Volume Calculator

Selecting the Shape

Choose from the available 3D shapes: sphere, cube, cylinder, cone, rectangular prism, ellipsoid, or capsule. Each shape requires different input parameters based on its geometry. The shape selector updates the input fields dynamically, showing only the dimensions relevant to your selection.

Entering Dimensions

Enter the required measurements for your chosen shape. For a sphere, you need only the radius. A cube needs only the edge length. A cylinder needs radius and height. A cone needs radius and height. A rectangular prism needs length, width, and height. An ellipsoid needs three semi-axis lengths. A capsule needs radius and cylindrical height. All dimensions accept decimal values for precise measurements.

Viewing Results

The calculator displays the volume immediately after you enter the dimensions. It also shows the formula used and provides context about the calculation. The result updates in real time as you adjust any input value.

Checking Your Answer

To verify a manual calculation, compare against the calculator output. For a sphere with radius 6 m, your manual calculation should yield V = (4/3)π × 216 ≈ 904.8 m³. Any significant discrepancy indicates an error in your manual formula or arithmetic.

Sphere

A sphere is a perfectly round 3D shape. The volume formula is:

Where r is the radius of the sphere. This formula shows that volume increases with the cube of the radius, meaning small radius increases cause large volume changes.

Example: A sphere with radius 3 cm: V = (4/3)π × 3³ = (4/3)π × 27 = 36π ≈ 113.1 cm³

Example: A sphere with radius 1.5 m: V = (4/3)π × (1.5)³ = (4/3)π × 3.375 = 4.5π ≈ 14.14 m³. This is roughly the volume of a small hot water storage tank.

Cube

A cube is a regular hexahedron with six equal square faces. The volume formula is:

Where a is the edge length of the cube. Since all edges are equal, volume is simply the edge cubed.

Example: A cube with edge 5 cm: V = 5³ = 125 cm³

Example: A shipping crate with edge length 1.2 m: V = (1.2)³ = 1.728 m³. This represents the maximum cargo volume inside the crate.

Cylinder

A cylinder has two parallel circular bases connected by a curved surface. The volume formula is:

Where r is the radius of the circular base and h is the height of the cylinder. This formula is essentially the area of the circular base multiplied by the height.

Example: A cylinder with radius 4 cm and height 10 cm: V = π × 4² × 10 = π × 16 × 10 = 160π ≈ 502.7 cm³

Example: A cylindrical water tank with radius 0.75 m and height 2 m: V = π × (0.75)² × 2 = π × 0.5625 × 2 = 1.125π ≈ 3.53 m³, which is equivalent to 3,530 liters.

Cone

A cone has a circular base and a curved surface that tapers to a single point (the apex). The volume formula is:

Where r is the radius of the circular base and h is the height of the cone. A cone's volume is exactly one-third of the equivalent cylinder.

Example: A cone with radius 3 cm and height 6 cm: V = (1/3)π × 3² × 6 = (1/3)π × 9 × 6 = 18π ≈ 56.5 cm³

Example: A traffic cone with base radius 0.2 m and height 0.7 m: V = (1/3)π × (0.2)² × 0.7 = (1/3)π × 0.04 × 0.7 = (0.028/3)π ≈ 0.0293 m³, or about 29.3 liters of material.

Rectangular Prism (Tank)

A rectangular prism has six rectangular faces. The volume formula is:

Where l is length, w is width, and h is height. This simple formula multiplies the three dimensions.

Example: A box with length 8 cm, width 5 cm, and height 4 cm: V = 8 × 5 × 4 = 160 cm³

Example: A concrete footing with length 2.4 m, width 0.4 m, and height 0.3 m: V = 2.4 × 0.4 × 0.3 = 0.288 m³. At a standard concrete density of 2,400 kg/m³, this footing weighs approximately 691 kg.

Pyramid (Square Base)

A square pyramid has a square base and four triangular faces meeting at an apex. The volume formula is:

Where a is the length of the square base edge and h is the height (perpendicular distance from base to apex).

Example: A pyramid with base side 6 cm and height 9 cm: V = (1/3) × 6² × 9 = (1/3) × 36 × 9 = 108 cm³

Example: A pyramid with base side 10 m and height 15 m: V = (1/3) × 10² × 15 = (1/3) × 100 × 15 = 500 m³. For scale, the Great Pyramid of Giza has a base side of about 230 m and an original height of 146.6 m, yielding a volume of roughly 2.6 million m³.

Ellipsoid

An ellipsoid is a stretched or compressed sphere with three different axis lengths. The volume formula is:

Where a, b, and c are the lengths of the three semi-axes.

Example: An ellipsoid with axes 5 cm, 4 cm, and 3 cm: V = (4/3)π × 5 × 4 × 3 = (4/3)π × 60 = 80π ≈ 251.3 cm³

Example: A rugby ball approximated as an ellipsoid with semi-axes a = 14 cm, b = 9 cm, c = 9 cm: V = (4/3)π × 14 × 9 × 9 = (4/3)π × 1134 = 1512π ≈ 4,750 cm³ or 4.75 liters.

Capsule

A capsule is a cylinder with hemispherical ends. The volume formula is:

Where r is the radius of the circular ends and h is the height of the cylindrical middle section.

Example: A capsule with radius 2 cm and cylindrical height 6 cm: V = π × 2² × 6 + (4/3)π × 2³ = π × 4 × 6 + (4/3)π × 8 = 24π + (32/3)π = (72/3 + 32/3)π = (104/3)π ≈ 108.9 cm³

Example 1: Sphere (Basketball)

Problem: A regulation men's basketball has a diameter of 24.26 cm. What is its volume?

Step 1: Find the radius. r = d/2 = 24.26/2 = 12.13 cm

Step 2: Apply the sphere formula. V = (4/3)πr³ = (4/3)π × (12.13)³

Step 3: Calculate r³. 12.13³ = 12.13 × 12.13 × 12.13 ≈ 1,785.4

Step 4: Multiply. V = (4/3)π × 1,785.4 = 2,380.5π/3 ≈ 7,476 cm³

A basketball contains about 7.5 liters of air when fully inflated.

Example 2: Cylinder (Soup Can)

Problem: A soup can has a radius of 3.5 cm and a height of 10.5 cm. What is its volume?

Step 1: Identify the input values. r = 3.5 cm, h = 10.5 cm

Step 2: Apply the cylinder formula. V = πr²h = π × (3.5)² × 10.5

Step 3: Calculate r². 3.5² = 12.25

Step 4: Multiply. V = π × 12.25 × 10.5 = 128.625π ≈ 404.1 cm³

The can holds approximately 404 mL, which is close to a standard 400 mL soup can.

Example 3: Cone (Party Hat)

Problem: A conical party hat has a base radius of 7 cm and a slant height of 25 cm. First find the perpendicular height, then the volume.

Step 1: Find the perpendicular height using the Pythagorean theorem. r = 7 cm, slant height l = 25 cm: h = √(l² - r²) = √(625 - 49) = √576 = 24 cm

Step 2: Apply the cone formula. V = (1/3)πr²h = (1/3)π × 7² × 24

Step 3: Calculate r². 7² = 49

Step 4: Multiply. V = (1/3)π × 49 × 24 = (1/3)π × 1,176 = 392π ≈ 1,231.5 cm³

A party hat of these dimensions contains about 1.23 liters of space.

For more information, see the Pythagorean Theorem Calculator.

Engineering and Manufacturing

Engineers calculate volume when designing containers, pipes, and storage tanks. Volume determines capacity, material requirements, and shipping dimensions. Even small miscalculations can lead to significant problems in large-scale projects. For example, a cylindrical pressure vessel with radius 1 m and length 5 m has a volume of about 15.7 m³. A 1% error in radius measurement results in roughly 2% error in the calculated volume, potentially compromising the vessel's design specifications.

Construction

Builders calculate concrete volume for foundations, beams, and columns. Accurate volume calculations ensure enough material is ordered while minimizing waste. The formula for a rectangular prism is used most frequently in construction. A concrete slab measuring 10 m by 6 m by 0.15 m thick requires 9 m³ of concrete. Ordering 10% extra (0.9 m³) accounts for spillage, uneven ground, and reinforcement displacement.

Medical Fields

Healthcare professionals calculate medication doses based on volume. Intravenous fluids, anesthesia, and many pharmaceuticals require precise volume measurements. The body's blood volume is a critical health indicator. An average adult has approximately 5 liters of blood. A cylindrical IV drip bag with radius 6 cm holds roughly 1 liter of saline solution when filled to 8.8 cm height.

Cooking and Food Industry

Recipes often specify volume measurements for ingredients. Food manufacturers calculate package sizes and production volumes. Understanding volume helps scale recipes up or down appropriately. A hemispherical mixing bowl with radius 15 cm can hold about 7 liters of dough. Scaling a recipe from 4 servings to 12 servings requires tripling each volumetric ingredient measure.

Science and Research

Scientists measure volumes of liquids, gases, and irregular objects in laboratories. Precise volume measurements are essential for chemical reactions, biological experiments, and physical measurements. A graduated cylinder's accuracy depends on its diameter: a narrow 10 mL cylinder (1 cm diameter) has a resolution of about 0.08 mL per millimeter of height, while a wider 100 mL cylinder (3 cm diameter) resolves roughly 0.7 mL per millimeter.

Volume

Volume measures the interior space occupied by a 3D object, measured in cubic units (cm³, m³, in³, etc.). It represents how much something can hold or contain.

Surface Area

Surface area measures the total area of all outer surfaces, measured in square units (cm², m², etc.). It represents how much exposed area an object has. See the Surface Area Calculator for more details.

The Relationship

As objects grow larger, volume increases faster than surface area. This principle explains why cells divide as they grow, why large animals have different cooling mechanisms than small animals, and why heat dissipation depends on surface area to volume ratio. For a cube of side length 1 cm, the surface area is 6 cm² and the volume is 1 cm³, giving a ratio of 6:1. For a cube of side length 10 cm, the surface area is 600 cm² and the volume is 1,000 cm³, giving a ratio of 0.6:1. This tenfold size increase reduces the surface-area-to-volume ratio by a factor of ten.

For more information, see the Area Calculator.

Same Base and Height: Cylinder vs. Cone

A cylinder and cone sharing the same base radius and height demonstrate a fundamental geometric relationship. The cone always has exactly one-third the volume of the cylinder. With radius 4 cm and height 9 cm, the cylinder volume is π × 16 × 9 = 144π ≈ 452.4 cm³, while the cone volume is 48π ≈ 150.8 cm³. This 3:1 ratio holds for any matching cylinder-cone pair.

Same Cross-Sectional Area: Cylinder vs. Rectangular Prism

A cylinder and a rectangular prism with equal base areas and heights have identical volumes. A cylinder with radius 3 cm has a base area of 9π ≈ 28.27 cm². A rectangular prism with length 5.66 cm and width 5 cm (also 28.3 cm²) and the same height of 10 cm has nearly the same volume: approximately 282.7 cm³ vs. 283.0 cm³.

Same Radius: Sphere vs. Cylinder vs. Cone

With a fixed radius of 5 cm and cylinder/cone height equal to the diameter (10 cm), the sphere volume is (4/3)π × 125 ≈ 523.6 cm³. The cylinder volume is π × 25 × 10 ≈ 785.4 cm³. The cone volume is (1/3) × 785.4 ≈ 261.8 cm³. The sphere is exactly two-thirds of the cylinder's volume, and the cone is one-third. This 3:2:1 ratio (cylinder : sphere : cone) is a classic geometric result.

Which Shape Holds the Most?

For a given surface area, a sphere encloses the maximum volume of any 3D shape. This principle, known as the isoperimetric inequality in three dimensions, explains why bubbles form spheres and why spherical storage tanks are material-efficient. A sphere with surface area 100 m² has a volume of about 94 m³, while a cube with the same surface area has a volume of only about 68 m³.

Units Consistency

Always use consistent units. If dimensions are in centimeters, volume will be in cubic centimeters. Mixing units produces incorrect results. Convert all inputs to the same unit system before entering them.

Using π

For most practical purposes, using π to 3.14 or 3.14159 provides sufficient precision. For very precise scientific work, use the full π value or a calculator. [nctm] A practical rule: for three significant figures, 3.14 is enough; for four significant figures, use 3.142; for five, use 3.1416.

Partial Volumes

Sometimes you need to calculate partial volumes, such as how much liquid is in a partially filled tank. In such cases, use the appropriate portion of the formula. For a half-filled horizontal cylindrical tank, multiply the full volume by the fill fraction. For spherical tanks, the partial volume depends on the fill depth and requires segment volume formulas.

Converting Units

Volume conversions involve cubed units. Converting from cubic centimeters to cubic meters requires dividing by 1,000,000 (not 1,000). [crc-press] Common conversions: 1 m³ = 1,000 L, 1 L = 1,000 cm³ = 0.001 m³, 1 gallon (US) ≈ 3.785 L, 1 cubic foot ≈ 28.317 L, 1 cubic inch ≈ 16.387 cm³.

Significant Figures

Report volume results with an appropriate number of significant figures. If input dimensions are given to two significant figures, the calculated volume should not have more than two significant figures. For example, dimensions 4.0 cm and 6.0 cm each have two significant figures, so the volume 301.59289 cm³ should be rounded to 300 cm³ (two significant figures).

Complex Shapes

This calculator handles only basic geometric shapes. Irregular objects require water displacement methods or 3D scanning technology. For composite objects made of multiple basic shapes, calculate each portion separately and sum the results.

Units

The calculator does not convert between unit systems. Convert all inputs to consistent units before calculation. Always double-check that you have not accidentally mixed metric and imperial units.

Maximum Values

Extremely large or very small numbers may cause display issues. For very large volumes, consider using scientific notation. Volumes exceeding 10&sup9; cubic units may lose precision in display due to floating-point limitations.

Approximation for Real Objects

Real-world objects rarely match perfect geometric shapes. A storage tank might have rounded corners, slight curvature, or internal structures that reduce usable volume. The calculated volume represents an idealized value; actual usable volume may be 5-15% less depending on the object's design.