Polygons: Types, Properties and Formulas for Aptitude Tests

Placement aptitude papers test polygon geometry through one formula applied repeatedly: the sum of interior angles of any polygon equals (n-2) multiplied by 180°, where n is the number of sides. Get this formula right and you handle almost every polygon question in TCS NQT, AMCAT, and Infosys InfyTQ before the rest of the section.

This article covers the full toolkit: what polygons are, how to classify them, the two core angle formulas, area and perimeter for common shapes, and four worked examples drawn from the question patterns that appear most often in placement papers.

What is a polygon

A polygon is a closed, flat figure formed entirely by straight-line segments connected end to end. Each segment is a side (also called an edge). Each point where two sides meet is a vertex (plural: vertices). The flat region enclosed is the interior.

Three properties define a polygon: it must be closed (the last side connects back to the first), it must be flat (a 2D plane figure, not 3D), and every boundary segment must be straight. A circle is not a polygon. An ellipse is not a polygon. Any shape with even one curved side does not qualify.

The minimum number of sides for a polygon is three, giving a triangle. Placement tests typically go up to 12 sides, though problems involving a 15-sided polygon (pentadecagon) occasionally appear in competitive aptitude sections. For all polygons, n is always at least 3.

Polygon types by number of sides

Every polygon has a standard name based on its side count. The table below covers the range that appears in placement aptitude papers, along with the angle values computed by applying Sum = (n-2) × 180° to each row.

Polygon	Sides (n)	Sum of Interior Angles	Each Interior Angle (Regular)
Triangle	3	180°	60°
Quadrilateral	4	360°	90°
Pentagon	5	540°	108°
Hexagon	6	720°	120°
Heptagon	7	900°	approximately 128.6°
Octagon	8	1,080°	135°
Nonagon	9	1,260°	140°
Decagon	10	1,440°	144°

The “Each Interior Angle (Regular)” column assumes all sides and angles are equal. For irregular polygons, individual angles vary, but their total always equals (n-2) × 180° regardless.

Among these, triangles, quadrilaterals, hexagons, and octagons appear most frequently in placement test geometry sections. The hexagon (120° per angle) and the octagon (135° per angle) are test favourites because their angle values are clean integers.

Regular, irregular, convex, and concave

These four terms classify polygons along two separate axes.

Regular vs irregular. A regular polygon has all sides equal in length and all interior angles equal. A square is a regular quadrilateral. A regular hexagon has six equal sides and six equal 120° angles, making it the shape of a honeycomb cell. An irregular polygon has at least one side or angle that differs from the others. A rectangle with sides 3 cm and 5 cm is irregular because its sides are unequal, even though all four angles are 90°. A scalene triangle is also irregular.

Convex vs concave. A convex polygon has all interior angles measuring less than 180°. No vertex points inward. Every regular polygon is convex. A concave polygon has at least one interior angle greater than 180° (called a reflex angle), making at least one vertex appear to point inward. The exterior-angle-sum formula of 360° applies to convex polygons. Placement test polygon questions almost exclusively involve regular or convex shapes.

Two rules follow from the definitions: all regular polygons are convex, and a polygon can be irregular yet still convex (a scalene triangle is both irregular and convex simultaneously).

Core angle formulas

Two formulas account for almost all polygon angle problems in placement papers.

Sum of interior angles

For a polygon with n sides:

Sum = (n-2) × 180°

This works because any polygon can be divided into exactly (n-2) non-overlapping triangles by drawing diagonals from a single vertex. Each triangle contributes 180° to the total angle sum.

Polygon	Calculation	Sum
Triangle	(3-2) × 180°	180°
Quadrilateral	(4-2) × 180°	360°
Pentagon	(5-2) × 180°	540°
Hexagon	(6-2) × 180°	720°
Octagon	(8-2) × 180°	1,080°

Each interior angle of a regular polygon

For regular polygons, divide the total angle sum evenly among n equal angles:

Each angle = (n-2) × 180° / n

For a regular hexagon: (6-2) × 180° / 6 = 720° / 6 = 120°. For a regular octagon: (8-2) × 180° / 8 = 1080° / 8 = 135°. Both values appear frequently as answer choices in placement papers.

Sum of exterior angles

For any convex polygon, regardless of the number of sides:

Sum of all exterior angles = 360°

This result is constant. A triangle’s exterior angles sum to 360°. A decagon’s exterior angles also sum to 360°. For a regular polygon, each exterior angle = 360° / n.

At each vertex of a convex polygon, the interior angle and the exterior angle are supplementary: interior + exterior = 180°. This relationship lets you jump between the two angle types without restarting a calculation.

Number of diagonals

For a polygon with n sides:

Diagonals = n(n-3) / 2

A hexagon has 6(6-3) / 2 = 9 diagonals. An octagon has 8(8-3) / 2 = 20 diagonals. This formula appears more often in Mu Sigma and DE Shaw aptitude papers than in standard service-tier tests, but it rounds out the toolkit.

Area and perimeter formulas

Polygon	Area (side = a)	Perimeter
Equilateral triangle	(√3 / 4) × a²	3a
Square	a²	4a
Regular hexagon	(3√3 / 2) × a²	6a
Regular octagon	2(1 + √2) × a²	8a

For the regular hexagon, a useful shortcut: dividing the hexagon into 6 equilateral triangles (each with side a) gives Area = 6 × (√3 / 4) × a², which simplifies to (3√3 / 2) × a². Both forms are equivalent; the simplified version is faster under timed conditions.

For the regular octagon, the formula 2(1 + √2) × a² has an approximate value of 4.828 × a². It can be re-derived by dividing the octagon into a central rectangle and isosceles right triangles at each corner.

Worked examples from placement papers

All four examples use the same four-step structure: state what is given, what is asked, which formula applies, then substitute.

Example 1: Angle sum for a 9-sided polygon

• Given: polygon with n = 9 (nonagon)
• Formula: Sum = (n-2) × 180°
• Substitute: (9-2) × 180° = 7 × 180° = 1,260°
• Answer: 1,260°

Example 2: Find n from a given interior angle

• Given: each interior angle of a regular polygon = 150°
• Formula: each angle = (n-2) × 180° / n
• Set up: 150 = (n-2) × 180 / n
• Cross-multiply: 150n = 180n - 360
• Rearrange: 360 = 30n
• Solve: n = 12 (a regular dodecagon)
• Verify: (12-2) × 180 / 12 = 10 × 180 / 12 = 1,800 / 12 = 150°

Example 3: Interior angle is 5 times the exterior angle

• Given: interior angle = 5 × exterior angle for a regular polygon
• Let exterior angle = x, so interior angle = 5x
• Relationship: interior + exterior = 180°
• Set up: 5x + x = 180°
• Solve: 6x = 180°, so x = 30° (exterior angle)
• Number of sides: n = 360° / 30° = 12

Example 4: Area of a regular hexagon with side 4 cm

• Given: regular hexagon with side a = 4 cm
• Formula: Area = (3√3 / 2) × a²
• Substitute: (3√3 / 2) × 16 = 24√3 cm²
• Answer: 24√3 cm² (approximately 41.57 cm²)

Polygon questions in placement tests

Polygon geometry appears across the major placement aptitude tests. The format and frequency depend on the company and the test level.

The TCS National Qualifier Test Numerical Ability section includes geometry questions drawing from polygon properties. Question patterns include calculating the angle sum for a polygon with a given n, finding n from a given interior angle, and applying the hexagon or octagon area formula. The section runs 26 questions in 40 minutes, so a 90-second budget per question applies.

The AMCAT Quantitative Ability module covers a broad geometry bank including polygon properties, angle sums, and area calculations alongside lines, circles, and coordinate geometry. Angle-sum problems appear more frequently in AMCAT polygon questions than area problems.

Infosys InfyTQ and Wipro NLTH both include geometry in their quantitative sections at a similar difficulty level to the examples above.

For a baseline on where polygon geometry sits in your overall quant performance, FACE Prep’s Campus Placement Evaluation Test gives you a percentile score across all major aptitude topics, including geometry. For polygon geometry and other quant topics like time and work, timed practice is more effective than repeated theory review. For structured study materials covering the full placement quant syllabus, the aptitude books guide for placement preparation covers options across budget and difficulty levels.

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