Cylinder Volume Calculator

**Cylinder formulas:** For a right circular cylinder with: - r = base radius - h = height (perpendicular distance between bases) **Volume:** V = π × r² × h **Lateral (curved side) surface area:** A_lateral = 2 × π × r × h **Base area (each circle):** A_base = π × r² **Total surface area:** A_total = 2 × π × r² + 2 × π × r × h = 2 × π × r × (r + h) **Worked example: r = 5, h = 10** V = π × 25 × 10 = 250π ≈ 785.4 cubic units A_lateral = 2π × 5 × 10 = 100π ≈ 314.2 A_base × 2 = 2 × 25π = 50π ≈ 157.1 A_total = 50π + 100π = 150π ≈ 471.2 **Cylinder vs cone (same base and height):** V_cylinder = πr²h V_cone = (1/3)πr²h Cone has exactly 1/3 the volume of cylinder with same base and height. **Cylinder vs sphere (same diameter):** Sphere fits inside cylinder of same diameter and height = diameter: V_cylinder = πr²(2r) = 2πr³ V_sphere = (4/3)πr³ Ratio: V_sphere/V_cylinder = 2/3 Sphere fills 2/3 of equivalent cylinder. Archimedes' famous discovery. **Common cylindrical objects:** | Object | Approximate dimensions | |---|---| | AA battery | r = 0.7 cm, h = 5 cm | | AAA battery | r = 0.5 cm, h = 4.5 cm | | Soda can (12 oz) | r = 3.3 cm, h = 12.2 cm | | Beer bottle | r = 2.7 cm, h = 23 cm | | Pringles can | r = 4 cm, h = 24 cm | | Toilet paper roll | r = 5 cm, h = 10 cm | | Paper towel roll | r = 5 cm, h = 28 cm | | Standard PVC pipe (4") | r = 5.4 cm, h varies | | 55-gallon drum | r = 28 cm, h = 88 cm | | Concrete column | varies — typically r = 30-50 cm, h = 3-12 m | | Grain silo | typically r = 5 m, h = 20 m | | Oil storage tank | r = 5-25 m, h = 5-15 m | **Soda can volume verification:** Standard 12 oz can: r ≈ 3.3 cm, h ≈ 12.2 cm. V = π × 10.89 × 12.2 ≈ 417 cm³ = 417 mL. 12 oz ≈ 355 mL... discrepancy because can is not full to top, has tapered sections. **Cylinder unrolled:** The curved (lateral) surface unrolls into a rectangle: - Width = circumference of base = 2πr. - Height = h (cylinder height). - Area = 2πrh. Like unrolling a label off a can. **Hollow cylinder (annular cylinder, pipe wall):** For pipe with outer radius R and inner radius r, length h: V = π(R² - r²)h The cross-section is an annulus (ring). **Pipe flow:** For fluid flow in pipe: - Cross-section area: πr². - Flow rate Q = velocity × area. - For long pipe length L: total internal volume = πr²L. **Cylindrical liquid capacity:** For tall thin cylinders (graduated cylinders, test tubes): 1 cm height adds π × r² mL volume. For r = 2 cm: each cm height = 4π ≈ 12.57 mL. For r = 5 cm: each cm height = 25π ≈ 78.54 mL. **Storage tank examples:** Cylindrical water tank, r = 0.5 m, h = 2 m (typical residential): V = π × 0.25 × 2 = 0.5π ≈ 1.57 m³ = 1,570 L. Large industrial tank, r = 10 m, h = 20 m: V = π × 100 × 20 ≈ 6,283 m³ = 6.28 million liters. **Oblique cylinder:** Same volume as right cylinder with same base area and perpendicular height. V = base area × perpendicular height = πr² × h_perp. Surface area is more complex for oblique. **Truncated cylinder (cylinder cut by tilted plane):** If cut creates a heights ranging from h_min to h_max: V = πr² × (h_max + h_min)/2 The volume equals a regular cylinder with height equal to the average. **Cylinder cross-sections:** Cut perpendicular to axis: circle (radius r). Cut along axis: rectangle (2r × h). Cut at angle: ellipse. **Surface-to-volume ratio:** SA/V = (2πr² + 2πrh) / (πr²h) = 2/h + 2/r For tall thin cylinders: SA/V dominated by 2/r (large for small r). For short fat cylinders: SA/V dominated by 2/h. Minimum SA for given V: h = 2r (height = diameter). Like a tin can. **Common applications:** - **Beverages**: cans, bottles, kegs. - **Pipes and plumbing**: water, gas, oil. - **Structural**: columns, supports. - **Hydraulics**: pistons, cylinders. - **Engine cylinders**: internal combustion. - **Storage tanks**: liquids, gases, grains. - **Pharmaceutical**: vials, ampoules, syringes. - **Pressure vessels**: gas tanks, propane cylinders. - **Construction**: rebar, columns, drilled piers. **Cylinder packing:** Cylinders can be stacked in: - **Square**: 4 cylinders touching = π/4 ≈ 78.5% efficient. - **Hexagonal close packing**: π/(2√3) ≈ 90.7% efficient. Honeycomb pattern fits more. **Concrete column:** For round concrete column r = 0.3 m, h = 3 m: V = π × 0.09 × 3 ≈ 0.85 m³. For typical concrete (2,400 kg/m³): ~2,030 kg = 2 tonnes per column. **Software:** - **CAD**: parametric cylinder modeling. - **CFD**: pipe flow simulations. - **Structural analysis**: column buckling, stress. - **Tank design**: API 650 for petroleum storage tanks.