Geometry

Geometry is the oldest written part of mathematics; the version we teach in school is essentially the one Euclid wrote down 2,300 years ago. Most of its school content is about measurement — lengths, angles, areas, volumes — and about the small set of facts (the Pythagorean theorem, similar triangles, the basic angle relations) that let you compute one of those measurements from another.

What makes the topic occasionally frustrating is that the techniques are not as algorithmic as in algebra. A geometry problem usually starts with a diagram, and the first task is to see what is going on — which sides are equal, which angles are right, which triangles are similar — before any formula gets used. The articles in this section try to make that “seeing” step explicit rather than assumed.

Articles in this section

• The Pythagorean theorem, four different proofs

  A diagram-driven walk through the most famous identity in geometry — visual rearrangement, algebraic, similar-triangle, and converse — with a calculator that handles the three usual cases (find the hypotenuse, find a leg, test for right-angledness).

  By Daniel Whitman · 14 min read

• Area and perimeter of the standard shapes

  Triangles, rectangles, parallelograms, trapeziums, circles. Each formula is short to memorise but easy to misapply (which length is the “height”? which radius is squared?). With a multi-shape calculator and a units-checking note.

  By Daniel Whitman · 11 min read

• The sine and cosine laws for non-right triangles

  What to do when the Pythagorean theorem stops applying. The two laws, when each one is the right tool (SAS, SSS, ASA, SSA), and the “ambiguous case” that catches every student at least once.

  By Daniel Whitman · 12 min read

A reasonable order to read them in

The Pythagorean theorem is the foundation; almost everything else either uses it directly or generalises it. After that the order is up to you, but area and perimeter is the lighter read, and the sine and cosine laws are the most useful piece for trigonometry-heavy exam questions at GCSE higher tier and beyond.

What this section does not cover

Coordinate geometry (lines, circles and conics in the $xy$-plane) is better thought of as a part of algebra and is not in this section. Solid geometry beyond the standard volume formulas, transformations, and projective and non-Euclidean geometries are also out of scope — useful and beautiful, but not what most students have to study.