Volume and Surface Area Calculator
- Free
- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This volume calculator finds the volume and surface area of seven solids: cube, rectangular prism (box), cylinder, cone, sphere, hemisphere and rectangular pyramid. Pick the solid, enter its dimensions, and you get the volume, the total surface area and, for cylinders and cones, the curved surface area. Answers involving \( \pi \) are shown exactly (for example \( 45\pi \)) and as decimals.
It’s free, runs in your browser and needs no sign-up.
How to use the volume calculator
- Choose the solid. The menu says which letters it uses: for a cylinder, a is the radius and h is the height.
- Enter the dimensions in one unit. Boxes the solid doesn’t need are ignored.
- Press Calculate. Volume comes out in cubic units (cm³, m³) and surface area in square units (cm², m²).
Volume and surface area formulas
| Solid | Volume | Surface area |
|---|---|---|
| Cube (edge a) | \( a^3 \) | \( 6a^2 \) |
| Rectangular prism | \( lwh \) | \( 2(lw + lh + wh) \) |
| Cylinder | \( \pi r^2h \) | \( 2\pi r^2 + 2\pi rh \) |
| Cone | \( \frac13\pi r^2h \) | \( \pi r^2 + \pi rl \), \( l = \sqrt{r^2 + h^2} \) |
| Sphere | \( \frac43\pi r^3 \) | \( 4\pi r^2 \) |
| Hemisphere | \( \frac23\pi r^3 \) | \( 3\pi r^2 \) (with the flat base) |
| Rectangular pyramid | \( \frac13lwh \) | base + four triangular faces |
The volume formulas guide explains where each one comes from.
Worked example: a cylinder
A can has radius 3 cm and height 5 cm (as loaded above).
$$V = \pi(3)^2(5) = 45\pi \approx 141.37 \text{ cm}^3$$
$$SA = 2\pi(3)^2 + 2\pi(3)(5) = 18\pi + 30\pi = 48\pi \approx 150.80 \text{ cm}^2$$
The \( 30\pi \) part is the label wrapped around the side; the \( 18\pi \) is the top and bottom.
Worked example: a cone
A cone with radius 3 and height 4 has slant height \( l = \sqrt{9 + 16} = 5 \), so
$$V = \tfrac13\pi(9)(4) = 12\pi \approx 37.70, \qquad SA = \pi(3)^2 + \pi(3)(5) = 24\pi \approx 75.40$$
A cone holds exactly one third of the cylinder with the same base and height.
Worked example: a sphere
A ball of radius 6 cm has volume \( \frac43\pi(6)^3 = 288\pi \approx 904.78 \) cm³ and surface area \( 4\pi(6)^2 = 144\pi \approx 452.39 \) cm².
Tips and common mistakes
- Radius, not diameter. Halve the diameter first.
- Height vs slant height. For cones and pyramids, h is the vertical height; the calculator works out the slant height for the surface area.
- Units. 1 litre \( = 1000 \) cm³, and 1 m³ \( = 1000 \) litres.
- Scaling. Doubling every dimension multiplies the volume by 8 and the surface area by 4.
Where the formulas come from
The cone and sphere formulas come from calculus: slice the solid into thin discs and add up their volumes with an integral. That’s the disk and washer method, and the disk method calculator does it for any curve. For hollow solids, see the shell method.
Related guides
Further reading
- Volume (Wikipedia) — definitions, units and formulas for common solids.
- Surface area (Wikipedia) — surface area formulas and their derivations.
FAQ
How do you find the volume of a cylinder?
\( V = \pi r^2h \): the area of the circular base times the height.
What is the volume of a cone?
\( V = \frac13\pi r^2h \), one third of the matching cylinder.
What is the formula for the volume of a sphere?
\( V = \frac43\pi r^3 \).
How do you convert cubic centimetres to litres?
Divide by 1000: 1000 cm³ is 1 litre.