3D Volume Calculator

About 3D Volume Calculator

Volume and Surface Area Formulas

Each 3D shape has a specific formula for both volume (the space it occupies) and surface area (the total area covering its outer faces). Here is every formula this calculator uses:

Shape	Volume	Surface Area
Sphere	(4/3)πr³	4πr²
Cylinder	πr²h	2πr² + 2πrh
Cone	(1/3)πr²h	πr² + πr√(r² + h²)
Cube	s³	6s²
Rectangular prism	lwh	2(lw + lh + wh)
Pyramid	(1/3)lwh	lw + l√((w/2)² + h²) + w√((l/2)² + h²)

The cone and pyramid both carry a factor of 1/3 compared to their "full" cylinder and prism counterparts. This is not a coincidence - it is a fundamental geometric relationship that holds for any shape tapering to a point from a flat base.

How Does Volume Calculation Work?

Volume measures the three-dimensional space enclosed by a shape, expressed in cubic units (cm³, m³, ft³, etc.). For regular shapes, the volume can be found by multiplying a base area by a height, sometimes with a scaling factor. A cylinder, for example, is just a circle (area = πr²) extended through a height h, giving V = πr²h. A cone is the same base and height but tapers to a point, which is why it holds exactly one-third the volume: V = (1/3)πr²h.

Surface area is the total area of all outer faces. For a cube with side length s, each of its 6 identical square faces has area s², so the total surface area is 6s². For curved shapes like spheres and cylinders, the formulas involve π because the surfaces wrap around circular cross-sections.

Worked Examples with Full Working

Sphere (r = 8 cm):

• Volume = (4/3) x π x 8³ = (4/3) x 3.14159 x 512 = 2,144.66 cm³
• Surface area = 4 x π x 8² = 4 x 3.14159 x 64 = 804.25 cm²

Cylinder (r = 5 cm, h = 12 cm):

• Volume = π x 5² x 12 = 3.14159 x 25 x 12 = 942.48 cm³
• Surface area = 2πr² + 2πrh = 2 x 3.14159 x 25 + 2 x 3.14159 x 5 x 12 = 157.08 + 376.99 = 534.07 cm²

Cone (r = 4 cm, h = 9 cm):

• Volume = (1/3) x π x 4² x 9 = (1/3) x 3.14159 x 16 x 9 = 150.80 cm³
• Slant height l = √(r² + h²) = √(16 + 81) = √97 = 9.849 cm
• Surface area = πr² + πrl = 3.14159 x 16 + 3.14159 x 4 x 9.849 = 50.27 + 123.72 = 173.98 cm²

Rectangular prism (l = 10 cm, w = 6 cm, h = 4 cm):

• Volume = 10 x 6 x 4 = 240 cm³
• Surface area = 2(10x6 + 10x4 + 6x4) = 2(60 + 40 + 24) = 2 x 124 = 248 cm²

Pyramid (l = 6 cm, w = 4 cm, h = 8 cm):

• Volume = (1/3) x 6 x 4 x 8 = 64 cm³
• Slant along length = √(8² + 2²) = √68 = 8.246 cm
• Slant along width = √(8² + 3²) = √73 = 8.544 cm
• Surface area = 6x4 + 6x8.246 + 4x8.544 = 24 + 49.48 + 34.18 = 107.65 cm²

The Archimedes Sphere-Cylinder Relationship

One of the most famous results in mathematics connects the volumes of a sphere, cone, and cylinder. Archimedes proved in his work On the Sphere and Cylinder (around 225 BC) that a sphere inscribed inside a cylinder - where the sphere's diameter equals both the cylinder's diameter and height - occupies exactly 2/3 of the cylinder's volume. He also showed that a cone with the same base and height as the cylinder holds exactly 1/3 of the cylinder's volume. Together: cone + sphere = cylinder.

Shape	Factor	Example (r = 5, h = 10)
Cylinder	1x (baseline)	πr²h = 785.40 cm³
Cone (same base and height)	1/3 of cylinder	(1/3)πr²h = 261.80 cm³
Sphere (diameter = height = 2r)	2/3 of cylinder	(4/3)πr³ = 523.60 cm³

Archimedes considered this his greatest mathematical achievement and reportedly asked for a diagram of a sphere inscribed in a cylinder to be carved on his tombstone. The Roman orator Cicero later found and identified the tomb by that very carving in 75 BC, over a century after Archimedes' death in 212 BC. The proof technique Archimedes used - slicing solids into parallel cross-sections of equal area - anticipated what would later be formalised as Cavalieri's principle in 1635 by the Italian mathematician Bonaventura Cavalieri.

Everyday Volume Reference

Putting volume formulas into context with real objects helps build intuition for how big a given number of cubic centimetres or cubic metres actually is.

Object	Shape	Dimensions	Volume
Tennis ball	Sphere	r = 3.35 cm (ITF spec: 6.54-6.86 cm diameter)	157 cm³
Standard soda can	Cylinder	r = 3.3 cm, h = 12.2 cm	418 cm³ (holds 355 mL)
NBA basketball (Size 7)	Sphere	r = 12.1 cm (29.5 in circumference)	7,424 cm³
20 ft shipping container	Rectangular prism	5.9 x 2.34 x 2.39 m (internal)	33.2 m³ (1,172 ft³)
Olympic swimming pool	Rectangular prism	50 x 25 x 2 m (FINA minimum depth)	2,500 m³ (2.5 million litres)
Earth	Sphere	r = 6,371 km (volumetric mean, per NASA)	1.083 x 10¹² km³

The ITF specifies tennis ball diameter as 6.54 to 6.86 cm. Using the midpoint of 6.70 cm (r = 3.35 cm) gives a volume of about 157 cm³. An NBA Size 7 basketball has a circumference of 29.5 inches (74.93 cm), giving a diameter of roughly 24.2 cm and a volume of about 7,424 cm³ - around 47 times the volume of a tennis ball despite being only 3.6 times wider in diameter. That is the power of cubic scaling: doubling a radius increases volume eightfold.

Unit Conversions for Volume

Volume units scale cubically. One metre is 100 centimetres, but one cubic metre is 1,000,000 cubic centimetres (100³). This catches people out regularly - a box that seems small in metres can hold a surprisingly large number of litres.

From	To	Multiply by
1 cm³	mL	1 (they are equal)
1 cm³	litres	0.001
1 m³	litres	1,000
1 in³	cm³	16.387
1 ft³	litres	28.317
1 US gallon	litres	3.785
1 imperial gallon	litres	4.546
1 m³	ft³	35.315

Note that US gallons and imperial gallons are different. One imperial gallon is about 20% larger than a US gallon. When working with recipes, fuel, or pool volumes, always check which gallon system is being used.

What Is the Difference Between Volume and Capacity?

Volume and capacity are closely related but not identical. Volume is the amount of three-dimensional space an object occupies, measured in cubic units like cm³ or m³. Capacity is the amount of substance (usually liquid) a container can hold, measured in litres, millilitres, or gallons. For practical purposes, 1 cm³ = 1 mL and 1 m³ = 1,000 litres. A glass might have a volume of 300 cm³ (measuring the space inside), which means it has a capacity of 300 mL. The distinction matters mostly in scientific contexts where the wall thickness of a container creates a difference between total external volume and internal capacity.

Common Mistakes When Calculating Volume

• Using diameter instead of radius: The sphere formula uses radius (half the diameter). Plugging in the full diameter gives a volume 8 times too large because the error is cubed.
• Mixing units: If the radius is in centimetres and the height is in metres, the result will be wrong. Convert everything to the same unit before calculating.
• Forgetting the 1/3 factor: Cones and pyramids hold one-third the volume of their cylindrical or prismatic counterparts. Missing this factor triples the answer.
• Confusing surface area with volume: Surface area is measured in square units (cm²) and volume in cubic units (cm³). They answer different questions - how much material to cover something vs. how much space is inside.
• Rounding π too early: Using 3.14 instead of a more precise value of π introduces compounding rounding errors in multi-step calculations. This calculator uses the full-precision Math.PI constant (3.14159265...) to avoid this.

Practical Applications of Volume Formulas

• Shipping and logistics: Package dimensions (l x w x h) determine volumetric weight for shipping. A standard 20 ft shipping container has an internal volume of about 33.2 m³ (1,172 ft³). Carriers calculate charges based on actual weight or volumetric weight, whichever is greater.
• Aquariums and tanks: A 60 x 30 x 40 cm fish tank holds 72,000 cm³ = 72 litres. For cylindrical tanks, the tank volume calculator handles horizontal and vertical orientations with partial fill levels.
• Concrete and construction: A cylindrical pier footing with r = 15 cm and h = 60 cm needs πr²h = 42,412 cm³ = 0.042 m³ of concrete. The concrete calculator converts this directly into bags and cubic yards for ordering.
• Cooking: A round cake tin 20 cm across and 8 cm deep holds π x 10² x 8 = 2,513 cm³ of batter. Scaling a recipe from a square tin (l x w x h) to a round one requires comparing these volumes.
• Swimming pools: An Olympic pool (50 x 25 x 2 m per FINA standards) holds 2,500 m³, which is 2.5 million litres of water.

For 2D shape areas (rectangles, triangles, circles, and more), use the area calculator. For dedicated surface area calculations with additional shape options, see the surface area calculator.

Sources

• Wikipedia - On the Sphere and Cylinder (Archimedes)
• NASA - Earth Fact Sheet
• ITF - Approved Tennis Balls Technical Booklet
• Wikipedia - Olympic-size Swimming Pool (FINA standards)
• Wikipedia - Cavalieri's Principle