Quadrilateral and polygon questions are among the quickest in SSC geometry — if you know the properties. They rarely need long calculations. A regular polygon's angles follow from one formula; a rhombus's side follows from its diagonals by Pythagoras; a parallelogram's angles follow from the rule that adjacent angles are supplementary. Learn the properties below and practise recognising them.
Quadrilaterals
The angle sum of any quadrilateral is 360°.
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Parallelogram | Opposite sides equal and parallel | Opposite angles equal; adjacent angles sum to 180° | Bisect each other |
| Rectangle | Opposite sides equal | All 90° | Equal; bisect each other |
| Rhombus | All sides equal | Opposite angles equal | Bisect each other at right angles; bisect the angles |
| Square | All sides equal | All 90° | Equal; bisect each other at right angles |
| Trapezium | One pair of parallel sides | Angles on the same leg sum to 180° | — |
| Kite | Two pairs of adjacent equal sides | One pair of opposite angles equal | Perpendicular |
Rhombus from its diagonals
Since the diagonals bisect each other at right angles:
side² = (d₁/2)² + (d₂/2)², and area = ½ × d₁ × d₂.
Worked example: A rhombus has diagonals of 16 cm and 12 cm. Find its side and area.
side² = 8² + 6² = 100, so side = 10 cm; area = ½ × 16 × 12 = 96 cm².
Parallelogram angles
Worked example: In a parallelogram, one angle is 40° more than its adjacent angle. Find the angles.
x + (x + 40) = 180 → x = 70. The angles are 70°, 110°, 70°, 110°.
Trapezium
Area = ½ × (sum of parallel sides) × height.
Worked example: A trapezium has parallel sides of 12 cm and 18 cm and a height of 8 cm. Area = ½ × 30 × 8 = 120 cm².
Polygons
For a polygon with n sides:
| Quantity | Formula |
|---|---|
| Sum of interior angles | (n − 2) × 180° |
| Each interior angle (regular) | (n − 2) × 180°/n |
| Each exterior angle (regular) | 360°/n |
| Sum of exterior angles | 360° (always) |
| Number of diagonals | n(n − 3)/2 |
Worked example: Find each interior angle of a regular 12-sided polygon.
Exterior = 360/12 = 30°, so interior = 150°.
Worked example: Each interior angle of a regular polygon is 140°. How many sides does it have?
Exterior = 40°; n = 360/40 = 9.
Worked example: How many diagonals does a decagon have?
10 × 7/2 = 35.
Ratio of interior to exterior angle
If each interior angle is k times each exterior angle, then 180 = (k + 1) × exterior, so n = 2(k + 1).
Worked example: Each interior angle is 5 times each exterior angle. Then n = 2 × 6 = 12.
Regular hexagon
A regular hexagon of side a is made of six equilateral triangles:
- area = 6 × (√3/4)a² = (3√3/2)a²;
- the longest diagonal = 2a.
Worked example: Find the area of a regular hexagon of side 4 cm.
(3√3/2) × 16 = 24√3 cm².
Common traps
| Trap | Correct approach |
|---|---|
| Assuming a rhombus's diagonals are equal | Only a square's are |
| Forgetting that the exterior angle sum is always 360° | Use it for any convex polygon |
| Dividing the interior-angle sum wrongly | Use (n − 2) × 180, not n × 180 |
Practice
- Find the sum of the interior angles of an octagon.
- Each exterior angle of a regular polygon is 24°. Find the number of sides.
- A rhombus has a side of 13 cm and one diagonal of 24 cm. Find the other diagonal.
- How many diagonals does a polygon with 15 sides have?
- In a parallelogram ABCD, ∠A = 3∠B. Find ∠A.
- Each interior angle of a regular polygon is 8 times each exterior angle. Find the number of sides.
Answers: 1. 1,080°. 2. 15. 3. 10 cm. 4. 90. 5. 135°. 6. 18.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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