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Why Polygons Are Free Marks
Polygon questions in SSC, Railway and TRE-4 papers are almost never hard; they are fast if you hold three formulas and slow if you start drawing. For a polygon with n sides:
Sum of interior angles = (n − 2) × 180°.
Each interior angle of a regular polygon = (n − 2) × 180° ÷ n.
Each exterior angle of a regular polygon = 360° ÷ n, and the sum of exterior angles of any convex polygon is 360°.
Number of diagonals = n(n − 3) ÷ 2.
Why (n − 2)? From one corner you can draw diagonals that cut the polygon into n − 2 triangles, and each triangle contributes 180°.
Interior and exterior angles at a corner always add to 180°, which links the two formulas.
Worked example: a quadrilateral gives (4 − 2) × 180° = 360°; a pentagon gives 540°; a hexagon gives 720°.
1. What is the sum of the interior angles of a hexagon?
2. Each exterior angle of a regular polygon is 40°. How many sides does the polygon have?
3. How many diagonals does a decagon (10-sided polygon) have?
4. The diagonals of a rhombus are 16 cm and 12 cm. What is the length of its side?
5. Which quadrilateral always has equal diagonals that bisect each other but are not perpendicular in general?
6. What is each interior angle of a regular pentagon?
7. The parallel sides of a trapezium are 12 cm and 8 cm and the distance between them is 5 cm. Its area is
8. Two adjacent angles of a parallelogram are in the ratio 2 : 3. What is the measure of the larger angle?