Volume and Surface Area Calculator
Volume and surface area of ten solids, each one drawn and each formula shown with your numbers in it.
-
Volume 60
V = w × d × h
V = 3 × 4 × 5
-
Surface area 94
S = 2 × (wd + wh + dh)
S = 2 × (34 + 35 + 45)
-
Diagonal through the middle 7.071068
D = √(w² + d² + h²)
D = √(3² + 4² + 5²)
Pick a solid, give it its measurements, and get the volume, the surface area and whatever else those measurements determine — a slant height, a diagonal, the area of the sides without the ends.
Every answer is drawn, and every formula is shown twice: as it is written, and with your numbers in it. Nothing is a figure to be taken on trust.
How it works
The drawing is an isometric projection, which is worth knowing because it decides what the picture does and does not tell you. Lengths along all three axes keep their ratios exactly, so a box that is twice as tall as it is wide is drawn that way, and a cube of 1 and a cube of 1000 are the same picture. Angles are not kept: the right angle at the corner of a box comes out as 120 or 60 degrees, and a circle lying flat is drawn as an ellipse. Those are the projection rather than the solid.
Two measurements are easy to confuse and both are asked for by the name that tells them apart. The height of a cone is straight up from the middle of its base; the slant is the distance up the outside, and it is the slant the side area needs. Using the height there is the commonest mistake in a cone.
The truncated cone does not refuse equal radii. The formulas reduce to a cylinder on their own, which is the right answer rather than a case to be turned away.
Measurements have no unit. Give centimetres and the volume is in cubic centimetres and the surface in square centimetres.
Examples
| Case | Input | Result |
|---|---|---|
| A sphere | Sphere, radius 5 | Volume 523.5988, surface 314.1593 |
| A cylinder | Cylinder, radius 3, height 10 | Volume 282.7433, surface 245.0442 |
Frequently asked questions
Is the surface of an ellipsoid exact?
No, and it is the only figure on this page that is not. There is no formula for it in elementary functions. What is shown is Knud Thomsen’s approximation, which is fitted rather than derived. Measured against the exact formula for the ellipsoids that have one, it is out by about 0.06 per cent at two to one, 0.75 per cent at five to one, and never worse than 1.06 per cent however far it is stretched. That is far looser than the ellipse perimeter on the area page, and the two should not be read as equally trustworthy. The volume beside it is exact.
Why is the side area given separately?
Because it is usually the one you want. Wrapping a label around a tin, painting a silo, cutting a sleeve: none of those involves the ends. The total is given too, so neither has to be worked out from the other.
Height or slant height?
Height, and the slant comes back as an answer. The height is straight up from the middle of the base; the slant is up the outside. For a square pyramid there are two slants and both are given: up the middle of a face, which a net needs, and up a corner edge, which a frame needs.
Why does my solid look like a line?
Because it is one. The drawing keeps the ratios of all three dimensions, so a box of 1 by 1 by 10000 is drawn as a line. Nothing is widened to make it look more like a box, and the page says so when it happens.
What can the drawing not tell me?
Angles, and anything that depends on them. An isometric projection draws a right angle as 120 or 60 degrees and a flat circle as an ellipse. Measuring the picture with a protractor gives the projection, not the solid. Lengths along the three axes are the part that is faithful.
Good to know
- Every formula here is exact except the surface of an ellipsoid, which has no exact form and is marked as an approximation wherever it appears.
- The drawing keeps the ratios of the three dimensions and nothing else. It shows shape and proportion, never scale, and never an angle you could measure.