Quadrilateral and Polygon
~12 min read
- What: Properties of quadrilaterals (parallelogram, rhombus, rectangle, square, trapezium, kite) and regular polygons — interior and exterior angle sums, diagonals, and side-from-angle deductions.
- Why it matters: CDS averages 3–4 questions per paper on this chapter. Properties are short, formulaic, and recur every year.
- Key fact: For any convex polygon with \(n\) sides: sum of interior angles \(= (n - 2) \times 180°\); sum of exterior angles \(= 360°\) (always, regardless of \(n\)). Each interior angle of a regular \(n\)-gon \(= (n - 2) \times 180° / n\).
Quadrilateral and Polygon is a high-yield, low-effort CDS chapter. The properties of each shape are tight, and the polygon angle-sum formulas are universal. Get the property table memorised, drill 10 polygon-angle problems, and you collect 3–4 marks per paper here.
This page is built from CDS Previous Year Questions across 2007–2022, plus NCERT Class 8 Quadrilaterals. Pair with Lines and Angles, Triangles, and Area and Perimeter.
What This Topic Covers
CDS scope: (1) quadrilateral types — parallelogram, rectangle, rhombus, square, trapezium, kite, cyclic quadrilateral; (2) properties — sides, angles, diagonals of each; (3) polygon angle sums — interior and exterior; (4) regular polygon — each angle, number of sides from given angle, number of diagonals; (5) cyclic quadrilateral — opposite angles supplementary.
Why This Topic Matters
- 3–4 CDS questions per paper, formula-driven.
- Polygon angle sum is the single most-tested concept here.
- Quadrilateral property table is the highest-ROI memorisation.
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2007-II | 2 | Parallelogram, polygon angle |
| 2011-I/II | 3 | Rhombus diagonals, polygon |
| 2012-I/II | 3 | Cyclic, polygon |
| 2013-I/II | 3 | Square, regular polygon |
| 2014-I/II | 3 | Rectangle, polygon diagonals |
| 2015-I/II | 3 | Trapezium, regular hexagon |
| 2016-I/II | 3 | Cyclic, kite |
| 2017-I/II | 3 | Polygon angle from side count |
| 2018-I/II | 3 | Rhombus, square |
| 2019-II | 2 | Cyclic quadrilateral |
| 2020-I/II | 3 | Mixed |
| 2021-I/II | 3 | Polygon angle sum |
| 2022-I | 2 | Mixed |
Sum of exterior angles of any convex polygon = 360°. This is independent of the number of sides. So each exterior angle of a regular \(n\)-gon is \(360°/n\). For \(n = 12\) (dodecagon): each exterior = 30°.
Core Concepts
Quadrilateral Hierarchy
Trapezium ⊃ Parallelogram ⊃ Rectangle, Rhombus ⊃ Square (the most "constrained"). Each child inherits all parent properties plus extras.
Polygon Angle Formulas
Sum of exterior angles \(= 360°\) (always).
Each interior angle of regular \(n\)-gon \(= \tfrac{(n - 2) \times 180°}{n}\).
Each exterior angle of regular \(n\)-gon \(= \tfrac{360°}{n}\).
Number of Diagonals
So a quadrilateral has 2 diagonals; a pentagon, 5; a hexagon, 9; a decagon, 35.
A square is a special rectangle AND a special rhombus — but a rectangle is not always a square and a rhombus is not always a square. CDS occasionally tests "every rhombus is a square" type true/false statements.
Worked Examples
Example 1 — Polygon Interior Sum (2021-I)
Q: Find the sum of interior angles of an octagon.
- \((n - 2) \cdot 180° = 6 \cdot 180° = 1080°\).
Example 2 — Regular Polygon Angle (2017-I)
Q: Each interior angle of a regular polygon is 144°. Find the number of sides.
- Interior + exterior = 180°. Each exterior = 36°.
- \(n = 360° / 36° = 10\).
Example 3 — Number of Diagonals (2014-II)
Q: How many diagonals does a pentagon have?
- \(D = n(n-3)/2 = 5 \cdot 2/2 = 5\).
Example 4 — Cyclic Quadrilateral (2019-II)
Q: In a cyclic quadrilateral, one angle is 95°. Find its opposite angle.
- Opposite angles supplementary: \(180° - 95° = 85°\).
Example 5 — Rhombus Side from Diagonals (2018-II)
Q: A rhombus has diagonals 24 cm and 10 cm. Find its side length.
- Diagonals of rhombus are perpendicular bisectors. Each half-diagonal: 12 and 5.
- Side = \(\sqrt{12^2 + 5^2} = \sqrt{169} = 13\) cm. (5-12-13 triple recognised.)
Example 6 — Parallelogram Adjacent Angles (2013-I)
Q: One angle of a parallelogram is 110°. Find the others.
- Opposite angles equal: another 110°. Adjacent angles supplementary: \(180° - 110° = 70°\). Last angle: 70°.
Example 7 — Exterior Angle Polygon (2015-II)
Q: The exterior angle of a regular polygon is 24°. Find the sum of interior angles.
- Number of sides \(n = 360° / 24° = 15\).
- Interior sum \(= (15 - 2) \cdot 180° = 13 \cdot 180° = 2340°\).
How CDS Tests This Topic
Five archetypes: (1) sum of interior or exterior angles given \(n\), (2) find \(n\) from a given interior or exterior angle, (3) cyclic quadrilateral opposite-angle relation, (4) rhombus side from diagonals (Pythagoras on halves), (5) number of diagonals.
Exam Shortcuts (Pro-Tips)
Shortcut 1 — Exterior Angles Always 360°
Sum of exterior angles of any convex polygon = 360°. Use this to find the number of sides from a given regular-polygon exterior angle: \(n = 360°/\theta_{ext}\).
Shortcut 2 — Interior + Exterior = 180°
At each vertex of a convex polygon, interior + exterior = 180°. Equivalent definitions of interior and exterior angle.
Shortcut 3 — Rhombus = Two Half-Diagonals on Pythagoras
Side of rhombus = \(\sqrt{(d_1/2)^2 + (d_2/2)^2}\). Watch for Pythagorean triple recognition (e.g. 5-12-13 from diagonals 10 and 24).
Shortcut 4 — Cyclic = Opposite Supplementary
Cyclic quadrilateral ⇒ opposite angles sum to 180°. Conversely, any quadrilateral with this property is cyclic.
Shortcut 5 — Regular Polygon Diagonal Count
\(D = n(n-3)/2\). For a regular \(n\)-gon, also: number of lines of symmetry = \(n\); rotational symmetry of order \(n\).
Common Question Patterns
Pattern 1 — Interior Sum from \(n\)
\((n - 2) \cdot 180°\). Trivial application.
Pattern 2 — \(n\) from Given Angle
Use exterior \(= 360°/n\) → \(n = 360°/\theta_{ext}\). Or use interior \(= (n-2) \cdot 180°/n\) → solve for \(n\).
Pattern 3 — Cyclic Quadrilateral
Opposite angles supplementary. Find missing angle given another.
Pattern 4 — Rhombus / Square Diagonals
Pythagoras on half-diagonals. Recognise common triples.
Pattern 5 — Property Verification
"Which of these is true for a rhombus?" — apply property table.
Preparation Strategy
Week 1. Build the property table for all six quadrilaterals (parallelogram, rectangle, rhombus, square, trapezium, kite). Drill 15 property-verification problems.
Week 2. Polygon angle formulas. Drill 15 problems on finding sides from given angles, and interior sums. Cover cyclic quadrilateral.
Mock testing. Use CDS mock tests. Most slip-ups: confusing rectangle and rhombus properties; forgetting that exterior angles sum to 360°.
Drill Quadrilateral and Polygon
CDS mocks with polygon angle, rhombus diagonal, and cyclic quadrilateral problems. Five archetypes — clean formulas — reflex.
Start Free Mock TestFrequently Asked Questions
What's the sum of interior angles of a polygon?
For an \(n\)-sided polygon, the sum is \((n - 2) \times 180°\). Triangle: 180°. Quadrilateral: 360°. Pentagon: 540°. Hexagon: 720°. Octagon: 1080°.
What's the sum of exterior angles?
Always 360°, regardless of the number of sides. This is true for any convex polygon. Each exterior angle of a regular \(n\)-gon = \(360°/n\).
How do I find the number of sides from a given interior angle?
If the interior angle is \(\theta\), then the exterior is \(180° - \theta\), and \(n = 360°/(180° - \theta)\). Or directly: \(\theta = (n - 2) \cdot 180°/n \implies n = 360°/(180° - \theta)\).
What's the rule for a cyclic quadrilateral?
All four vertices lie on a single circle. The opposite angles are supplementary: \(\angle A + \angle C = 180°\) and \(\angle B + \angle D = 180°\). The converse: any quadrilateral with this property is cyclic.
How do I find the side of a rhombus given its diagonals?
The diagonals of a rhombus are perpendicular bisectors of each other. So each side is the hypotenuse of a right triangle with legs equal to half the diagonals. Side = \(\sqrt{(d_1/2)^2 + (d_2/2)^2}\). Often a Pythagorean triple appears.
How many diagonals does an \(n\)-sided polygon have?
\(D = n(n - 3)/2\). A pentagon has 5, a hexagon 9, an octagon 20, a decagon 35.
Which CDS Maths topics connect to Quadrilateral and Polygon?
Lines and Angles — angle relations. Triangles — quadrilateral diagonals split into triangles. Area and Perimeter — area formulas. Circles — cyclic quadrilateral.