One Equation Links the Corners, Edges and Faces of Any Solid

There is, and it is remarkably short. For any polyhedron,

$$F+V-E=2$$

A cube has 6 faces, 8 vertices and 12 edges, and $6+8-12=2$. A pyramid, a prism and far stranger shapes all obey the same equation.

This page covers everything in the ICSE Class 7 Mathematics chapter on solids: counting faces, edges and vertices, Euler's formula, nets, and representing solids in two dimensions.

A face is a flat or curved surface, an edge is where two faces meet, and a vertex is a corner where edges meet.

Solids with only flat faces are called polyhedra:

- Cube — 6 faces, 12 edges, 8 vertices. All faces are equal squares.
- Cuboid — 6 faces, 12 edges, 8 vertices. Faces are rectangles.
- Triangular prism — 5 faces, 9 edges, 6 vertices.
- Square pyramid — 5 faces, 8 edges, 5 vertices.
- Triangular pyramid (tetrahedron) — 4 faces, 6 edges, 4 vertices.

Solids with curved surfaces are not polyhedra:

- Cylinder — 3 faces (2 flat circles and 1 curved), 2 edges, 0 vertices.
- Cone — 2 faces (1 flat circle and 1 curved), 1 edge, 1 vertex at the apex.
- Sphere — 1 curved face, 0 edges, 0 vertices.

A matchbox is a cuboid, a tin of food a cylinder, an ice-cream cone a cone, and a football a sphere.

The count students get wrong is the cylinder's vertices. Its two circular rims are edges, but no corner is formed anywhere, so a cylinder has no vertices at all — and a sphere has neither edges nor vertices.

A net is a flat pattern that folds up into a solid, with no overlapping and no gaps.

A cube's net must have exactly 6 squares, since a cube has 6 faces. The commonest arrangement is a row of four squares with one square attached above and one below, in the shape of a cross.

There are eleven different nets of a cube, counting arrangements that cannot be turned or flipped onto one another.

Other simple nets:

- Cuboid — 6 rectangles, in matching pairs.
- Cylinder — 2 circles and 1 rectangle, where the rectangle's length equals the circle's circumference.
- Cone — 1 circle and 1 sector of a larger circle.
- Square pyramid — 1 square with 4 triangles attached to its sides.
- Triangular prism — 2 triangles and 3 rectangles.

An unfolded cardboard carton flattened for recycling is a net you can hold.

Six squares is necessary but not sufficient, which is exactly what these questions test. A vertical column of six squares in a straight line has the right number and still cannot fold into a cube — the squares would wrap round and overlap before closing. So check by mentally folding: each square must reach a different face of the cube.

Three methods appear in the syllabus, each showing something different.

Sketches (oblique or isometric). A cube is drawn as a square with a second square behind it, corners joined. The hidden edges are drawn as dashed lines. On isometric dot paper the lengths stay in proportion, so a $3\times 2\times 2$ cuboid can be drawn exactly.

Shadows. Light shone on a solid casts a flat shadow, and the shape depends on the direction. A cylinder standing upright casts a rectangle from the side and a circle from above. A cube casts a square from any face-on direction.

Views. A solid can be drawn as seen from the top, the front and the side. A cylinder lying down gives a rectangle from the front and a circle from the end.

Maps to scale. A stated scale converts drawing lengths to real ones. With 1 cm = 5 m, a line of 4 cm on the map represents

$$4\times 5=20\text{ m}$$

and a real distance of 35 m is drawn as $\frac{35}{5}=7$ cm.

A house plan or a school-ground sketch works exactly this way.

The rule that must not slip is the scale statement. A map without its scale written on it cannot be read at all, and the conversion runs in both directions — multiply to go from drawing to reality, divide to go the other way.