A polygon is a closed figure bounded by straight line segments, and a quadrilateral is simply a four-sided polygon. In the CDS / OTA paper this chapter delivers reliable marks because the questions rotate around a few angle-sum rules, property tables and area formulas. This Cavalier guide builds the topic from the basic definitions upward and drills the exact patterns the exam repeats.
Why quadrilaterals and polygons matter in CDS
Geometry is one of the largest blocks in CDS / OTA Maths, and quadrilaterals and polygons appear in almost every paper. The questions reward memory of a small set of rules more than long calculation, so a well-prepared candidate clears most of them in under a minute each.
The chapter also overlaps with triangles, area and perimeter and lines and angles, which means the few hours you invest here pay off across several topics. Once you know the angle-sum rules and the property table, even a wordy question reduces to a single substitution.
For the OTA paper especially, the cut-off rewards speed on these guaranteed questions, so leaving this chapter half-prepared is a costly gap. Treat it as a free block of marks rather than an optional extra. Most candidates who lose marks here do so not because the topic is hard, but because they confuse the property tables under time pressure. A clear mental map of the shapes, built once and revised often, removes that risk entirely.
Across the previous-year papers, three question types dominate: finding a missing angle, identifying a shape from a property, and computing an area. Each maps onto a single formula or rule covered below, so your job is simply to recognise which one a question is testing and substitute the numbers.
Every quadrilateral is a polygon, but not every polygon is a quadrilateral. A quadrilateral is the special four-sided case, and most CDS area questions sit here.
Polygons: basic vocabulary
A polygon is a closed plane figure formed by three or more straight line segments called sides. The point where two sides meet is a vertex, and the line joining two non-adjacent vertices is a diagonal.
Polygons are named by their number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9) and decagon (10).
- A convex polygon has every interior angle less than 180°; no diagonal falls outside it.
- A concave polygon has at least one interior angle greater than 180° (a reflex angle).
- A regular polygon has all sides equal and all angles equal; an irregular one does not.
Number of diagonals of an n-sided polygon = n(n − 3) ÷ 2. For a hexagon this is 6 × 3 ÷ 2 = 9 diagonals.
Interior and exterior angle sums
These two rules answer the majority of CDS polygon questions, so commit them to memory.
Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
Sum of exterior angles of any convex polygon = 360°, regardless of the number of sides.
For a regular polygon, every interior angle is equal, so each interior angle = (n − 2) × 180° ÷ n, and each exterior angle = 360° ÷ n. The interior and exterior angle at any vertex are supplementary: they add to 180°.
- Triangle (n = 3): interior sum = 180°.
- Quadrilateral (n = 4): interior sum = 360°.
- Pentagon (n = 5): interior sum = 540°.
- Hexagon (n = 6): interior sum = 720°; each interior angle = 120°.
If a question gives one exterior angle of a regular polygon, find the number of sides instantly: n = 360° ÷ (exterior angle). An exterior angle of 40° means a 9-sided polygon.
The quadrilateral angle sum
Put n = 4 into the interior-angle rule: (4 − 2) × 180° = 360°. So the four interior angles of any quadrilateral — whatever its shape — always add up to 360°.
This single fact lets you find a missing angle when the other three are known, or solve for an unknown in a ratio of angles.
Three angles of a quadrilateral are 80°, 95° and 100°. Find the fourth angle.
Do not use 180° for a quadrilateral — that is the triangle sum. A four-sided figure always totals 360°.
The family of special quadrilaterals
CDS loves property-matching questions, so learn what makes each shape special. The key idea is that the figures form a hierarchy, with the square at the top sharing every property below it.
- Trapezium: exactly one pair of opposite sides parallel.
- Parallelogram: both pairs of opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.
- Rhombus: a parallelogram with all four sides equal; diagonals bisect each other at right angles.
- Rectangle: a parallelogram with all angles 90°; diagonals equal and bisect each other.
- Square: all sides equal and all angles 90°; diagonals equal, bisect each other at right angles, and bisect the angles.
A square is simultaneously a rectangle (all angles 90°) and a rhombus (all sides equal). It inherits the diagonal properties of both.
In a rhombus the diagonals are perpendicular but not equal; in a rectangle they are equal but not perpendicular. Only the square has both. Mixing these up is the single most common slip in property questions.
Diagonal properties at a glance
Diagonal behaviour is a favourite CDS theme because a single property often identifies the shape. Use this checklist.
- Bisect each other: parallelogram, rhombus, rectangle, square.
- Equal in length: rectangle and square.
- Perpendicular to each other: rhombus and square (and a kite).
- Bisect the interior angles: rhombus and square.
A kite has two pairs of adjacent equal sides; one diagonal bisects the other at right angles, but the diagonals are not equal and only one of them is bisected. The kite is the odd one out in this family because its equal sides are adjacent rather than opposite, which is why its diagonals behave differently from a rhombus.
A quick way to keep the four parallelograms straight is to read the checklist from the top. Every parallelogram has diagonals that bisect each other. Add the condition that they are equal and you move to a rectangle. Add instead the condition that they are perpendicular and you move to a rhombus. Insist on both at once and you arrive at the square.
If a question says the diagonals are equal AND bisect each other at right angles, the figure must be a square. Equal-only points to a rectangle; perpendicular-only points to a rhombus.
Area formulas you must know
Area questions are guaranteed marks if you keep these formulas at your fingertips. Here b is base, h is height, d is a diagonal, and a is a side.
Square = a2 = ½ × (diagonal)2
Rectangle = length × breadth
Parallelogram = base × height
Rhombus = ½ × d1 × d2
Trapezium = ½ × (sum of parallel sides) × height
For the rhombus, the half-product of the diagonals works because the diagonals split it into four congruent right triangles. For the trapezium, the formula averages the two parallel sides and multiplies by the perpendicular distance between them. The square has two valid area formulas: use a2 when the side is given, and use half the diagonal squared when only the diagonal is supplied. Recognising which piece of data the question hands you saves a step.
A useful linking idea is that the parallelogram is the parent of three of these formulas. A rectangle is a parallelogram with a right angle, so base × height becomes length × breadth. A rhombus is a parallelogram with equal sides, and its base-times-height area equals the half-product of its diagonals. Seeing the formulas as variations of one rule means you carry fewer separate facts into the exam hall.
In a parallelogram the height is the perpendicular distance between the parallel sides, not the slant side. Multiplying base by the slant side overstates the area.
Perimeter and regular-polygon area
The perimeter of any polygon is the sum of its side lengths. For a regular polygon with n sides each of length a, perimeter = n × a.
- Square perimeter = 4a; rectangle perimeter = 2(length + breadth).
- Rhombus perimeter = 4a, since all four sides are equal.
- Regular hexagon perimeter = 6a.
Area of a regular polygon = ½ × perimeter × apothem, where the apothem is the perpendicular distance from the centre to a side. For an equilateral triangle of side a, area = (√3 ÷ 4) × a2, and a regular hexagon of side a has area = (3√3 ÷ 2) × a2.
A regular hexagon is made of six equilateral triangles. So its area is just six times (√3 ÷ 4) × a2, which simplifies to (3√3 ÷ 2) × a2. Deriving it this way saves memorising a separate formula.
Worked example: interior angle of a regular polygon
Polygon angle problems look intimidating but collapse into one or two substitutions. Work through this carefully and the pattern becomes clear.
Each interior angle of a regular polygon is 144°. How many sides does it have, and what is the sum of its interior angles?
Notice how using the exterior angle first (which always totals 360°) gave the number of sides in one step. Always reach for the exterior-angle route when an interior angle of a regular polygon is supplied. The alternative, setting (n − 2) × 180° ÷ n equal to 144° and solving for n, also works but takes longer and invites arithmetic slips under time pressure.
Worked example: area of a rhombus and trapezium
Area questions reward clean substitution. Keep your units consistent and the answer follows directly.
(a) Find the area of a rhombus whose diagonals are 16 cm and 12 cm. (b) Find the area of a trapezium whose parallel sides are 20 cm and 12 cm and whose height is 5 cm.
The rhombus formula needs both diagonals, not the side. If only the side and one diagonal are given, use the right-triangle relation: half-diagonals and the side form a right angle, so the other half-diagonal comes from the Pythagoras rule.
Previous-year style question
This is the exact flavour of question CDS sets on this chapter — a regular polygon with a stated angle, solved through the exterior-angle route.
Q. The ratio of the interior angle to the exterior angle of a regular polygon is 5 : 1. How many sides does the polygon have?
Answer: The interior and exterior angles are supplementary, so they add to 180°. With a ratio of 5 : 1, the parts total 6, giving exterior angle = 180° × (1 ÷ 6) = 30°. Number of sides = 360° ÷ 30° = 12. The polygon is a regular dodecagon.
Quick recap and revision
- Interior angle sum of an n-gon = (n − 2) × 180°; exterior angles always total 360°.
- A quadrilateral's four angles add to 360°.
- Diagonals: equal in rectangle and square; perpendicular in rhombus and square; bisect each other in every parallelogram.
- Square = ½ × diagonal2; rhombus = ½ × d1 × d2; trapezium = ½ × (parallel sides) × height.
- For a regular polygon, exterior angle = 360° ÷ n — the fastest route to the side count.
- Number of diagonals = n(n − 3) ÷ 2.
Revise the property table and the five area formulas the night before the exam, and rehearse the exterior-angle shortcut until it is automatic. With these locked in, the quadrilateral and polygon questions become some of the quickest marks on the whole paper.
Frequently asked questions
What is the sum of the interior angles of a quadrilateral?
Always 360°, found by putting n = 4 into (n − 2) × 180°. This holds for every quadrilateral regardless of its shape, which lets you find any missing fourth angle.
How do I quickly find the number of sides of a regular polygon?
Divide 360° by one exterior angle, since the exterior angles of any convex polygon total 360°. If only the interior angle is given, subtract it from 180° first to get the exterior angle.
What is the difference between a rhombus and a rectangle in CDS questions?
A rhombus has all sides equal with diagonals that are perpendicular but unequal; a rectangle has all angles 90° with diagonals that are equal but not perpendicular. Only a square has both equal and perpendicular diagonals.
Which area formula is used most often in the CDS exam?
The rhombus formula (½ × d1 × d2) and the trapezium formula (½ × sum of parallel sides × height) appear most often, alongside the standard square and rectangle areas.
How many diagonals does a polygon have?
An n-sided polygon has n(n − 3) ÷ 2 diagonals. For example, a hexagon has 6 × 3 ÷ 2 = 9 diagonals, and a pentagon has 5 diagonals.
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