Cylinder Volume Calculator
The volume (V) of a cylinder is the total space inside its 2 circular bases and its curved side. V = πr²h calculates that space by multiplying the base area (πr²) by the height (h), so the radius and height are the only 2 measurements required.
Volume of a Cylinder Definition
The formula needs only 2 measurements, works in every measurement system, and extends to hollow cylinders, oblique cylinders, and partly filled tanks.
The 3 main uses are sizing tanks and piping systems, finding the swept volume of an engine, and calculating the mass of steel bars from density. A cylinder has 4 main parts: the radius, the diameter (d), the height, and the base area (A), and pi (π ≈ 3.14159) links the radius to the area of a circle. The volume of a cylinder is the total space enclosed by the 2 parallel circular bases and the curved lateral surface of the cylinder. Solid geometry classifies the cylinder as a three-dimensional shape, and mensuration, the branch of mathematics that measures lengths, areas, and volumes, assigns the cylinder a volume in cubic units. Euclidean geometry treats the right circular cylinder as the standard cylinder: the axis, the line through the centers of both bases, stands perpendicular to the bases. A cylinder has 4 parts that set the volume: • Radius (r) measures the distance from the center of the base to the edge of the base. • Diameter (d) measures the width of the base through the center, so d = 2r. • Height (h) measures the perpendicular distance between the 2 bases. • Base area (A) equals the area of a circle, A = πr². Pi (π) is the ratio of a circle's circumference to its diameter, about 3.14159. Drinking glasses, cans, pipes, flower pots, and storage tanks are common cylinders, and the volume of each shape gives the capacity of the container for liquid, grain, or gas.
How to Calculate Volume of a Cylinder
To calculate the volume of a cylinder, follow these 3 steps. A can with a radius of 3 cm (1.18 in) and a height of 10 cm (3.94 in) has V = π × 3² × 10 = 90π ≈ 282.74 cm³. That volume equals 0.283 L, 282.74 mL, 17.25 in³, 0.075 US gal, and 9.56 US fluid ounces. The result carries the cubic unit of the input measurement: centimeters give cubic centimeters, meters give cubic meters, and inches give cubic inches. Common volume conversions: • 1 cm³ = 1 mL • 1 L = 1,000 cm³ = 0.001 m³ • 1 in³ = 16.387 cm³ • 1 ft³ = 28.317 L = 7.481 US gal • 1 US gal = 3.785 L = 231 in³ • 1 UK gal = 4.546 L = 277.42 in³
Volume of Cylinder Formula
V = πr²h, where V is the volume, π is pi (about 3.14159), r is the radius, and h is the height.
Squaring the radius and multiplying by π gives the area of the circular base, and multiplying the base area by the height gives the volume. The formula has 4 equivalent forms, one for each known measurement: • Radius form: V = πr²h • Diameter form: V = πd²h ÷ 4 • Circumference form: V = C²h ÷ (4π), where C is the circumference • Base area form: V = A × h A cylinder with a diameter of 10 in (25.4 cm) and a height of 24 in (61 cm) has V = π × 10² × 24 ÷ 4 = 600π ≈ 1,884.96 in³ (30,889 cm³). The lateral surface area of a cylinder is 2πrh, and the total surface area is 2πr² + 2πrh = 2πr(r + h). A cylinder with r = 3 cm and h = 10 cm has a lateral surface area of 188.50 cm², a total surface area of 245.04 cm², and a volume of 282.74 cm³. Surface area measures the covering in square units (cm², m²), and volume measures the contents in cubic units (cm³, m³). Calculus derives the formula by integration. Rotating a rectangle with width r and height h around one side forms a cylinder, a solid of revolution, and integrating the circular cross-section area along the height gives V = ∫ (from 0 to h) πr² dx = πr²h. Differentiation reverses the step: dV/dh = πr² returns the base area, and dV/dr = 2πrh returns the lateral surface area.
Volume of a Hollow Cylinder
The volume of a hollow cylinder is V = πh(R² − r²), where R is the outer radius, r is the inner radius, and h is the height. A hollow cylinder (a cylindrical shell, tube, or pipe) has material between the outer radius and the inner radius. The formula subtracts the volume of the inner cylinder from the volume of the outer cylinder, so the result is the volume of the wall material. The empty bore holds πr²h. The wall thickness is t = R − r, and the formula rewrites as V = πht(R + r). In calculus, the shell method sums thin cylindrical shells with V = ∫ 2πrh dr. A steel pipe with an outer radius of 6 cm (2.36 in), an inner radius of 5 cm (1.97 in), and a length of 200 cm (6.56 ft) has V = π × 200 × (6² − 5²) = 2,200π ≈ 6,911.50 cm³. Steel has a density of 7.85 g/cm³ (7,850 kg/m³), so the pipe weighs 6,911.50 × 7.85 ≈ 54,255 g, or 54.26 kg (119.6 lb). The bore holds π × 5² × 200 = 5,000π ≈ 15,707.96 cm³, or 15.71 L (4.15 US gal). Engineers use the wall volume to find pipe mass and the bore volume to find fluid capacity in piping systems. A 50 m (164 ft) run of pipe with an inner diameter of 100 mm (3.94 in) holds π × 0.05² × 50 ≈ 0.3927 m³, which is 392.7 L (103.7 US gal).
Volume of an Oblique Cylinder
The volume of an oblique cylinder is V = πr²h, where h is the perpendicular height between the 2 bases, not the slant length of the side. An oblique cylinder has 2 parallel circular bases joined by an axis that leans away from the perpendicular. Slanted architectural columns and sheared cylinders in solid geometry exercises are oblique cylinders. Cavalieri's principle explains the formula. Bonaventura Cavalieri, an Italian mathematician, published the principle in 1635: 2 solids with equal heights and equal cross-sectional areas at every level have equal volumes. Every horizontal slice of an oblique cylinder is a circle of radius r, the same circle that a right circular cylinder with the same radius and height shows, so the volumes match. To find the perpendicular height, use h = L × sin θ, where L is the slant length of the axis and θ is the angle between the axis and the base. An oblique cylinder with r = 3 cm (1.18 in), L = 10 cm (3.94 in), and θ = 60° has h = 10 × sin 60° ≈ 8.66 cm (3.41 in) and V = π × 3² × 8.66 ≈ 244.86 cm³ (14.94 in³). A right circular cylinder with the same radius and a 10 cm height holds 282.74 cm³, so substituting the slant length for the perpendicular height overstates the oblique volume by 15.5%.
Volume: Cylinder vs Cone
A cone holds exactly one-third of the volume of a cylinder with the same base radius and height. The cone volume formula is V = (1/3)πr²h, so 3 identical cones fill 1 cylinder. Euclid's Elements (Book XII) proves the ratio, and integration confirms the ratio: a cone cross-section at distance x from the tip has radius rx ÷ h, so V = ∫ (from 0 to h) π(rx ÷ h)² dx = (1/3)πr²h. Cavalieri's principle extends the 1:3 ratio to right and oblique pairs with the same base and height. A cylinder with r = 3 cm and h = 10 cm holds 282.74 cm³, and the matching cone holds 94.25 cm³. A hemisphere of radius r holds (2/3)πr³. A cylinder, a hemisphere, and a cone with the same radius r and height r have volumes in the ratio 3 : 2 : 1. For r = 3 cm, the volumes are 84.82 cm³, 56.55 cm³, and 28.27 cm³.
Cylinder vs Sphere Volume
A sphere fills two-thirds of the volume of the smallest cylinder that contains the sphere. The sphere volume is V = (4/3)πr³. The containing cylinder has radius r and height 2r, so the cylinder volume is πr² × 2r = 2πr³, and the ratio is (4/3) ÷ 2 = 2/3. The surface areas follow the same 2:3 ratio: the sphere has 4πr², and the cylinder with both bases has 6πr². Archimedes proved both ratios in On the Sphere and Cylinder and requested a sphere inside a cylinder as the carving on his tomb. For r = 5 mm (0.2 in), the sphere holds 523.60 mm³ (0.5236 cm³) and the cylinder holds 785.40 mm³ (0.7854 cm³). A steel ball bearing with this 10 mm (0.39 in) diameter weighs about 4.11 g (0.145 oz) at a density of 7.85 g/cm³. Tank designers and packaging engineers compare both shapes for the same contents: the sphere uses the least surface area for a given volume, and the cylinder fits flat surfaces and rectangular crates.
Cylinder Volume Calculators
Cylinder volume calculators apply V = πr²h to the measurements already known, and 6 calculator groups cover most cases: • Input calculators accept a radius, diameter, circumference, base area, or total surface area (SA), convert the input to a radius, and apply V = πr²h. A total surface area with a known radius gives the height: h = SA ÷ (2πr) − r. • Missing-value calculators find the height from the volume and radius with h = V ÷ (πr²), or the radius from the volume and height with r = √(V ÷ (πh)). • Shape calculators handle right cylinders, oblique cylinders, hollow cylinders, and open cylinders without a top and bottom. • Unit calculators return litres, milliliters, gallons, fluid ounces, cubic inches, cubic feet, and cubic meters from the same inputs. • Tank calculators size vertical cylinder tanks and horizontal cylinder tanks, including partly filled tanks. • Comparison calculators show the cylinder volume next to the cone volume ((1/3)πr²h) and the sphere volume ((4/3)πr³). A vertical cylinder tank with a diameter of 2 m (6.56 ft) and a height of 3 m (9.84 ft) holds π × 1² × 3 ≈ 9.42 m³, which equals 9,424.8 L, 2,489.8 US gal, 2,073 UK gal, and 332.8 ft³. A partly filled vertical tank holds πr² multiplied by the liquid depth. A horizontal cylinder tank of length L holds πr²L when full. A partly filled horizontal tank with liquid depth d holds V = L × [r² cos⁻¹((r − d) ÷ r) − (r − d) × √(2rd − d²)], where the inverse cosine returns radians. A horizontal tank with r = 0.5 m (1.64 ft), L = 2 m (6.56 ft), and d = 0.25 m (0.82 ft) holds about 0.307 m³ (307 L, 81.1 US gal), which is 19.6% of the full volume.
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Frequently Asked Questions
Where can you find cylinders in nature?
Cylindrical shapes occur in tree trunks, plant stems, bones, stalactites, the filament of a bacterial flagellum, and a thread of spider silk. Each shape approximates a cylinder, so V = πr²h estimates the volume. A tree trunk with a diameter of 0.5 m (1.64 ft) and a height of 10 m (32.8 ft) has a volume of π × 0.25² × 10 ≈ 1.96 m³ (69.3 ft³).
How do you calculate the volume of a cylinder in gallons?
To calculate the volume of a cylinder in gallons, divide the volume in cubic inches by 231 for US gallons, or by 277.42 for UK gallons. In metric units, divide litres by 3.785 for US gallons or by 4.546 for UK gallons. A cylinder with a diameter of 10 in (25.4 cm) and a height of 24 in (61 cm) has a volume of 1,884.96 in³, which is 8.16 US gal, 6.79 UK gal, and 30.89 L.
How do you calculate the volume of a cylinder in cubic feet?
To calculate the volume of a cylinder in cubic feet, measure the radius and the height in feet, then apply V = πr²h. For measurements in inches, divide the cubic inch result by 1,728. A cylinder with a radius of 2 ft (0.61 m) and a height of 5 ft (1.52 m) has V = π × 2² × 5 = 20π ≈ 62.83 ft³, which equals 1.78 m³, 1,779 L, and 470 US gal.
How do you find the volume of a cylinder using circumference?
To find the volume of a cylinder using circumference, divide the circumference (C) by 2π to get the radius, then apply V = πr²h, or use V = C²h ÷ (4π) directly. A cylinder with a circumference of 18.85 cm (7.42 in) and a height of 10 cm (3.94 in) has V = 18.85² × 10 ÷ (4π) ≈ 282.7 cm³.
Does doubling the radius double the volume of a cylinder?
No, doubling the radius multiplies the volume by 4, and doubling the height multiplies the volume by 2. A cylinder with a radius of 3 cm and a height of 10 cm holds 282.74 cm³, and the same cylinder with a radius of 6 cm holds 1,130.97 cm³.
How do you calculate the weight of a cylinder?
To calculate the weight of a cylinder, multiply the volume by the density of the material: mass = V × density. A solid steel rod with a radius of 2 cm (0.79 in) and a length of 50 cm (19.7 in) has V = π × 2² × 50 ≈ 628.32 cm³, so the mass is 628.32 × 7.85 g/cm³ ≈ 4,932 g, or 4.93 kg (10.87 lb).
How do you find the volume of an elliptical cylinder?
To find the volume of an elliptical cylinder, multiply the area of the elliptical base (πab) by the height: V = πabh, where a and b are the semi×major and semi×minor axes of the ellipse. An elliptical cylinder with a = 4 cm, b = 3 cm, and h = 10 cm holds π × 4 × 3 × 10 ≈ 376.99 cm³.
How do you find the volume of a truncated cylinder?
To find the volume of a truncated cylinder, a cylinder cut by a slanted plane, multiply the base area by the average of the longest and shortest side heights: V = πr² × (h₁ + h₂) ÷ 2. A truncated cylinder with r = 3 cm, h₁ = 10 cm, and h₂ = 6 cm holds π × 9 × 8 = 72π ≈ 226.19 cm³.
How do you calculate the swept volume of an engine cylinder?
To calculate the swept volume of an engine cylinder, multiply the bore area by the stroke: swept volume = π × (bore ÷ 2)² × stroke. Swept volume is the volume the piston displaces between bottom dead center and top dead center. An engine cylinder with an 86 mm (3.39 in) bore and an 86 mm stroke has a swept volume of π × 43² × 86 ≈ 499.56 cm³, and a 4×cylinder engine of that size displaces 1,998 cm³, about 2.0 L (121.9 in³).
How do you find the height or radius of a cylinder from the volume?
To find the height of a cylinder from the volume, divide the volume by πr²: h = V ÷ (πr²). To find the radius, divide the volume by πh and take the square root: r = √(V ÷ (πh)). A cylinder with V = 1,000 cm³ and r = 5 cm has h = 1,000 ÷ (π × 25) ≈ 12.73 cm (5.01 in), and a cylinder with V = 1,000 cm³ and h = 10 cm has r = √(1,000 ÷ (π × 10)) ≈ 5.64 cm (2.22 in).