylindrical cistern, the volume is:

V = π × r2 × L

• V: Volume (m³)
• r: Radius of the cylinder (m)
• L: Length of the cylinder (m)

When the cistern is partially filled, the volume of liquid depends on the liquid height h inside the cylinder. The formula for partial volume is:

V = L × (r2 × arccos((r – h) / r) – (r – h) × √(2rh – h2))

• h: Depth of liquid inside the cylinder (m)
• arccos: Inverse cosine function (radians)

3. Rectangular Cistern

The volume of a rectangular cistern is straightforward:

V = L × W × H

• V: Volume (m³)
• L: Length (m)
• W: Width (m)
• H: Height (m)

Typical dimensions vary widely, but common residential cisterns range from 1 m to 5 m in each dimension.

4. Spherical Cistern

The volume of a spherical cistern is given by:

V = (4 / 3) × π × r3

• r: Radius of the sphere (m)

Diameters typically range from 1 m to 6 m for small to medium cisterns.

5. Elliptical Cistern

For elliptical cisterns, the volume is:

V = π × a × b × h

• a: Semi-major axis of the ellipse (m)
• b: Semi-minor axis of the ellipse (m)
• h: Height of the cistern (m)

Common values for a and b range from 1 m to 5 m, with height varying similarly.

Detailed Real-World Examples of Cistern Volume Calculation

Example 1: Vertical Cylindrical Cistern for Rainwater Harvesting

A residential building requires a vertical cylindrical cistern to store rainwater. The design specifies a radius of 1.5 meters and a height of 4 meters. Calculate the total volume in cubic meters and liters.

Step 1: Identify variables:

• r = 1.5 m
• h = 4 m

Step 2: Apply the formula:

V = π × r2 × h = 3.1416 × (1.5)2 × 4

Step 3: Calculate:

1.52 = 2.25

V = 3.1416 × 2.25 × 4 = 3.1416 × 9 = 28.2744 m³

Step 4: Convert to liters:

1 m³ = 1000 liters

Volume = 28,274.4 liters

Result: The cistern can store approximately 28,274 liters of water.

Example 2: Partial Volume of a Horizontal Cylindrical Cistern

A horizontal cylindrical cistern has a radius of 1.2 meters and a length of 5 meters. It is partially filled to a depth of 0.8 meters. Calculate the volume of water inside.

Step 1: Identify variables:

• r = 1.2 m
• L = 5 m
• h = 0.8 m

Step 2: Calculate the volume of liquid using the partial volume formula:

V = L × (r2 × arccos((r – h) / r) – (r – h) × √(2rh – h2))

Step 3: Calculate intermediate values:

(r – h) = 1.2 – 0.8 = 0.4

(r – h) / r = 0.4 / 1.2 = 0.3333

arccos(0.3333) ≈ 1.230 radians

r2 = 1.44

2rh – h2 = 2 × 1.2 × 0.8 – 0.82 = 1.92 – 0.64 = 1.28

√1.28 ≈ 1.131

Step 4: Calculate volume:

V = 5 × (1.44 × 1.230 – 0.4 × 1.131) = 5 × (1.771 – 0.452) = 5 × 1.319 = 6.595 m³

Step 5: Convert to liters:

6.595 m³ = 6,595 liters

Result: The cistern contains approximately 6,595 liters of water at 0.8 m depth.

Additional Considerations and Best Practices

When calculating cistern volumes, consider the following factors to ensure accuracy and compliance with standards:

• Material Thickness: Account for wall thickness if internal volume is required.
• Shape Irregularities: Some cisterns have tapered or rounded edges; adjust calculations accordingly.
• Safety Margins: Include freeboard space to prevent overflow.
• Standards Compliance: Follow local and international standards such as ASTM D1998 or ISO 9001 for design and testing.
• Measurement Precision: Use calibrated instruments for dimension measurement.

Useful External Resources for Further Reading

• Engineering Toolbox – Water Tanks and Cisterns
• Whole Building Design Guide – Water Storage Tanks
• ISO 9001: Quality Management Systems
• ASTM D1998 – Standard Specification for Fiberglass Tanks

Summary of Key Points

• Volume calculation depends on cistern shape: cylindrical, rectangular, spherical, or elliptical.
• Precise formulas with clearly defined variables enable accurate volume determination.
• Partial volume calculations are critical for horizontal cylindrical cisterns with varying fill levels.
• Real-world examples demonstrate practical application of formulas and conversions.
• Consider material, shape irregularities, and safety margins for design compliance.

Mastering these calculations ensures optimal cistern design, efficient water management, and regulatory compliance in engineering projects.