Volume Calculator

About the Volume Calculator

Volume measures how much space a solid takes up: how much water a tank holds, how much concrete a foundation needs or how much air is in a balloon. Surface area measures the outside, which matters for painting, wrapping, heat loss and material cost. Most real problems need one or both.

This calculator covers the solids in standard geometry courses. It keeps π symbolic when the dimensions are exact, so a sphere of radius 3 gives a volume of exactly 36π, and it shows the slant height step for cones and pyramids, which is where surface area problems most often go wrong.

What are volume and surface area?

Volume is the number of unit cubes that fit inside a solid, measured in cubic units such as cm³ or m³. For a prism or cylinder, it is the area of the base times the height. Pointed solids such as cones and pyramids hold exactly one third of the prism or cylinder with the same base and height.

Surface area is the total area of all the faces or curved surfaces, measured in square units. For a closed solid it includes every face, so a closed cylinder has two circular ends plus the curved side, which unrolls into a rectangle of width $2\pi r$ and height $h$ .

How to use the Volume Calculator

Choose the solid from the menu.
Enter the dimensions that appear, in a single unit.
Use the perpendicular height for cones, cylinders and pyramids.
Press Calculate volume.
Read the volume in cubic units and the surface area in square units.

Formulas

Cube and rectangular prism

V = s^3,\quad V = lwh

Cylinder

V = \pi r^2 h,\quad S = 2\pi r^2 + 2\pi r h

Cone

V = \tfrac13 \pi r^2 h,\quad S = \pi r^2 + \pi r l,\ l = \sqrt{r^2 + h^2}

Sphere

V = \tfrac43 \pi r^3,\quad S = 4\pi r^2

Square pyramid

V = \tfrac13 s^2 h,\quad S = s^2 + 2sl

Worked example: Box 2 × 3 × 4

The worked example is a rectangular box 2 by 3 by 4. Its volume is $2 \times 3 \times 4 = 24$ cubic units, and its surface area is $2(6 + 8 + 12) = 52$ square units.

A cone with radius 3 and height 4 has volume $\frac13 \pi \times 9 \times 4 = 12\pi \approx 37.699$ . Its slant height is $\sqrt{9 + 16} = 5$ , so its surface area is $9\pi + 15\pi = 24\pi \approx 75.398$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

V = 24

How we read your problem

Compute the volume (cubic units) and total surface area (square units) of the solid from its dimensions in one length unit.

V = lwh

Step-by-step solution

Volume
$V = 2 \cdot 3 \cdot 4 = 24$
Surface area: three pairs of rectangles
$S = 2(6 + 8 + 12) = 52$

Formulas used

Volume
$V = lwh$
Surface area
$S = 2(lw + lh + wh)$

Volume counts unit cubes inside a solid; surface area counts unit squares needed to wrap it. Doubling every length multiplies volume by 8 and surface area by 4.

Verification

Volume 24 and surface area 52 are positive.

Detailed explanation

Why cones and pyramids have one third

A cone fits inside a cylinder with the same base and height, and it holds exactly one third as much. The same is true for a pyramid inside a prism. This can be shown with calculus, or demonstrated by pouring: three cones of water fill the matching cylinder.

So a square pyramid with base edge 6 and height 4 has volume $\frac13 \times 36 \times 4 = 48$ , one third of the 144 that a box of the same base and height would hold.

Slant height for surface area

The sloping surfaces of a cone or pyramid are measured along the slant height, not the vertical height. For a cone, the slant height is the hypotenuse of a right triangle with legs r and h. For a square pyramid, each triangular face has height $\sqrt{h^2 + (s/2)^2}$ .

With base edge 6 and height 4, the slant height is $\sqrt{16 + 9} = 5$ , so each triangular face has area $\frac12 \times 6 \times 5 = 15$ , and the total surface area is $36 + 4 \times 15 = 96$ .

Spheres and hemispheres

A sphere of radius r has volume $\frac43 \pi r^3$ and surface area $4\pi r^2$ . For $r = 3$ both happen to equal $36\pi$ , though they have different units; this coincidence occurs only at radius 3.

A solid hemisphere has half the sphere's volume, $\frac23 \pi r^3$ , but its surface includes the flat circular face as well as half the curved surface, giving $2\pi r^2 + \pi r^2 = 3\pi r^2$ .

Scaling and unit conversions

When every length of a solid is multiplied by a factor k, its surface area is multiplied by $k^2$ and its volume by $k^3$ . A ball with twice the radius needs four times as much leather but holds eight times as much air. This is why large animals and large tanks lose heat more slowly relative to their size.

The same rule governs conversions. One metre is 100 cm, so one cubic metre is $100^3 = 1{,}000{,}000$ cm³, or 1000 litres. Convert lengths before cubing rather than after, and the units will take care of themselves.

Common mistakes

Using the slant height instead of the vertical height in a volume formula.
Forgetting the ⅓ for cones and pyramids.
Using the diameter as the radius.
Leaving out the flat base when finding the surface area of a closed cone or hemisphere.
Giving volume in square units or surface area in cubic units.

Tips

Volume of any prism or cylinder is base area times height.
One litre is 1000 cm³, and one cubic metre is 1000 litres.
Doubling every length multiplies volume by eight.
Use the Pythagorean theorem calculator to find a slant height exactly.
Use the area calculator to find the base area of an unusual prism first.

Frequently asked questions

How do you find the volume of a cylinder?

Multiply the area of the circular base by the height: V = πr²h. A cylinder with radius 2 and height 5 holds 20π ≈ 62.83 cubic units.

What is the formula for the volume of a cone?

V = ⅓πr²h, one third of the cylinder with the same base and height.

How do I calculate the volume of a sphere?

Use V = (4/3)πr³. A sphere with radius 3 has a volume of 36π ≈ 113.1 cubic units.

What is the difference between volume and surface area?

Volume measures the space inside a solid in cubic units; surface area measures the total area of its outside in square units.

How do I find the slant height of a cone?

Use the Pythagorean theorem with the radius and the vertical height: l = √(r² + h²).

How many litres are in a cubic metre?

There are 1000 litres in one cubic metre, and one litre equals 1000 cubic centimetres.

Does the calculator include the base in the surface area?

Yes. It gives the total surface area of the closed solid, including every flat base. Subtract a base if your problem describes an open container.

Calculate a volume

Choose a solid above and enter its dimensions to get the volume and surface area with every step.

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