Volume Calculator
Enter dimensions to compute the volume of a three-dimensional shape. The primary result is the rectangular box (cuboid) volume — length × width × height — which applies to rooms, shipping containers, and any rectangular object. The calculator also shows sphere and cylinder volumes using the length dimension as the diameter for comparison. All three formulas are explained in detail below.
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Formula
Box: V = L×W×H | Sphere: V = (4/3)πr³ | Cylinder: V = πr²h
A rectangular box (cuboid) has volume equal to length times width times height — the intuitive 'how many unit cubes fit inside?' For a sphere with radius r (diameter = 2r), the volume is (4/3)πr³. This formula integrates the area of circular cross-sections across the full diameter. For a cylinder with radius r and height h, the volume is the area of the circular base (πr²) multiplied by the height. A cone with the same base and height holds exactly one-third the volume of the cylinder: V = (1/3)πr²h.
How to use the Volume Calculator
- 1
Enter your primary shape
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Enter your length (or diameter)
- 3
Enter your width
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Enter your height
- 5
Read your results instantly
Results update in real time as you type.
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Volume formulas for common shapes
Volume measures the amount of three-dimensional space a solid occupies. The simplest case is the rectangular box (cuboid): V = l × w × h. This formula is intuitive — layer l×w unit squares, h layers high. A cube is the special case where l = w = h, giving V = s³.
A sphere is the set of all points within distance r of a center. Its volume (4/3)πr³ was first derived by Archimedes using the method of exhaustion. A sphere fits snugly inside a cylinder with the same diameter and height equal to the diameter; the sphere occupies exactly 2/3 of that cylinder's volume.
A cylinder's volume is πr²h — the base area times the height. This formula works because every cross-section is the same circle, so stacking them gives the full volume. A cone with the same base and height holds (1/3)πr²h — exactly one-third of the cylinder.
Units and unit conversion
Volume units are cubic length units. If you measure in inches, volume is in cubic inches (in³). If you measure in centimeters, volume is in cubic centimeters (cm³), where 1 cm³ = 1 milliliter. This relationship between cm³ and mL is exact by definition and is extremely useful in chemistry and cooking.
Common conversions: • 1 L = 1000 cm³ = 1000 mL • 1 gallon (US) = 231 in³ ≈ 3.785 L • 1 cubic foot = 1728 in³ ≈ 28.317 L • 1 cubic meter = 1,000,000 cm³ = 1000 L
When computing volumes of rooms for heating/cooling loads, materials needed for construction, or shipping container capacity, make sure all three dimensions are in the same unit before multiplying. A common mistake is mixing feet and inches.
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Practical applications of volume calculation
Volume calculations arise in nearly every field. In construction, you need the volume of a concrete pour (length × width × depth) to order the right amount of material. A typical residential slab at 4 inches deep under a 20 × 30 foot room needs 20 × 30 × (4/12) = 200 cubic feet ≈ 7.4 cubic yards of concrete.
In medicine, dosage calculations sometimes require knowing the volume of a body cavity or organ. Radiologists estimate organ volumes from measured dimensions using the prolate ellipsoid formula V = (4/3)π × (L/2) × (W/2) × (H/2).
In cooking and baking, understanding volume helps when scaling recipes. A mixing bowl's volume determines whether a doubled recipe will fit. A cylindrical cake pan with diameter 9 inches and height 2 inches holds π × 4.5² × 2 ≈ 127 in³ ≈ 2.2 L — useful when the recipe calls for a specific pan size.
Tips & Insights
Use consistent units
All three dimensions must be in the same unit before multiplying. If length is in meters and height is in centimeters, the result will be wrong by a factor of 100. Convert everything to one unit first, then calculate.
The sphere-cylinder relationship
A sphere fits exactly inside a cylinder with the same diameter and a height equal to that diameter. The sphere's volume is 2/3 of the cylinder's. Archimedes considered this his greatest discovery. You can use it as a quick sanity check: the sphere volume should equal (2/3) × π × r² × (2r).
Estimate irregular volumes by displacement
For irregular objects, place them in a graduated container partially filled with water and measure the rise in water level. Volume of object = change in water level × base area of container. This is Archimedes' displacement principle.
Worked Examples
Shipping box capacity
V = 18 × 12 × 10 = 2160 in³ ≈ 1.25 ft³ ≈ 35.4 L. This is the box's total internal volume before packing material is added.
Aquarium water volume
V = 36 × 18 × 16 = 10,368 in³. Dividing by 231 in³/gallon ≈ 44.9 gallons. A standard '40-gallon breeder' aquarium holds about 45 gallons.
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Frequently Asked Questions
What is the difference between volume and capacity?
Volume is a geometric property of a solid — the total 3D space it occupies. Capacity usually refers to the maximum amount a container can hold. For a hollow container, capacity equals the interior volume. They use the same formulas but apply to different objects.
How do I calculate the volume of an irregular shape?
For objects with complex shapes, use water displacement (Archimedes' principle): submerge the object and measure the volume of water displaced. For digital 3D models, software can integrate volume directly. For shapes with known cross-sections, integrate the cross-sectional area along the height.
What units are used for volume?
Any cubic length unit: cm³, m³, in³, ft³. Liquid volumes use liters (L) and milliliters (mL), where 1 mL = 1 cm³. US customary liquid volumes use gallons, quarts, pints, and fluid ounces.
How much bigger is a sphere than a cube with the same side length?
A sphere inscribed in a cube (diameter = side length s) has volume (4/3)π(s/2)³ = πs³/6 ≈ 0.524s³, while the cube has volume s³. The sphere occupies about 52.4% of the cube's volume. If the sphere has the same diameter as the cube's side, it fits inside and occupies just over half the cube.
How do I convert cubic feet to gallons?
1 cubic foot = 7.48052 US gallons (exactly 1728 in³ ÷ 231 in³/gallon). For quick mental math, multiply cubic feet by 7.5. For example, a 10 cubic foot space holds about 75 gallons of water.
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