Polygons
A polygon is any closed figure bounded by straight line segments — triangles, quadrilaterals, pentagons, hexagons and beyond. In CAT, XAT and SNAP the polygon questions almost never test definitions; they test whether you can deploy three or four compact angle and diagonal formulas under time pressure, and whether you can tell a regular polygon (all sides and angles equal) from an irregular one. The whole topic rests on a single elegant fact: any n-sided polygon splits into (n−2) triangles, so its interior angles sum to (n−2)×180°. From that one idea flow the interior angle of a regular polygon, the always-constant 360° exterior-angle sum, and the diagonal count n(n−3)/2. CAT loves to disguise these as "find the number of sides when each interior angle is 156°" or "how many diagonals does a 20-gon have", and to pair them with mensuration when a regular hexagon or octagon needs its area. This chapter builds that toolkit from the ground up — angle theory first, regular-polygon symmetry second, diagonals and area last — with worked CAT-style examples, the fastest exam method, and the traps that catch careless students.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Sum of interior angles = (n − 2) × 180°. Memorise the first few: triangle 180°, quad 360°, pentagon 540°, hexagon 720°.
- Exterior angles ALWAYS sum to 360°. To find the number of sides of a regular polygon, do 360 / (each exterior angle) — one division.
- For a regular polygon, work via the exterior angle (360/n); it is smaller and divides 360 cleanly. Interior = 180 − exterior.
- Diagonals = n(n − 3)/2. Reverse it by solving n(n − 3) = 2D — guess near √(2D) and adjust.
- A regular hexagon = six equilateral triangles, so its area is (3√3/2)a²; its long diagonal = 2a and short diagonal = √3·a.
- Sanity check a "regular" claim: 360/(180 − interior) must be a whole number, or the polygon cannot be regular.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Using n × 180° instead of (n − 2) × 180° for the sum of interior angles.
- Forgetting that the exterior-angle sum is a fixed 360° regardless of the number of sides.
- Omitting the ÷2 in the diagonal formula and reporting n(n − 3) instead of n(n − 3)/2.
- Treating an irregular polygon as regular — only regular polygons have equal angles, so 360/n gives each angle only when sides AND angles are equal.
- Accepting a non-integer number of sides; if 360/(180 − interior) is not whole, the regular polygon described does not exist.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Polygons when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Angles in a polygon
| Sum of interior angles | (n − 2) × 180° |
|---|---|
| Each interior angle (regular) | (n − 2) × 180° / n |
| Sum of exterior angles | 360° (any convex polygon) |
| Each exterior angle (regular) | 360° / n |
| Interior + exterior (at a vertex) | 180° |
Diagonals & area
| Number of diagonals | n(n − 3) / 2 |
|---|---|
| Sides from interior angle | n = 360 / (180 − interior) |
| Area of a regular polygon | (1/2) × perimeter × apothem |
| Area via side a | (n a² / 4) × cot(180°/n) |
| Area of a regular hexagon | (3√3 / 2) × a² |