What is the formula for volume?

There is no single formula — each solid has its own. A cube is V = s cubed, a box is V = length x width x height, a sphere is V = (4/3) x pi x radius cubed, a cylinder is V = pi x radius squared x height, and a cone is one-third of that.

How do you calculate the volume of a cylinder?

Multiply the area of the circular base by the height: V = pi x radius squared x height. For a cylinder with radius 3 and height 10, V = pi x 9 x 10, which is about 282.7 cubic units.

What is the difference between volume and surface area?

Volume measures the space inside a solid and is given in cubic units. Surface area measures the total area of its outer faces and is given in square units. This calculator returns both at once.

How do I convert cubic feet to gallons or liters?

One cubic foot equals about 7.481 US gallons or about 28.317 liters. To convert cubic feet to gallons, multiply by 7.481; to convert to liters, multiply by 28.317.

How do you find the volume of an irregular shape?

Use water displacement. Submerge the object in a measured container of water and record how much the water rises. The increase in water volume equals the object's volume — the method Archimedes is famous for.

What units is volume measured in?

Volume is measured in cubic units such as cubic centimeters, cubic meters, cubic inches, or cubic feet. Capacity for liquids is often given in milliliters, liters, or gallons instead.

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

A cone with the same base radius and height as a cylinder fills exactly one-third of it. Three identical cones of liquid fill the matching cylinder completely, which is why the cone formula has the factor of one-third.

Volume & Surface Area Calculator

Formula

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Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

This free volume calculator finds the volume and surface area of a cube, sphere, cylinder, cone, rectangular prism, triangular prism, and more — with 3D diagrams and a full step-by-step solution for your own numbers. Pick a shape, type in your measurements, and read the answer instantly. No sign-up, no ads in the way, and every result shows the working so you can put it straight onto your homework. It is one of the free math calculators on MathInSite.

What Is Volume?

Volume is the amount of three-dimensional space a solid object occupies, measured in cubic units. If you imagine filling a shape with tiny unit cubes, the volume is simply how many of those cubes fit inside. A box that is 2 units long, 2 wide, and 2 tall holds 8 unit cubes, so its volume is 8 cubic units.

Volume is easy to confuse with two related ideas. Area is a two-dimensional measure — the space inside a flat shape like a square or circle — and it is given in square units; you can explore it with the area and perimeter of 2-D shapes tool. Surface area is the total area of all the outer faces of a 3-D solid, also in square units. Volume alone tells you how much a shape can hold; surface area tells you how much material wraps around it.

How to Calculate Volume (Formulas by Shape)

Every solid has its own volume formula, but they all describe the same thing: how much space is inside. Below are the formulas this calculator uses, each with a quick plain-English description and a worked number you can check. For any shape that uses a circle — the sphere, cylinder, and cone — you can find the radius, diameter, and circumference first with the circle calculator.

Cube — V = s³

A cube has six equal square faces, so you cube the side length s. A cube with side 5 gives V = 5³ = 125 cubic units.

V = s³

Rectangular Prism (Box) — V = l × w × h

Multiply length by width by height. A box 4 by 3 by 2 gives V = 4 × 3 × 2 = 24 cubic units. This is the most common everyday shape — rooms, tanks, and shipping boxes.

V = l × w × h

Sphere — V = (4/3)πr³

A perfectly round ball uses the radius r. A sphere with radius 3 gives V = (4/3)π(3³) = (4/3)π(27) ≈ 113.10 cubic units.

V = (4/3)πr³

Cylinder — V = πr²h

A cylinder is a circle pushed straight up, so multiply the circular base area πr² by the height. Radius 3, height 10 gives V = π(9)(10) ≈ 282.74 cubic units.

V = πr²h

Cone — V = (1/3)πr²h

A cone fills exactly one-third of the cylinder that surrounds it. Radius 3, height 4 gives V = (1/3)π(9)(4) ≈ 37.70 cubic units. Its slant height uses the Pythagorean theorem: l = √(r² + h²).

V = (1/3)πr²h

Triangular Prism — V = ½ × b × h × l

Find the triangular end's area (½ × base × height), then multiply by the prism length. With base 6, height 4, length 10: V = ½ × 6 × 4 × 10 = 120 cubic units.

V = ½ × b × h × l

Square Pyramid — V = (1/3) × b² × h

Like the cone, a pyramid is one-third of its enclosing box. With base side 6 and height 9: V = (1/3)(6²)(9) = (1/3)(36)(9) = 108 cubic units.

V = (1/3) × b² × h

Hemisphere — V = (2/3)πr³

A hemisphere is half a sphere, so it is two-thirds of the cylinder that bounds it. Radius 3 gives V = (2/3)π(27) ≈ 56.55 cubic units — exactly half the full sphere above.

V = (2/3)πr³

Solid	Volume formula	Surface area formula
Cube	V = s³	SA = 6s²
Rectangular 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
Triangular prism	V = ½ × b × h × l	SA = (b × h) + (perimeter × l)
Square pyramid	V = (1/3) × b² × h	SA = b² + 2b × (slant height)
Hemisphere	V = (2/3)πr³	SA = 3πr²

Tip: In the cone and pyramid surface-area formulas, l is the slant height (the distance up the sloping face), not the vertical height. For a cone, l = √(r² + h²).

Where the Formulas Come From (Why a Cone Is ⅓ of a Cylinder)

The formulas are not arbitrary — they follow from one simple idea: how a curved or pointed solid compares to the box or cylinder around it. Understanding this makes the formulas easy to remember.

A cone fills exactly one-third of the cylinder that has the same base radius and height. If you fill a cone with water and pour it into the matching cylinder, it takes three full cones to fill the cylinder — so the cone is ⅓πr²h. The same is true for a pyramid inside its enclosing box, which is why both formulas carry the factor ⅓.

The sphere fills two-thirds of its bounding cylinder — the cylinder that just contains it, with height equal to the diameter. Archimedes proved this over 2,000 years ago and was so proud of it that the result was carved on his tomb. From that ⅔ relationship the familiar V = (4/3)πr³ falls right out.

Volume vs Surface Area (and the Surface-Area-to-Volume Ratio)

This calculator returns both volume and surface area for every shape. Volume (cubic units) is the space inside; surface area (square units) is the skin around it. They answer different questions — how much it holds versus how much it is exposed.

The surface-area-to-volume ratio (SA:V) is the surface area divided by the volume, and it shrinks as an object gets larger. A small object has a high ratio — lots of surface for its size — which is why ice cubes melt fast, small animals lose heat quickly, and biological cells stay tiny so nutrients can cross their membranes fast enough. Engineers use the same idea when designing radiators, packaging, and cooling fins.

Volume Units & Conversions

Volume is reported in cubic units — cm³, m³, in³, ft³, yd³ — while the capacity of liquids is usually given in milliliters, liters, or gallons. Because this tool is unit-agnostic, just enter every dimension in the same unit and the answer comes out in that unit cubed. Use this table to convert the result into the unit you need.

Convert	Equals
1 cubic foot (ft³)	1,728 cubic inches (in³)
1 cubic yard (yd³)	27 cubic feet (ft³)
1 cubic meter (m³)	1,000 liters (L)
1 liter (L)	1,000 cubic centimeters (cm³)
1 US gallon	231 cubic inches (in³)
1 cubic foot (ft³)	≈ 7.481 US gallons

How to Find the Volume of an Irregular Object

For an object with no neat formula — a rock, a key, a sculpture — use water displacement. Fill a measuring container with a known amount of water, submerge the object completely, and read the new water level. The rise in water volume equals the object's volume, because the object pushes aside exactly its own volume of water. This is the displacement method Archimedes discovered, and 1 mL of water rise equals 1 cm³ of object.

Real-World Uses of Volume Calculations

Pools & tanks: work out how many liters or gallons a swimming pool, water tank, or fuel drum holds.
Aquariums: size the heater, filter, and fish stocking from the tank's volume.
Concrete & soil: order the right number of cubic yards for a slab, footing, or raised garden bed.
Shipping & cargo: compute box and container volume to plan loads and dimensional shipping weight.
Cooking & baking: scale recipes and pick the right pan by its capacity.

How to Use This Volume Calculator

Pick a shape. Choose the cube, sphere, cylinder, cone, prism, or pyramid you are measuring — the matching 3D diagram and formula appear automatically.
Enter your dimensions in one consistent unit. Type the side, radius, height, or length. All dimensions must share the same unit (all cm, or all inches) for the answer to be correct.
Press Calculate. Read the volume and surface area instantly, or load a preset example to see how it works first.
Open the step-by-step solution. Expand the Steps tab to see the formula filled in with your numbers, plus a sense-check explaining why the answer makes sense.

Common Mistakes

Mistake — mixing unitsEntering one dimension in inches and another in feet gives a meaningless answer. Convert everything to a single unit before you calculate.

Mistake — forgetting the ⅓ on cones and pyramidsA cone is not πr²h — that is the full cylinder. Multiply by one-third to get (1/3)πr²h, and do the same for pyramids.

Mistake — using diameter instead of radiusSphere, cylinder, and cone formulas need the radius. If you measured across the full width (the diameter), halve it before substituting.

Mistake — reporting volume in square unitsVolume is always in cubic units (cm³, ft³). Square units belong to area and surface area. Cube the unit, not square it.

Mistake — using slant height as vertical heightA cone's volume uses the vertical height h, not the slant height l along the side. They are different: l = √(r² + h²).

Frequently Asked Questions

What is the formula for volume?

There is no single formula — each solid has its own. A cube is V = s³, a box is V = l × w × h, a sphere is V = (4/3)πr³, a cylinder is V = πr²h, and a cone is one-third of that, V = (1/3)πr²h.
How do you calculate the volume of a cylinder?

Multiply the area of the circular base by the height: V = πr²h. For a cylinder with radius 3 and height 10, V = π × 9 × 10 ≈ 282.7 cubic units.
What is the difference between volume and surface area?

Volume measures the space inside a solid and is given in cubic units. Surface area measures the total area of its outer faces and is given in square units. This calculator returns both at once.
How do I convert cubic feet to gallons or liters?

One cubic foot equals about 7.481 US gallons or about 28.317 liters. To convert cubic feet to gallons, multiply by 7.481; to convert to liters, multiply by 28.317.
How do you find the volume of an irregular shape?

Use water displacement. Submerge the object in a measured container of water and record how much the water rises. The increase in water volume equals the object's volume — the method Archimedes is famous for.
What units is volume measured in?

Volume is measured in cubic units such as cubic centimeters, cubic meters, cubic inches, or cubic feet. Capacity for liquids is often given in milliliters, liters, or gallons instead.
Why is the volume of a cone one-third of a cylinder?

A cone with the same base radius and height as a cylinder fills exactly one-third of it. Three identical cones of liquid fill the matching cylinder completely, which is why the cone formula has the factor of one-third.