What is the formula for the volume of a cube?

Volume of a cube = side length cubed (s^3). All six faces of a cube are equal squares, so you only need one measurement. Example: a cube with side 5 cm has volume = 5 x 5 x 5 = 125 cubic centimeters.

How do you calculate the volume of a sphere?

Volume of a sphere = (4/3) x pi x r^3, where r is the radius. Example: sphere with radius 6 cm = (4/3) x 3.14159 x 216 = 904.78 cubic centimeters. If you know the diameter, radius = diameter divided by 2.

What is the volume formula for a cylinder?

Volume of a cylinder = pi x r^2 x h, where r is the base radius and h is the height. Example: cylinder with radius 4 cm and height 10 cm = 3.14159 x 16 x 10 = 502.65 cubic centimeters. This formula applies to cans, pipes, tanks, and circular columns.

What is the volume of a rectangular prism (box)?

Volume = Length x Width x Height. This is the most commonly used volume formula. Example: a box 10 cm x 5 cm x 3 cm = 150 cubic centimeters. Used for shipping boxes, rooms, tanks, and storage containers.

What is the difference between volume and surface area?

Volume measures the three-dimensional space inside a shape (cubic units: cm3, m3, in3). Surface area measures the total area of all outer faces (square units: cm2, m2, in2). A swimming pool with volume 50,000 liters uses volume for water capacity and surface area for tiles needed on the walls and floor.

How do you calculate the volume of a pyramid?

Volume of a square pyramid = (1/3) x Base Area x Height. For a square base: (1/3) x side^2 x height. Example: pyramid with square base 6 cm per side and height 9 cm = (1/3) x 36 x 9 = 108 cubic centimeters. Like a cone, a pyramid holds one-third the volume of a prism with the same base.

What is the volume of a triangular prism?

Volume = (1/2) x base x height x length, where base and height define the triangular face and length is the depth of the prism. Example: triangular prism with triangle base 4 cm, triangle height 3 cm, and prism length 10 cm = (1/2) x 4 x 3 x 10 = 60 cubic centimeters.

How do you convert between volume units?

Common conversions: 1 liter = 1000 cubic centimeters (cm3). 1 cubic meter = 1000 liters. 1 US gallon = 231 cubic inches = 3.785 liters. 1 cubic foot = 28.317 liters = 7.481 US gallons. 1 cubic inch = 16.387 cubic centimeters.

How is volume used in real life?

Volume is used everywhere: calculating how much water a tank holds, how much concrete for a foundation, how many boxes fit in a truck, liquid capacity of bottles and cans, fuel tank size, swimming pool capacity, medication dosage in medicine, and material quantity in manufacturing.

What is the volume of a capsule?

A capsule (cylinder with two hemispherical ends) has volume = pi x r^2 x h + (4/3) x pi x r^3, where r is the radius and h is the cylindrical body height. Simplified: pi x r^2 x (h + 4r/3). Used for pharmaceutical capsule sizing and storage tank design.

Why does a cone have one-third the volume of a cylinder?

A cone has volume (1/3) x pi x r^2 x h while a cylinder with the same base and height has volume pi x r^2 x h. The cone tapers to a point, so it encloses one-third the space. This can be demonstrated by filling a cone-shaped container three times to fill an equal cylinder.

How do you find volume when you know the surface area?

For a cube: if surface area = 6s^2, then s = sqrt(SA/6), and volume = s^3. For a sphere: if surface area = 4 x pi x r^2, then r = sqrt(SA / (4 x pi)), and volume = (4/3) x pi x r^3. Each shape has its own relationship between surface area and volume.

What unit is volume measured in?

Volume is measured in cubic units: cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), cubic feet (ft3). For liquids, liters (L) and milliliters (mL) are used, where 1 mL = 1 cm3 exactly. Always specify the unit when reporting volume.

Volume Calculator -- All Shapes -- CalculatorCove

📦 Select a Shape & Enter Dimensions

V = s³

Side Length (s)

All 6 faces are equal squares

Enter a valid side length.

V = L × W × H

Length (L)

Enter valid length.

Width (W)

Enter valid width.

Height (H)

Enter valid height.

V = (4/3) × π × r³

Radius (r)

Half the diameter

Enter a valid radius.

OR Diameter

Overrides radius if entered

V = π × r² × h

Radius (r)

Base circle radius

Enter a valid radius.

Height (h)

Perpendicular height

Enter a valid height.

V = (1/3) × π × r² × h

Base Radius (r)

Enter a valid radius.

Height (h)

Enter a valid height.

V = (1/2) × b × h × L

Triangle Base (b)

Enter valid base.

Triangle Height (h)

Enter valid height.

Prism Length (L)

Enter valid length.

V = (1/3) × s² × h

Base Side (s)

Square base side length

Enter a valid side.

Height (h)

Perpendicular height to apex

Enter a valid height.

V = π × r² × (h + 4r/3)

Radius (r)

Radius of both hemispherical ends

Enter a valid radius.

Body Height (h)

Cylindrical body length only

Enter a valid height.

Units:

Volume

—

⚠️ Disclaimer: Volume is mathematically exact for entered dimensions. For real-world projects add 5–10% for material waste and measurement tolerance. Always verify measurements before purchasing materials.

Was this calculator helpful?

Sources & Methodology

✓All volume formulas verified against Euclid’s Elements geometry definitions and NIST Mathematical Functions. Pi = 3.14159265358979 (IEEE 754 double precision). Capsule formula = cylinder + sphere (two hemispheres). All calculations guard with isFinite.

📘

Khan Academy — Solid Geometry

Khan Academy’s complete solid geometry curriculum covering volumes and surface areas of all standard 3D shapes. The globally used standard for mathematics education on this topic.

🏛️

NIST Digital Library of Mathematical Functions

U.S. government authoritative mathematical reference. Sphere, cylinder, cone, and prism volume formulas are verified against NIST geometric definitions and constants including pi.

Formulas used (all verified):
Cube: s³ | Rectangular Prism: L×W×H | Sphere: (4/3)πr³ Cylinder: πr²h | Cone: (1/3)πr²h | Tri. Prism: (1/2)bhL Sq. Pyramid: (1/3)s²h | Capsule: πr²(h + 4r/3) Surface area bonus calculated for each shape. All inputs use parseFloat(x)||0. isFinite guard on every result. Pi = Math.PI (IEEE 754).

Last reviewed: April 2026

Volume Formulas for All 3D Shapes — Complete Reference Guide

Volume is the measure of three-dimensional space enclosed within a solid object. It is always expressed in cubic units — cubic centimeters (cm³), cubic meters (m³), cubic inches (in³), or cubic feet (ft³). Volume calculations are essential across engineering, construction, medicine, chemistry, cooking, packaging, and everyday life. Knowing the volume of a shape tells you how much it can hold, how much material it contains, or how much space it occupies.

Worked Example: Volume of a Cylindrical Water Tank

A cylindrical tank has radius 0.5 m and height 2 m.
Formula: V = π × r² × h
Step 1: r² = 0.5² = 0.25 m²
Step 2: V = 3.14159 × 0.25 × 2 = 1.5708 m³
Convert to liters: 1 m³ = 1,000 L → 1,570.8 liters
Add 10% safety margin: 1,570.8 × 1.10 = 1,727.9 liters recommended tank size

Volume Formulas and Their Derivations

Every volume formula connects to a geometric principle. The rectangular prism (V = L×W×H) is the foundation — all other formulas can be understood relative to it. A cylinder is like a circular prism: base area (πr²) times height. A cone holds exactly one-third the volume of a cylinder with the same base and height. A pyramid holds exactly one-third the volume of a prism with the same base and height. A sphere fits inside a cylinder of radius r and height 2r with volume two-thirds of that cylinder.

All Volume Formulas Reference Table

Shape	Formula	Variables	Example
Cube	V = s³	s = side length	5³ = 125 cm³
Rectangular Prism	V = L × W × H	Length, Width, Height	10×5×3 = 150 cm³
Sphere	V = (4/3)πr³	r = radius	(4/3)π×6³ = 904.8 cm³
Cylinder	V = πr²h	r = radius, h = height	π×4²×10 = 502.7 cm³
Cone	V = (1/3)πr²h	r = base radius, h = height	(1/3)π×3²×8 = 75.4 cm³
Triangular Prism	V = (1/2)bhL	b,h = triangle, L = length	(1/2)×4×3×10 = 60 cm³
Square Pyramid	V = (1/3)s²h	s = base side, h = height	(1/3)×6²×9 = 108 cm³
Capsule	V = πr²(h + 4r/3)	r = radius, h = body height	π×3²×(8+4) = 339.3 cm³

Volume of Cube vs Rectangular Prism

A cube is a special rectangular prism where all three dimensions are equal. The simplest volume formula is V = s³ — just cube the side length. Doubling the side length of a cube does not double the volume: it increases it by 8 times (2³ = 8). This is why storage units and shipping containers use rectangular prisms — the three independent dimensions allow more flexibility in optimizing space. Real use: a room 4 m × 5 m × 3 m has volume = 60 m³. At 0.12 kg/m³ air density, that room contains about 7.2 kg of air.

💡 Volume vs Capacity vs Weight: Volume is the space a solid object occupies (cm³). Capacity is how much liquid a hollow object holds (liters, mL). For water: 1 cm³ = 1 mL = 1 gram exactly. So a cylinder with volume 500 cm³ can hold 500 mL of water weighing 500 g. For other liquids, multiply by their density. This relationship (1 cm³ = 1 mL for water) is why metric units are so convenient in science and cooking.

Volume of Sphere, Cylinder, and Cone Relationships

The three most important volume relationships in geometry: a cylinder with radius r and height 2r has volume πr²(2r) = 2πr³. A sphere of radius r has volume (4/3)πr³ — exactly 2/3 of the cylinder that contains it. A cone with the same base and height as the cylinder has volume (1/3)πr²h = one-third of the cylinder. Archimedes discovered the sphere-cylinder relationship and considered it his greatest achievement.

Real-World Applications of Volume Calculations

Construction & concrete: Volume of a foundation slab = L × W × thickness. At 2,400 kg/m³ concrete density, you can calculate weight and material cost.
Water tanks & pools: Cylinder or rectangular prism volume gives capacity in liters (1 m³ = 1,000 L).
Shipping & packaging: Box volume determines how many units fit per shipping container.
Medicine: Capsule volume determines drug dosage. Syringe barrel is a cylinder volume.
Cooking: Baking pan volume (rectangular prism) determines recipe batch size.
Earth moving: Excavation volume in m³ determines truck loads and disposal cost.
Aquariums: Volume in liters determines fish stocking density and filter size.

How to Calculate the Volume of a Box (Rectangular Prism)

The rectangular box, also called a rectangular prism or cuboid, is the most commonly calculated volume in everyday life. To find its volume, multiply the three dimensions: length, width, and height. The formula V = L × W × H works for boxes, rooms, shipping containers, tanks, and swimming pools. If you have a cardboard box measuring 30 cm long, 20 cm wide, and 15 cm tall, its volume is 30 × 20 × 15 = 9,000 cm³, or exactly 9 liters.

How to Find the Volume of a Sphere

The sphere is one of the most elegant shapes in mathematics. Its volume formula V = (4/3)πr³ requires only the radius — the distance from the center to the surface. If you know the diameter instead, divide by 2 to get the radius. A soccer ball with a diameter of 22 cm has a radius of 11 cm, giving a volume of about 5,575 cm³. The sphere is the shape that encloses the maximum volume for a given surface area — which is why bubbles, planets, and water droplets naturally form spheres.

Cylinder Volume: The Tank Formula

Cylinders appear everywhere in real life — cans, pipes, storage tanks, and drinking glasses. The volume formula V = πr²h multiplies the area of the circular base (πr²) by the height. Note that this uses the radius, not the diameter. If a can has a diameter of 8 cm and a height of 12 cm, the radius is 4 cm and the volume is π × 4² × 12 ≈ 603 cm³, which is about 603 milliliters — slightly more than a standard 500 ml can.

Cone Volume and Why It Is One-Third of a Cylinder

The cone volume formula V = (1/3)πr²h is exactly one-third the volume of a cylinder with the same base and height. This is a proven geometric relationship: if you fill a cone with water three times and pour each into a same-sized cylinder, you fill it exactly. This relationship is used in ice cream cone manufacturing, funnel design, and grain storage calculations. The "height" in the formula is the vertical height from the center of the base to the apex, not the slant height along the side.

Cube Volume: When All Sides Are Equal

A cube is a special case of a rectangular box where all three dimensions are equal. The formula V = s³ simply cubes the side length. A Rubik's cube with a side of 5.7 cm has a volume of 5.7³ ≈ 185 cm³. The cube is the shape used to define cubic units themselves — a "cubic centimeter" is literally the volume of a cube with 1 cm sides.

Triangular Prism Volume for Construction and Architecture

A triangular prism has two triangular faces connected by three rectangular faces. Its volume is calculated by finding the area of the triangular cross-section and multiplying by the length. The formula V = (1/2) × base × height × length is used in roof design, bridge construction, and calculating concrete for triangular structural elements. A roof section that is 4 m at the base, 2.5 m tall at the peak, and 10 m long holds V = (1/2) × 4 × 2.5 × 10 = 50 m³ of space.

Real-World Applications of Volume Calculations

Volume formulas are not just classroom exercises — they solve real problems daily:

Construction: Calculating concrete needed for foundations, columns, and slabs
Shipping and packaging: Determining how much product fits in a container
Cooking and baking: Scaling recipes by adjusting pan volumes
Aquariums and pools: Calculating water volume for chemical treatment
Medicine: Measuring drug dosages in syringes and capsules
Engineering: Designing tanks, pipes, and structural components
Environmental science: Estimating reservoir capacity and rainfall runoff

Volume Units and Conversions

Volume measurements use different unit systems depending on the context. The metric system uses cubic meters (m³), cubic centimeters (cm³), and liters, where 1 liter = exactly 1,000 cm³. The imperial system uses cubic inches (in³), cubic feet (ft³), gallons, and fluid ounces. One U.S. gallon equals 231 cubic inches or 3.785 liters. When mixing systems, always convert all dimensions to the same unit before calculating.

📏 Archimedes' Displacement Method

For irregular objects without a defined formula, the ancient water displacement method works perfectly: submerge the object in a known volume of water and measure the rise. The volume of water displaced equals the volume of the object. This method was famously discovered by Archimedes in 250 BC and is still used in labs today.

Surface Area vs. Volume: Understanding Both Measures

Surface area and volume are related but measure different things. Surface area measures the total area of all outer faces (in square units), while volume measures the 3D space inside (in cubic units). A box can have a large surface area but small volume if it is flat and wide. Engineers must calculate both — surface area determines how much material is needed to build something, while volume determines how much it can hold.

Frequently Asked Questions

V = s³ (side length cubed). A cube with side 5 cm has volume = 125 cm³. All three dimensions are equal, so you only need one measurement.

V = (4/3) × π × r³. Enter the radius (half the diameter). Sphere with radius 6 cm = (4/3) × 3.14159 × 216 = 904.78 cm³.

V = π × r² × h. Cylinder with radius 4 cm and height 10 cm = 3.14159 × 16 × 10 = 502.65 cm³. Used for tanks, pipes, cans, and columns.

V = (1/3) × π × r² × h. A cone holds one-third the volume of a cylinder with the same base and height. Cone with radius 3 cm, height 8 cm = 75.40 cm³.

V = L × W × H. Most commonly used volume formula. Box 10 cm × 5 cm × 3 cm = 150 cm³. Used for rooms, storage, and shipping.

Volume = 3D space inside (cubic units: cm³, ft³). Surface area = total area of all outer faces (square units: cm², ft²). This calculator shows both volume AND surface area for every shape.

V = (1/3) × s² × h for a square pyramid. Like a cone, a pyramid holds one-third the volume of a prism with the same base. Example: base 6 cm, height 9 cm = (1/3) × 36 × 9 = 108 cm³.

V = (1/2) × b × h × L where b and h are the triangle’s base and height, and L is the prism’s length. Example: triangle base 4, height 3, length 10 = (1/2) × 4 × 3 × 10 = 60 cm³.

1 liter = 1,000 cm³ = 0.001 m³. So 500 cm³ = 0.5 liters. 1 m³ = 1,000 liters. For water specifically: 1 cm³ = 1 mL = 1 gram.

Concrete slab volume = L × W × thickness. A 5 m × 3 m × 0.15 m slab = 2.25 m³. At 2,400 kg/m³ density = 5,400 kg of concrete. Volume determines material quantities, cost, and truck loads for delivery.

A cone tapers from full base area to a point, enclosing only (1/3) the space of a cylinder with the same base and height. Demonstrable by filling a cone-shaped container three times to fill an equal cylinder.

Volume is in cubic units: cm³, m³, in³, ft³. For liquids: liters (L) and milliliters (mL) where 1 mL = 1 cm³. Always include the unit when reporting volume to avoid confusion between shapes and scales.

A capsule is a cylinder with two hemispherical ends. V = πr²(h + 4r/3). Used for pharmaceutical capsule sizing, storage tank design, and pill packaging. The hemispherical ends distribute pressure evenly, making capsule-shaped tanks stronger than flat-ended cylinders.

Decompose it into known shapes, calculate each volume, and add them. Or use water displacement: submerge the object in water and measure the volume of water displaced (1 mL displaced = 1 cm³ object volume). This is Archimedes’ principle.

Related Math Calculators