Toolbox
A formula sheet plus reference tools for calculator, equation solver, units, elements, graphs and molecules.
Solid geometry calculator
Cuboid, cube, cylinder, cone, sphere, regular prism and pyramid: base, lateral and total surface area plus volume, with an isometric sketch.
Pick a shape and fill in only the sizes it needs; the base of a regular prism or pyramid has 3 to 24 sides.
Isometric sketch (dashed edges are hidden)
- Space diagonal
- 5.3852 cm
- Faces
- 6
- 1Formula
Write down the formulas for volume and surface area.
- 2Substitute
Compute the base and lateral areas separately.
- 3Solve
Base times height gives the volume; base plus lateral gives the surface area.
- 4Conclusion
Round at the end and remember that volume uses cubic units.
Concept
Surface area adds up every face while the lateral area counts only the sides. Unrolling a cylinder gives a rectangle and a cone gives a sector, which is where the formulas come from.
Volumes share one idea: a prism is base times height, a cone or pyramid is a third of the matching prism, and a sphere is remembered separately.
Going further
- Slant height of a cone
- The radius, the height and the slant height form a right triangle with the slant as hypotenuse, so it equals the square root of r squared plus h squared.
Common mistakes
- Convert to one unit first; mixing centimetres and metres is off by a factor of a hundred.
- Sanity-check with an estimate: a cone with r = 3 and h = 4 holds about 37.7, less than a cube of edge 4.