Geometry
Volume and surface area calculator
Pick the solid, enter the measurements and get volume, total surface and lateral surface, plus the quantities problems usually hide: slant height, base area, diagonal.
Volume, total and lateral surface
Volume and surface answer different questions: volume says how much fits inside — litres of water in a tank, cubic metres of concrete — surface says how much material it takes to cover it. They scale in a way that is not intuitive: double every measurement and the surface is four times larger while the volume is eight times larger, which is why one big container beats many small ones.
The lateral surface is given separately because it is the one needed most often in practice: painting the sides of a silo, cladding a column, sizing the label that wraps a can. For the sphere and the regular tetrahedron it does not appear: there is no base to tell apart from the sides.
For the cone and the pyramid the decisive measurement is the slant height, the sloping line from the apex to the edge of the base. It is not the height and has to be found with Pythagoras: the calculator works it out and shows it, because that is the step where the mistake usually happens.
Common mistakes
- Using the height instead of the slant height for the lateral surface of a cone or pyramid: the slant height is always the larger of the two.
- Forgetting that the total surface of a cylinder includes two bases, not one: the formula is 2πr(r+h).
- Mixing units: a volume in cubic centimetres and a surface in square metres cannot be compared. Bring every measurement to the same unit before calculating.
Frequently asked questions
What is the difference between total and lateral surface?
The lateral surface is the sides without the bases; the total adds the bases. For a cylinder the lateral is 2πrh and the total is 2πr(r+h): the difference is the two circular bases.
How do I get the litres in a cylindrical tank?
Enter the radius and height in decimetres: the volume comes out in cubic decimetres, and one cubic decimetre is exactly one litre. It is the quickest route from measurements to litres with no conversion.
Is the slant height of a pyramid the same as its height?
No. The height runs from the apex to the centre of the base, perpendicular to it; the slant height runs from the apex to the midpoint of a base edge, along a face. The slant height is larger, and it is the one the lateral surface uses.
How this calculation works
Cube: V = l³, S = 6l², diagonal = l√3. Cuboid: V = a·b·c, S = 2(ab+ac+bc). Sphere: V = 4/3·πr³, S = 4πr². Cylinder: V = πr²h, lateral = 2πrh, S = 2πr(r+h). Cone: slant = √(r²+h²), V = πr²h/3, lateral = πr·slant, S = πr(r+slant). Square pyramid: slant = √(h²+(l/2)²), V = l²h/3, lateral = 2·l·slant, S = l² + lateral. Regular tetrahedron: V = l³/(6√2), S = √3·l².