What is the relationship between volume and capacity?

Volume is the 3D space a shape occupies (in cubic units like m^3 or ft^3). Capacity is the amount it holds, often in fluid units (liters, gallons). They are the same measurement: 1 liter = 1 cubic decimeter (dm^3) = 1,000 cm^3. 1 US gallon = 231 in^3 = 3.785 liters.

Why is a cone volume one-third of a cylinder?

Archimedes proved this geometrically: any cone fits exactly one-third of the volume of its enclosing cylinder (same base and height). Similarly, a sphere fits exactly two-thirds of its enclosing cylinder. These elegant 1/3 and 2/3 ratios arise from calculus integration.

How do I convert cubic feet to gallons?

1 cubic foot = 7.4805 US gallons. So divide cubic feet by 7.4805 to get US gallons, or multiply by 7.4805 to convert gallons to ft^3. For liters: 1 ft^3 = 28.317 liters.

What is displacement in engines?

Engine displacement is the total volume swept by all pistons (cylinder volume x number of cylinders). A 2.0L engine has cylinders with combined swept volume of 2,000 cm^3 = 2 liters. Larger displacement generally means more power potential.

What is the difference between volume and capacity?

Volume refers to the space a solid occupies; capacity refers to how much a container can hold. They often have the same numeric value but different contexts.

How do I calculate the volume of an irregular shape?

Use water displacement (Archimedes principle): submerge the object and measure how much water is displaced. That volume equals the object's volume.

What units should I use for volume?

Use cubic versions of your length unit. Feet gives cubic feet (ft3), meters gives cubic meters (m3). 1 m3 equals 1,000 liters.

Volume Calculator — Calculover

Volume & Surface Area Formulas

Shape	Volume	Surface Area
Cube	`V = s³`	`SA = 6s²`
Rect. Prism	`V = l×w×h`	`SA = 2(lw+lh+wh)`
Sphere	`V = (4/3)πr³`	`SA = 4πr²`
Cylinder	`V = πr²h`	`SA = 2πr²+2πrh`
Cone	`V = (1/3)πr²h`	`SA = πr²+πrl, l=√(r²+h²)`
Pyramid	`V = (1/3)×base×h`	`SA = base + ½×perimeter×slant`
Torus	`V = 2π²Rr²`	`SA = 4π²Rr`
Capsule	`V = πr²(4r/3+h)`	`SA = 2πr(2r+h)`

Shape

Volume

Surface Area

Cube

V = s³

SA = 6s²

Rect. Prism

V = l×w×h

SA = 2(lw+lh+wh)

Sphere

V = (4/3)πr³

SA = 4πr²

Cylinder

V = πr²h

SA = 2πr²+2πrh

Cone

V = (1/3)πr²h

SA = πr²+πrl, l=√(r²+h²)

Pyramid

V = (1/3)×base×h

SA = base + ½×perimeter×slant

Torus

V = 2π²Rr²

SA = 4π²Rr

Capsule

V = πr²(4r/3+h)

SA = 2πr(2r+h)

How to Use This Calculator

Select the Shape

Choose from box (rectangular prism), sphere, cylinder, or cone.

Enter the Dimensions

Provide the required measurements: length/width/height for a box, radius for a sphere, radius and height for a cylinder or cone.

Read the Volume

The calculated volume is displayed with unit-cubed notation and a step-by-step solution.

Formula & Methodology

Box Volume

V = l × w × h

Multiply length, width, and height to get the volume of a rectangular prism.

Sphere Volume

V = (4/3)πr³

Four-thirds pi times the radius cubed gives the volume of a sphere.

Cylinder Volume

V = πr²h

Pi times the radius squared times the height gives the volume of a cylinder.

Shape	Formula	Variables
Box	l × w × h	length, width, height
Sphere	(4/3)πr³	radius
Cylinder	πr²h	radius, height
Cone	(1/3)πr²h	radius, height
Triangular Prism	½bh × l	base, height, length

Shape

Formula

Variables

Box

l × w × h

length, width, height

Sphere

(4/3)πr³

radius

Cylinder

πr²h

radius, height

Cone

(1/3)πr²h

radius, height

Triangular Prism

½bh × l

base, height, length

Key Terms

Volume

The amount of three-dimensional space enclosed by a solid shape, measured in cubic units.

Rectangular Prism

A box-shaped solid with six rectangular faces; also called a cuboid.

Sphere

A perfectly round solid where every point on the surface is equidistant from the center.

Cylinder

A solid with two parallel circular bases connected by a curved surface.

Cone

A solid with a circular base that tapers to a single point (apex); its volume is one-third of the corresponding cylinder.

Volume Calculations in Design and Engineering

Why Volume Matters

Volume calculations are critical in shipping (box dimensions determine cost), manufacturing (material volume drives raw-material estimates), civil engineering (concrete pours require precise cubic-yard quantities), and medicine (dosage calculations may depend on body volume). Accurate volume computation prevents waste, reduces costs, and ensures structural integrity.

The Cone-Cylinder-Sphere Relationship

A cone, a cylinder, and a sphere all with the same radius and height (where the sphere's diameter equals the height) have a beautiful volume ratio of 1 : 3 : 2. Archimedes considered this relationship one of his greatest discoveries. It means a cone is exactly one-third the volume of the cylinder that encloses it, and a sphere is exactly two-thirds of that cylinder.

Volume Calculator

Volume & Surface Area Formulas

How to Use This Calculator

Select the Shape

Enter the Dimensions

Read the Volume

Formula & Methodology

Box Volume

Sphere Volume

Cylinder Volume

Volume Formulas for Common Solids

Key Terms

Real-World Examples

Volume Calculations in Design and Engineering

Why Volume Matters

The Cone-Cylinder-Sphere Relationship

Volume Calculator

Volume & Surface Area Formulas

How to Use This Calculator

Select the Shape

Enter the Dimensions

Read the Volume

Formula & Methodology

Box Volume

Sphere Volume

Cylinder Volume

Volume Formulas for Common Solids

Key Terms

Real-World Examples

Volume Calculations in Design and Engineering

Why Volume Matters

The Cone-Cylinder-Sphere Relationship

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