Volume Calculator
Pick the solid, enter the measurements and find the volume, the total surface area and how many liters fit inside, with the math shown.
Educational use: results are estimates made for study and reference. Checking the answer and deciding what to do with it is the sole responsibility of whoever uses the tool. Read the legal notice.
On this page
- How to use the volume calculator
- Solved examples
- A beautiful relationship between three solids
- The volume and surface area formulas
- From volume to capacity
- Accuracy, limits and responsibility
- Mistakes that get in the way
- Where volume matters
- A little history
- Questions about volume and capacity
- Words from solid geometry
- Fun facts
- Related tools
How to use the volume calculator
- Select the solid. Each one needs different measurements, and the form adjusts.
- Type the measurements in the same unit. For a cylinder, cone or sphere, use the radius (half the diameter).
- Choose the unit. If you pick meters, decimeters, centimeters or millimeters, the tool also converts the result to liters.
- Calculate. You get the volume, the total surface area and the step-by-step work.
A common question is the difference between volume and capacity. Volume is the space a solid takes up. Capacity is how much fits inside a container. For a tank, the two numbers coincide, and the only difference is the unit we usually talk in: cubic meters for volume and liters for capacity.
Solved examples
1. The water tank
A tank measures 2 m long, 1.5 m wide and 1 m high. How much does it hold?
It holds 3,000 liters when filled to the brim.
2. The cylindrical can
A can has a radius of 4 cm and a height of 12 cm.
A little over 600 milliliters. That is why cans this size are usually sold as "600 mL" or close to it.
3. The ball
A ball has a radius of 11 cm. How much air fits inside?
4. The ice cream cone
A cone with a radius of 3 cm and a height of 10 cm.
About 94 mL, one third of what a cylindrical cup with the same opening and the same height would hold.
5. Square-based pyramid
A base of 6 m × 6 m and a height of 4 m.
A beautiful relationship between three solids
Take a cylinder whose height equals its diameter (in other words, a cylinder that fits snugly around a sphere). Its volume splits in a curious way: the sphere takes up 2/3 and a cone with the same base and height takes up 1/3.
The volume and surface area formulas
| Solid | Volume | Total surface area |
|---|---|---|
| Cube (edge a) | a³ | 6a² |
| Box (l, w, h) | l × w × h | 2(lw + lh + wh) |
| Cylinder (r, h) | π r² h | 2πr (r + h) |
| Cone (r, h) | π r² h ÷ 3 | πr (r + s), with s = √(r² + h²) |
| Sphere (r) | 4/3 π r³ | 4π r² |
| Pyramid (base a × b, height h) | a × b × h ÷ 3 | base + four triangular faces |
These two patterns cover most cases: solids that keep the same cross-section from one end to the other (boxes, cylinders) use base × height. Solids that end in a point (pyramids, cones) use one third of that. The sphere is a case of its own, with the famous 4/3 π r³.
The s for the cone is the slant height: the length of the slanted side, from the edge of the base to the tip. You find s with the Pythagorean theorem, from the radius and the height.
From volume to capacity
| Volume | Equivalent capacity |
|---|---|
| 1 cm³ | 1 milliliter (mL) |
| 1 dm³ (a 10 cm cube) | 1 liter (L) |
| 1,000 cm³ | 1 liter |
| 1 m³ | 1,000 liters |
| 1 mm³ | 0.001 mL |
| 1 cubic foot | about 28.32 liters |
| 1 gallon (US) | about 3.785 liters |
The point that confuses people most: 1 m³ is not 100 liters, it is 1,000. A cube 1 m on each side measures 10 decimeters in every direction, and 10 × 10 × 10 = 1,000 dm³, which is 1,000 liters.
Accuracy, limits and responsibility
The formulas hold for perfect, right-angled solids. The pyramid considered here is a right pyramid, with the apex directly above the center of the rectangular base. If your object is slanted, rounded, tapered or irregular, the real volume may differ from the calculated one.
The value in liters is the theoretical capacity, up to the brim. Real containers have wall thickness, lids, rounded corners and a margin so they do not overflow. For purchases (concrete, sand, water) or projects, add a suitable allowance or consult a professional.
Bottom line: this tool is educational. Results do not replace your own checking, and how you use them is entirely your responsibility. More details in the legal notice.
Mistakes that get in the way
Using the diameter instead of the radius
In the cylinder, cone and sphere formulas, r is the radius. Using the diameter makes the volume come out 4 times too large for the cylinder and the cone, and 8 times too large for the sphere.
Confusing height with slant height
In a pyramid or a cone, the height is the vertical distance from the base to the tip. The slanted side is longer. For the volume, you use the height.
Forgetting the ÷ 3
Cones and pyramids have one third of the volume of the matching prism or cylinder. Forgetting that division triples the result.
Converting cubic units incorrectly
Because there are three dimensions, the conversion factor is cubed. 1 m = 100 cm, but 1 m³ = 1,000,000 cm³ (100³).
Confusing inside and outside volume
A water tank with thick walls has a smaller inside volume than outside volume. For capacity, use the inside measurements.
Where volume matters
- Home and construction: volume of water tanks, pools, concrete for slabs and footings, sand and gravel.
- Kitchen: capacity of baking pans, pots and containers.
- Logistics: fitting cargo and packages into boxes, trucks and shipping containers.
- Aquariums and gardening: amount of water, soil or substrate.
- Physics and chemistry: calculating density (mass ÷ volume) and the concentration of solutions.
- School: solid geometry is a recurring topic on exams and college entrance tests.
A little history
The Egyptians knew how to calculate the volume of truncated pyramids almost four thousand years ago: the Moscow Papyrus contains a problem that, when solved today, matches the formula for the frustum of a square pyramid.
The relationship between the sphere, the cone and the cylinder shown on this page was one of Archimedes' favorite discoveries. According to tradition, he asked for a sphere inscribed in a cylinder to be engraved on his tomb. Later, in the 17th century, Cavalieri's principle, which says that two solids with the same height and the same cross-section areas at every level have the same volume, gave an elegant justification for all of these formulas and paved the way for integral calculus.
Questions about volume and capacity
How do I calculate the volume of a box?
Multiply length, width and height. A box of 2 × 1.5 × 1 m has 3 m³, or 3,000 liters.
How many liters are in 1 cubic meter?
1,000 liters. A cube 1 m on a side measures 10 dm on each edge, and 10 × 10 × 10 = 1,000 dm³, which is 1,000 liters.
How do I calculate the volume of a cylinder?
Multiply the base area (π × radius²) by the height. With a radius of 4 cm and a height of 12 cm, the volume is approximately 603 cm³.
Why does a cone have one third of the volume of a cylinder?
It is a geometric property that can be proved, for example, with Cavalieri's principle. A cone with the same base and height as a cylinder takes up exactly 1/3 of its space.
What is the formula for the volume of a sphere?
V = 4/3 × π × r³. The volume grows with the cube of the radius, so doubling the radius multiplies the volume by 8.
What is the difference between volume and capacity?
Volume is the space a solid takes up. Capacity is how much fits inside a container. Volume is usually measured in m³ and capacity in liters, but 1 dm³ = 1 L.
How do I calculate the volume of a horizontal cylindrical tank?
When it is completely full, the volume is that of the cylinder (π r² × length). If it is partly full, the calculation is more complex and depends on the level, and this tool does not cover that case.
Can I use measurements in different units?
No. Convert everything to the same unit before calculating. If you need to, use the site's unit converter.
How do I convert cm³ to mL?
They are equal: 1 cm³ = 1 mL. And 1,000 cm³ = 1 liter.
Does the calculator's pyramid work for any pyramid?
It works for pyramids with a rectangular base. The volume holds for any pyramid with that base and height, but the surface area assumes a right pyramid.
Words from solid geometry
- Volume
- A measure of the space a solid occupies, in cubic units (m³, cm³).
- Capacity
- The amount that fits in a container, usually in liters or milliliters.
- Edge
- The line where two faces of a solid meet.
- Base
- The face the solid "rests on," used as the reference for the calculation.
- Height
- The perpendicular distance between the base and the opposite face or tip.
- Slant height
- The slanted segment along the side of a cone, from the edge of the base to the apex.
- Total surface area
- The sum of the areas of all the faces of the solid, like the paper needed to wrap it.
- Prism
- A solid with two equal, parallel bases joined by rectangular faces.
Fun facts
- Shape and economy. Of all solids with the same volume, the sphere has the smallest surface area, which is why drops and bubbles are round. For stacking and transport, though, rectangular boxes win.
- Soda cans are cylinders because they stand up well to pressure and use little material, with a height of about twice the diameter.
- Doubling the radius of a sphere multiplies the volume by 8 (2³), not by 2. That is why a basketball holds far more air than two tennis balls.