Quadrilaterals: Types, Properties and Formulas

A quadrilateral is any closed shape with four sides and four angles, and its interior angles always add up to 360°.

That single property, the angle-sum 360°, is the fastest shortcut when a placement test hides one angle and gives you three others. But each of the six standard types has additional properties and formulas worth knowing cold before you sit a Numerical Ability section.

What Is a Quadrilateral

A quadrilateral is a polygon with four vertices, four sides, and four interior angles. “Polygon” here just means the sides are straight lines that close into a shape (no curves).

The angle-sum rule holds for every quadrilateral, regardless of shape. Any four-sided closed figure, however irregular, has interior angles that total exactly 360°. The proof is straightforward: draw one diagonal to split the quadrilateral into two triangles. Each triangle has angles summing to 180°. Two triangles: 2 × 180° = 360°.

This makes angle-sum problems the most testable quadrilateral concept on campus aptitude tests. If three angles are given, the fourth is always 360° minus their sum.

Six Types and Their Properties

Square

A square has all four sides equal and all four angles equal at 90°.

• Sides: All four sides equal (side = a)
• Angles: All 90°
• Diagonals: Equal in length, bisect each other at 90°, each diagonal = a × √2

Rectangle

A rectangle is a parallelogram with all four right angles. Opposite sides are equal, not all four sides.

• Sides: Opposite sides equal (length = l, breadth = b)
• Angles: All 90°
• Diagonals: Equal in length, bisect each other but not at right angles. Each diagonal = √(l² + b²)

Parallelogram

A parallelogram has two pairs of opposite sides that are parallel and equal.

• Sides: Opposite sides equal and parallel
• Angles: Opposite angles equal; adjacent angles supplementary (sum to 180°)
• Diagonals: Bisect each other (not necessarily equal, not necessarily perpendicular)
• Diagonal relation: AC² + BD² = 2(AB² + BC²) — the parallelogram law

Rhombus

A rhombus is a parallelogram where all four sides are equal. The diagonals are perpendicular and bisect each other, dividing the shape into four congruent right-angled triangles.

• Sides: All four equal (side = a)
• Angles: Opposite angles equal; adjacent angles supplementary
• Diagonals: Perpendicular bisectors of each other (diagonals d1 and d2)
• Side from diagonals: a = √((d1/2)² + (d2/2)²)

Trapezium

A trapezium (called a trapezoid in North American usage) has exactly one pair of parallel sides, called the bases. The non-parallel sides are called legs or lateral sides.

• Parallel sides (bases): a and b
• Height: Perpendicular distance between the bases (h)
• Midline: The segment joining the midpoints of the lateral sides equals (a + b) / 2
• Isosceles trapezium: When the lateral sides are equal, the base angles are equal and the diagonals are equal in length

Kite

A kite has two pairs of adjacent sides that are equal. It is not a parallelogram; opposite sides are not parallel.

• Sides: Two pairs of consecutive equal sides (AB = AD, CB = CD)
• Angles: One pair of opposite angles is equal (the angles between the unequal sides)
• Diagonals: Perpendicular to each other; the longer diagonal bisects the shorter one
• Symmetry: One axis of symmetry along the longer diagonal

Formulas at a Glance

Shape	Area	Perimeter
Square	a²	4a
Rectangle	l × b	2(l + b)
Parallelogram	base × height	2(a + b)
Rhombus	(d1 × d2) / 2	4a
Trapezium	(a + b) × h / 2	a + b + c + d
Kite	(d1 × d2) / 2	2(a + b)

Where: a = side, l = length, b = breadth/base, h = height, d1 and d2 = diagonals, c and d = lateral sides of trapezium.

Note: rhombus and kite share the same area formula, diagonal product divided by 2. They differ in structure (rhombus has all sides equal; kite has adjacent-pairs equal) but the diagonals in both cases are perpendicular, which is why the formula is the same.

Worked Examples

These are re-derived from first principles. Do not rely on memory for the steps; trace the logic each time in an exam to avoid copying an error.

Example 1: Missing Angle from Ratio

• Problem: In quadrilateral PQRS, angle P = 60°. The remaining angles are in the ratio Q : R : S = 2 : 3 : 7. Find angle S.
• Step 1: The four angles sum to 360°. So Q + R + S = 360° − 60° = 300°.
• Step 2: Express the ratio. Let Q = 2x, R = 3x, S = 7x.
• Step 3: 2x + 3x + 7x = 300°, so 12x = 300°, giving x = 25°.
• Step 4: S = 7 × 25° = 175°.
• Answer: Angle S = 175°.

Example 2: Rhombus Area from Diagonals

• Problem: A rhombus has diagonals of 12 cm and 16 cm. Find its area.
• Step 1: Area of rhombus = (d1 × d2) / 2.
• Step 2: Area = (12 × 16) / 2 = 192 / 2 = 96 sq cm.
• Answer: 96 sq cm.

Example 3: Trapezium Area

• Problem: A trapezium has parallel sides of 9 cm and 15 cm and a height of 8 cm. Find its area.
• Step 1: Area of trapezium = (a + b) × h / 2.
• Step 2: Area = (9 + 15) × 8 / 2 = 24 × 8 / 2 = 192 / 2 = 96 sq cm.
• Answer: 96 sq cm.

Both Example 2 and Example 3 gave 96 sq cm but via different shapes and formulas. This is coincidental; do not infer a pattern. Worked examples are practice, not mnemonics.

Example 4: Rectangle Diagonal

• Problem: A rectangle has length 6 cm and breadth 8 cm. Find the length of its diagonal.
• Step 1: Diagonal of rectangle = √(l² + b²).
• Step 2: Diagonal = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
• Answer: 10 cm.
• Note: The 6-8-10 triple is a Pythagorean triple. The diagonal formula for a rectangle is simply the hypotenuse of the right triangle formed by the two sides.

Quadrilaterals in Campus Placement Tests

Quadrilateral questions appear in the Geometry sub-topic of the quantitative aptitude sections in TCS NQT, Infosys InfyTQ, and AMCAT. The IndiaBix Area section gives a reliable sample of the question types and difficulty bands these tests draw from.

Four patterns account for most placement test questions on this topic:

• Angle-sum problems: One or more angles given, find the missing one using 360°. Difficulty: easy to medium.
• Area from dimensions: Sides or diagonals given, apply the correct formula. Difficulty: easy once you know which formula to use for each shape.
• Property identification: “Which quadrilateral has perpendicular diagonals that bisect each other?” Tests whether you have memorised properties, not just formulas.
• Midline and isosceles trapezium: Questions about the midline segment, or about equal diagonals in an isosceles trapezium, appear at medium difficulty in AMCAT and Infosys InfyTQ.

The third and fourth types are where students lose marks. Area formulas are practised repeatedly; property questions and midline-based problems are skipped. The properties listed under each type above, specifically which diagonals are perpendicular and which bisect each other, cover the standard gap.

Geometry is one quantitative aptitude topic among several in these tests. Quantitative aptitude questions on Time and Work follow the same preparation pattern: one core method covers the main question variants. For a structured overview of what these tests assess end to end, the campus placement evaluation test article covers the full section-by-section breakdown. Students who want a curated list of books that cover aptitude geometry alongside the full quantitative syllabus can check placement preparation resources.

Clearing the geometry section in aptitude tests, with angle sums, area formulas, and diagonal properties memorised, handles one layer of the placement screen. TinkerLLM at ₹299 is where engineering students start building the AI-literacy layer on top of that foundation, the layer that placement evaluations are increasingly adding alongside the standard aptitude screen.