Naming shapes precisely

In Discover you met shapes the friendly way: by looking, touching and folding. Now we sharpen the words. In mathematics a name is a promise: if someone says a shape is a rhombus, you know exactly which facts are guaranteed and which are not.

That precision is what lets a carpenter in Jaipur and an engineer in Chennai talk about the same shape without a picture, and it is what lets us answer questions like is a square a rectangle? without arguing. This layer gives each shape a proper definition (the smallest set of facts that decides whether something belongs to the family), lists its properties (everything else that follows), and clears up the mix-ups that trip most learners.

A line segment is the straight path between two points. A polygon is a simple closed figure made only of line segments, where:

1. each segment meets exactly two others, one at each end;
2. the segments meet only at their endpoints (no crossing);
3. the figure closes up.

The segments are sides, the meeting points are vertices, and at each vertex the two sides make an interior angle. Two sides that share a vertex are adjacent sides; two vertices at the ends of one side are adjacent vertices. The fewest sides possible is 3, because two segments cannot enclose anything.

A regular polygon has all sides equal and all angles equal. Both conditions matter:

• A rhombus that is not a square has equal sides but unequal angles, so it is not regular.
• A rectangle that is not a square has equal angles but unequal sides, so it is not regular.
• Only the square has both, so the square is the one regular quadrilateral.

For triangles something special happens: if all three sides are equal, the angles are automatically equal too (each 60°). So an equilateral triangle is always regular.

A diagonal is a line segment joining two vertices that are not next to each other. Sides join neighbouring vertices; diagonals join the others.

A triangle has no diagonals: every vertex is next to both of the others. A quadrilateral has 2. A pentagon has 5, which together draw the famous five-pointed star inside it. As the number of sides grows, the diagonals multiply quickly.

Every triangle can be described in two separate ways at once: by its sides and by its angles.

By sides: equilateral (all three equal), isosceles (two equal), scalene (all different).

By angles: acute-angled (all three angles less than 90°), right-angled (one angle exactly 90°), obtuse-angled (one angle more than 90°).

So a full description sounds like a right-angled isosceles triangle (the shape of a folded square scarf) or an obtuse-angled scalene triangle.

Here are the definitions used in Indian school textbooks. Each is the smallest test; the table after it lists the properties that follow.

• Trapezium: a quadrilateral with a pair of parallel sides.
• Parallelogram: a quadrilateral with both pairs of opposite sides parallel.
• Rhombus: a parallelogram with all four sides equal (equivalently, any quadrilateral with four equal sides).
• Rectangle: a parallelogram with a right angle (then all four angles are right angles).
• Square: a rectangle with all sides equal (equivalently, a rhombus with a right angle).
• Kite: a quadrilateral with two pairs of equal sides next to each other (adjacent), such as AB = AD and CB = CD.

A circle is the set of all points in a plane that are the same distance from a fixed point, the centre. That distance is the radius (r).

The words for the parts of a circle are worth learning exactly, because they appear in engineering, astronomy and sport:

The perimeter of a closed figure is the total length of its boundary: the distance an ant walks to go once all the way round. For a polygon, just add the lengths of all the sides. For a circle, the perimeter has its own name: the circumference.

Perimeter answers questions like: how much fencing for this garden? how much lace for the edge of this dupatta? how far is one lap of the school ground?

A polyhedron (plural polyhedra) is a solid whose surface is made entirely of flat polygons. Its faces are those polygons, its edges are the segments where two faces meet, and its vertices are the points where three or more edges meet. Cubes, cuboids, prisms and pyramids are polyhedra. Cylinders, cones and spheres are not, because they have curved surfaces.

Two big families of polyhedra are named after the shape of their base:

• A prism has two identical, parallel ends (the bases) joined by rectangles. A prism with triangle ends is a triangular prism; with hexagon ends, a hexagonal prism. A cuboid is a rectangular prism, and a cube is a square prism with square sides too.
• A pyramid has one base, and triangles rising from every side of the base to a single point called the apex. A pyramid with a square base is a square pyramid; with a triangle base, a triangular pyramid (or tetrahedron).

A net is a flat, connected arrangement of polygons (or other shapes) that folds, along its edges, into the surface of a solid with no gaps and no overlaps.

A net must have exactly the right faces: a cube net has 6 equal squares; a square pyramid net has 1 square and 4 identical triangles. But having the right faces is not enough; they must also be joined in the right arrangement. Six squares in one long row do not fold into a cube, because the ends overlap and two faces are missing on the sides.

When architects and engineers draw a building, they use three flat views:

• the top view or plan: looking straight down from above;
• the front view or front elevation: looking straight at the front;
• the side view or side elevation: looking straight at one side.

Each view flattens the solid into a 2D shape. None of them alone tells you the whole shape, but together they usually do. A map is a top view of the land, drawn to scale; that is why roads look like lines and buildings look like rectangles.

A figure has line symmetry (or reflection symmetry) if there is a line such that folding along it makes one half land exactly on the other. That line is a line of symmetry or axis of symmetry. Every point on one side has a partner at the same distance on the other side, and the segment joining them crosses the line at a right angle.

Some shapes have one line, some several, some none at all, and a circle has infinitely many.