Volume and surface area
Pick a solid, enter the measurements, and volume and surface area appear straight away.
Measurements
Results
Fill in every measurement.
How it works
Volume is measured in cubic units and area in square units. The tool works without units, so the answer follows whatever unit you entered.
The volume of a sphere is four thirds pi times the radius cubed, and the surface area is four pi times the radius squared. That the two happen to match at a radius of three is a coincidence in the numbers, not a rule.
A cone has exactly one third the volume of a cylinder with the same radius and height. The same ratio holds between a pyramid and a cuboid.
The slant height of a cone is the distance from the tip out to the edge along the surface, that is the hypotenuse of a right triangle with the radius and the height as its legs.
A torus is measured by its major radius, from the centre to the middle of the tube, and the radius of the tube itself. The outer radius is the two added together. The volume is two pi squared times both radii, with the tube radius squared. The tube radius cannot be larger than the major radius, because the tube would then pass through itself.
The formulas for the other solids
A cube has the volume s³, the surface area 6s² and the space diagonal s√3. A cuboid has the volume l × w × h, the surface area 2(lw + lh + wh) and the diagonal √(l² + w² + h²), which runs through the solid from one corner to the opposite one. A cuboid of 2, 3 and 6 has the diagonal √(4 + 9 + 36) = 7. The cylinder’s surface area 2πr(r + h) includes both ends, while the lateral area 2πrh is the side alone. The cone’s surface area πr(r + s) includes its base.
The pyramid has a square base of side b with its apex straight above the centre. Its volume is b² × h / 3. The slant height is √((b / 2)² + h²), measured from the apex to the middle of a base edge, and the surface area is b² + 2 × b × the slant height.
Example: the pyramid
A pyramid with base 6 and height 4 has the slant height √(3² + 4²) = 5. Its volume is 36 × 4 / 3 = 48 and its surface area 36 + 2 × 6 × 5 = 96. The edge from the apex down to a corner is longer, √(3² + 3² + 4²) = √34 ≈ 5.83, so do not measure that if it is the slant height you need.
Litres and scale
The SI Brochure defines the litre as a cubic decimetre, 10⁻³ m³, and says that a prefix is raised to the power together with its unit: 2.3 cm³ is 2.3 × 10⁻⁶ m³. A cylinder with a radius of 5 cm and a height of 10 cm has a volume of 785.39816 cm³, or about 0.785 litres, since a litre is 1,000 cm³.
For the same reason volume grows with the cube of the scale. Double every measurement, to a radius of 10 and a height of 20, and the volume becomes eight times as large and the surface area four times. A model at 1:10 scale holds a thousandth of the original.