Angles and polygons: deducing what the diagram refuses to show

5 min read·Updated 5 September 2026

Angle chasing in a polygon is the most deduction-heavy topic in Paper 2 that involves no algebra at all. You are given a figure, a few marked facts, and asked for an angle — and the entire difficulty is that the figure is drawn to suggest an answer while the marked facts force a different one.

Diagrams in a multiple-choice paper are schematic. They show you which points join to which, and nothing more. Treating them as measured drawings is the single largest source of lost marks in the topic.

Misconception 1: judging acute or obtuse by eye

An angle drawn looking comfortably acute can be forced obtuse by the rest of the information — most often through a supplementary pair, where the angle you can compute directly is the other one on a straight line, and the answer is $180^\circ$ minus it. Students compute correctly, then quietly report the angle that matches the picture instead of the one their working produced.

The defence is mechanical: finish the arithmetic, write the number, and only then look at the diagram — and if the two disagree, believe the arithmetic.

Misconception 2: mixing up a regular polygon's three angles

A regular $n$-gon has three different angles that all get called "the angle" in casual speech:

• the exterior angle, $360^\circ / n$;
• the interior angle, $180^\circ - 360^\circ/n$;
• the angle at the centre subtended by one side, which is also $360^\circ / n$ but lives in a different triangle.

The exterior and central angles being numerically equal is exactly what makes this confusing, because a student who reaches for the wrong one is right half the time by accident and cannot tell which half they are in.

Misconception 3: reading equal lengths off the picture

Two segments that look the same length are not equal unless something says so: tick marks, a stated regular polygon, an isosceles triangle you have proved, or a property such as radii of one circle. "It looks isosceles" is not a reason, and the base angles you then write down are not facts.

A worked example

A regular polygon has an exterior angle of $24^\circ$. Three consecutive vertices are labelled A, B and C. Find angle BAC.

The exterior angle gives the number of sides:

so the interior angle at each vertex is

Triangle ABC is isosceles — and here that is a reason, because AB and BC are sides of a regular polygon and are therefore equal. Its apex angle is the interior angle at B, so

Two wrong turns are common. The first is to take the interior angle as $180^\circ / 15 = 12^\circ$, which is a half-remembered formula that happens to produce a number that also appears in this problem — a coincidence that hides the error completely. The second is to stop at $180^\circ - 156^\circ = 24^\circ$ and report that as angle BAC, forgetting that the two base angles share it. Both answers are the kind that sit comfortably among four options.

Check yourself

Two habits. First: the arithmetic outranks the picture, always. Second: before writing any interior angle, say which of the three angles you want and which formula produces it — if the sentence "this is the interior angle, because..." does not finish, you have reached for the wrong one.

The drill below redraws the polygon with a new number of sides each time, so the formula has to be chosen rather than recalled.

Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.