Olympiad geometry at this level is mostly bookkeeping: know which shape has which property, and know the two angle-sum formulas. Everything else follows.
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Two formulas answer almost every angle question in the paper.
📐 The two formulas
Interior angles of an n-sided polygon add up to (n − 2) × 180°. For a quadrilateral that is 360°; for a pentagon 540°; for a hexagon 720°.
Exterior angles always add up to 360°, whatever the number of sides. So each exterior angle of a regular n-gon is 360° ÷ n.
For a regular polygon, each interior angle = 180° − (360° ÷ n).
The number of diagonals is n(n − 3) ÷ 2 — an octagon has 8 × 5 ÷ 2 = 20.
Work through the exterior angle. To find each interior angle of a regular octagon, do 360 ÷ 8 = 45 first, then 180 − 45 = 135°. It is faster and safer than (8 − 2) × 180 ÷ 8.
Each shape inherits every property of the one above it.
👪 Who has what
| Shape | Sides | Diagonals |
|---|---|---|
| Trapezium | one pair of parallel sides | nothing special |
| Parallelogram | both pairs parallel and equal | bisect each other |
| Rhombus | a parallelogram with all sides equal | bisect each other at 90° |
| Rectangle | a parallelogram with four right angles | bisect each other and are equal |
| Square | all sides equal and all angles 90° | equal, and bisect at 90° |
| Kite | two pairs of equal adjacent sides | cross at 90°; one bisects the other |
In a parallelogram, adjacent angles are supplementary. If one angle is 70°, the next is 110°, and the opposite one is 70° again. One angle therefore fixes all four.
Every square is a rhombus and a rectangle, but not the other way round. "All rhombuses are squares" is false and is offered as an option constantly.
The same three question shapes come round every year.
Angles in a ratio. "The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4." Add the parts: 1 + 2 + 3 + 4 = 10, so one part is 360 ÷ 10 = 36°, and the angles are 36°, 72°, 108° and 144°.
Find n from an angle. "Each interior angle of a regular polygon is 150°." Then each exterior angle is 30°, and n = 360 ÷ 30 = 12 sides.
The unknown fourth angle. Three angles of a quadrilateral are 80°, 95° and 112°. The fourth is 360 − (80 + 95 + 112) = 73°.
12 Olympiad questions on quadrilaterals and polygons. 🍀
