Solid shapes
A three-dimensional (3D) shape has length, width and height. Shapes with only flat faces and straight edges are called polyhedra: cubes, cuboids, prisms and pyramids. Shapes with curved surfaces, such as cylinders, cones and spheres, are not polyhedra. A prism has the same cross-section all along its length, while a pyramid or cone narrows to a point called the apex.
Faces, edges and vertices
A face is a flat surface, an edge is where two faces meet, and avertex (plural vertices) is a corner. In 1750 Leonhard Euler noticed that for every convex polyhedron:
F − E + V = 2
A cube has 6 − 12 + 8 = 2; a tetrahedron 4 − 6 + 4 = 2. This simple rule was one of the first results intopology, the mathematics of shape.
Volume
Volume is the amount of space a shape fills, measured in cubic units such as cm³. For any prism (including cuboids and cylinders), volume = area of cross-section × length. A pyramid or cone holds exactly one third of the prism or cylinder with the same base and height, which you can show by filling one with sand or water three times. Archimedes proved that a sphere has two thirds of the volume of the cylinder that just encloses it, and was so proud of it that he asked for the diagram on his tomb.
Surface area
Surface area is the total area of all the outside surfaces, in square units such as cm². For polyhedra, add up the areas of all the faces; it helps to imagine unfolding the shape into a flatnet. The curved surface of a cylinder unrolls into a rectangle whose length is the circumference, 2πr, giving 2πrh. A cone’s curved surface is a sector of a circle with area πrl, where l is the slant height.
Formula sheet
| Shape | Volume | Surface area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid | l × w × h | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | ⅓πr²h | πr² + πrl |
| Sphere | ⁴⁄₃πr³ | 4πr² |
| Square pyramid | ⅓a²h | a² + 2as |
Cross-sections
A cross-section is the flat shape you get when you slice through a solid. Slices of a prism parallel to its end are all identical. Slices of a pyramid or cone parallel to the base are the same shape as the base but shrink towards the apex: halfway up, the side length is halved and the area is a quarter. Slices of a sphere are always circles, largest through the centre. Cross-sections are used in medicine (CT scans), engineering and 3D printing, which builds objects one slice at a time.
Scaling and the square–cube law
If every length of a shape is multiplied by a scale factor k, its surface area is multiplied by k² and its volume by k³. Double the size and you get 4 times the area but 8 times the volume. This square–cube law explains why small animals lose heat quickly, why cells divide rather than growing huge, and why giant insects from films could not exist: their legs could not support their weight.
Key terms
- Polyhedron
- A solid with flat polygon faces, straight edges and vertices.
- Net
- A flat pattern that folds up to make a 3D shape.
- Slant height (l or s)
- The distance from the apex to the edge of the base, measured along the surface.
Common misconceptions
- “Doubling the sides doubles the volume.” It multiplies the volume by 8.
- “A cylinder has no edges.” It has two curved edges where the flat faces meet the curved surface.
- “Surface area and volume use the same units.” Area is in square units (cm²), volume in cubic units (cm³).
Key takeaways
- Volume measures space inside (cm³); surface area measures the outside covering (cm²).
- Prisms: V = cross-section × length. Pyramids and cones: one third of the matching prism or cylinder.
- For polyhedra, F − E + V = 2.
- Scaling lengths by k scales area by k² and volume by k³.
Model note: curved shapes are drawn with many small flat panels, but the readouts use the exact formulas.