Polygon Notes for Class 8 CBSE - Ultimate Guide

Polygon Notes for Class 8 CBSE - Ultimate Guide with 55 Practice Questions

Introduction to Polygons

A polygon is a simple closed curve made up of only line segments. The word "polygon" comes from Greek words "poly" (meaning many) and "gon" (meaning angles). Polygons are fundamental shapes in geometry that form the building blocks for more complex figures.

Key Definitions

Regular Polygon: A polygon where all sides are equal and all angles are equal.
Irregular Polygon: A polygon where sides or angles are not all equal.
Convex Polygon: A polygon where all interior angles are less than 180°.
Concave Polygon: A polygon where at least one interior angle is greater than 180°.
Diagonal: A line segment connecting two non-adjacent vertices of a polygon.

Detailed Explanations of Polygon Types

Below, we explain each major type of polygon in detail, including properties, formulas, and examples.

Triangle (3 Sides)

A triangle is the simplest polygon with three sides and three angles. It's the building block for all polygons.

Properties: Sum of interior angles is 180°. Can be equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides different). All triangles are convex.
Formulas: Area = (base × height)/2; Perimeter = sum of sides.
Example: An equilateral triangle has each angle 60° and is regular.

Quadrilateral (4 Sides)

A quadrilateral has four sides and four angles, with sum of interior angles 360°. It can be divided into subtypes.

Square

A square is a regular quadrilateral with all sides equal and all angles 90°.

Properties: Opposite sides parallel, diagonals equal and bisect at 90°, four lines of symmetry.
Formulas: Area = side²; Perimeter = 4 × side; Diagonal = side × √2.
Example: A chessboard tile is a square.

Rectangle

A rectangle has opposite sides equal and all angles 90°.

Properties: Opposite sides parallel, diagonals equal and bisect each other, two lines of symmetry.
Formulas: Area = length × width; Perimeter = 2(length + width); Diagonal = √(length² + width²).
Example: A book cover is often a rectangle.

Parallelogram

A parallelogram has opposite sides parallel and equal.

Properties: Opposite angles equal, consecutive angles supplementary, diagonals bisect each other.
Formulas: Area = base × height; Perimeter = 2(length + width).
Example: A slanted rectangle, like a diamond shape.

Rhombus

A rhombus has all sides equal, opposite sides parallel.

Properties: Diagonals bisect at 90° and bisect angles, two lines of symmetry.
Formulas: Area = (diagonal1 × diagonal2)/2; Perimeter = 4 × side.
Example: A diamond in playing cards.

Trapezium (Trapezoid)

A trapezium has at least one pair of parallel sides.

Properties: Non-parallel sides are legs; isosceles if legs equal.
Formulas: Area = (sum of parallel sides × height)/2.
Example: A table with slanted legs.

Kite

A kite has two pairs of adjacent sides equal.

Properties: Diagonals intersect at 90°, one line of symmetry.
Formulas: Area = (diagonal1 × diagonal2)/2.
Example: A flying kite shape.

Pentagon (5 Sides)

A pentagon has five sides. Regular pentagons are common in nature.

Properties: Sum of interior angles 540°. Regular version has equal sides and angles (108° each).
Formulas: Number of diagonals = 5; Area of regular = (5/4) × side² × tan(54°).
Example: The Pentagon building in the USA.

Hexagon (6 Sides)

A hexagon has six sides. Regular hexagons tile perfectly.

Properties: Sum of interior angles 720°. Regular has 120° angles.
Formulas: Number of diagonals = 9; Area of regular = (3√3/2) × side².
Example: Honeycomb cells.

Heptagon (7 Sides)

A heptagon has seven sides. Less common, often irregular.

Properties: Sum of interior angles 900°. Regular angles ≈128.57°.
Formulas: Number of diagonals = 14.
Example: Some coins or architectural designs.

Octagon (8 Sides)

An octagon has eight sides. Regular ones are symmetric.

Properties: Sum of interior angles 1080°. Regular angles 135°.
Formulas: Number of diagonals = 20; Area of regular = 2 × (1 + √2) × side².
Example: Stop signs.

Nonagon (9 Sides)

A nonagon has nine sides.

Properties: Sum of interior angles 1260°. Regular angles 140°.
Formulas: Number of diagonals = 27.
Example: Some religious symbols.

Decagon (10 Sides)

A decagon has ten sides.

Properties: Sum of interior angles 1440°. Regular angles 144°.
Formulas: Number of diagonals = 35.
Example: Some stars or coins.

Hendecagon (11 Sides)

A hendecagon has eleven sides, often called an 11-gon.

Properties: Sum of interior angles 1620°. Regular angles ≈147.27°.
Formulas: Number of diagonals = 44.
Example: Rare in everyday objects.

Dodecagon (12 Sides)

A dodecagon has twelve sides.

Properties: Sum of interior angles 1800°. Regular angles 150°.
Formulas: Number of diagonals = 54.
Example: Some clocks or tiles.

For polygons with more than 12 sides, they are generally called "n-gon" where n represents the number of sides.

Important Polygon Formulas

1. Sum of Interior Angles

\( \text{Sum of Interior Angles} = (n-2) \times 180^\circ \)

2. Each Interior Angle of Regular Polygon

\( \text{Each Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \)

3. Each Exterior Angle of Regular Polygon

\( \text{Each Exterior Angle} = \frac{360^\circ}{n} \)

4. Sum of Exterior Angles

\( \text{Sum of Exterior Angles} = 360^\circ \)

5. Number of Diagonals

\( \text{Number of Diagonals} = \frac{n(n-3)}{2} \)

Properties of Polygons

A polygon must have at least 3 sides
The sum of exterior angles is always 360° regardless of the number of sides
In a regular polygon, all interior angles are equal and all exterior angles are equal
Each exterior angle and its corresponding interior angle are supplementary (sum to 180°)

Practice Questions with Solutions

Section A: Basic Concepts (Questions 1-5)

Q1. What is a polygon?

Solution: A polygon is a simple closed curve made up of only line segments.

Q2. Name the polygon with 7 sides.

Solution: Heptagon (or Septagon)

Q3. What is the difference between regular and irregular polygons?

Solution: Regular polygons have all sides equal and all angles equal. Irregular polygons do not have all sides and angles equal.

Q4. What is the difference between convex and concave polygons?

Solution: Convex polygons have all interior angles less than 180°. Concave polygons have at least one interior angle greater than 180°.

Q5. What is a diagonal of a polygon?

Solution: A diagonal is a line segment that connects two non-adjacent vertices of a polygon.

Section B: Sum of Interior Angles (Questions 6-15)

Q6. Find the sum of interior angles of a triangle.

Solution: Sum = (n-2) × 180° = (3-2) × 180° = 1 × 180° = 180°

Q7. Find the sum of interior angles of a quadrilateral.

Solution: Sum = (n-2) × 180° = (4-2) × 180° = 2 × 180° = 360°

Q8. Find the sum of interior angles of a pentagon.

Solution: Sum = (n-2) × 180° = (5-2) × 180° = 3 × 180° = 540°

Q9. Find the sum of interior angles of a hexagon.

Solution: Sum = (n-2) × 180° = (6-2) × 180° = 4 × 180° = 720°

Q10. Find the sum of interior angles of an octagon.

Solution: Sum = (n-2) × 180° = (8-2) × 180° = 6 × 180° = 1080°

Q11. Find the sum of interior angles of a nonagon.

Solution: Sum = (n-2) × 180° = (9-2) × 180° = 7 × 180° = 1260°

Q12. Find the sum of interior angles of a decagon.

Solution: Sum = (n-2) × 180° = (10-2) × 180° = 8 × 180° = 1440°

Q13. Find the sum of interior angles of a heptagon.

Solution: Sum = (n-2) × 180° = (7-2) × 180° = 5 × 180° = 900°

Q14. Find the sum of interior angles of a dodecagon.

Solution: Sum = (n-2) × 180° = (12-2) × 180° = 10 × 180° = 1800°

Q15. Find the sum of interior angles of a hendecagon.

Solution: Sum = (n-2) × 180° = (11-2) × 180° = 9 × 180° = 1620°

Section C: Interior Angles of Regular Polygons (Questions 16-25)

Q16. Find each interior angle of a regular triangle.

Solution: Each interior angle = [(n-2) × 180°]/n = [(3-2) × 180°]/3 = 60°

Q17. Find each interior angle of a regular quadrilateral (square).

Solution: Each interior angle = [(n-2) × 180°]/n = [(4-2) × 180°]/4 = 90°

Q18. Find each interior angle of a regular pentagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(5-2) × 180°]/5 = 108°

Q19. Find each interior angle of a regular hexagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(6-2) × 180°]/6 = 120°

Q20. Find each interior angle of a regular octagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(8-2) × 180°]/8 = 135°

Q21. Find each interior angle of a regular heptagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(7-2) × 180°]/7 ≈ 128.57°

Q22. Find each interior angle of a regular nonagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(9-2) × 180°]/9 = 140°

Q23. Find each interior angle of a regular decagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(10-2) × 180°]/10 = 144°

Q24. Find each interior angle of a regular hendecagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(11-2) × 180°]/11 ≈ 147.27°

Q25. Find each interior angle of a regular dodecagon.

Solution: Each interior angle = [(n-2) × 180°]/n = [(12-2) × 180°]/12 = 150°

Section D: Exterior Angles (Questions 26-35)

Q26. Find each exterior angle of a regular triangle.

Solution: Each exterior angle = 360°/n = 360°/3 = 120°

Q27. Find each exterior angle of a regular quadrilateral.

Solution: Each exterior angle = 360°/n = 360°/4 = 90°

Q28. Find each exterior angle of a regular pentagon.

Solution: Each exterior angle = 360°/n = 360°/5 = 72°

Q29. Find each exterior angle of a regular hexagon.

Solution: Each exterior angle = 360°/n = 360°/6 = 60°

Q30. Find each exterior angle of a regular octagon.

Solution: Each exterior angle = 360°/n = 360°/8 = 45°

Q31. Find each exterior angle of a regular heptagon.

Solution: Each exterior angle = 360°/n = 360°/7 ≈ 51.43°

Q32. Find each exterior angle of a regular nonagon.

Solution: Each exterior angle = 360°/n = 360°/9 = 40°

Q33. Find each exterior angle of a regular decagon.

Solution: Each exterior angle = 360°/n = 360°/10 = 36°

Q34. Find each exterior angle of a regular hendecagon.

Solution: Each exterior angle = 360°/n = 360°/11 ≈ 32.73°

Q35. Find each exterior angle of a regular dodecagon.

Solution: Each exterior angle = 360°/n = 360°/12 = 30°

Section E: Number of Diagonals (Questions 36-45)

Q36. Find the number of diagonals in a quadrilateral.

Solution: Number of diagonals = n(n-3)/2 = 4(4-3)/2 = 4×1/2 = 2

Q37. Find the number of diagonals in a pentagon.

Solution: Number of diagonals = n(n-3)/2 = 5(5-3)/2 = 5×2/2 = 5

Q38. Find the number of diagonals in a hexagon.

Solution: Number of diagonals = n(n-3)/2 = 6(6-3)/2 = 6×3/2 = 9

Q39. Find the number of diagonals in a heptagon.

Solution: Number of diagonals = n(n-3)/2 = 7(7-3)/2 = 7×4/2 = 14

Q40. Find the number of diagonals in an octagon.

Solution: Number of diagonals = n(n-3)/2 = 8(8-3)/2 = 8×5/2 = 20

Q41. Find the number of diagonals in a nonagon.

Solution: Number of diagonals = n(n-3)/2 = 9(9-3)/2 = 9×6/2 = 27

Q42. Find the number of diagonals in a decagon.

Solution: Number of diagonals = n(n-3)/2 = 10(10-3)/2 = 10×7/2 = 35

Q43. Find the number of diagonals in a hendecagon.

Solution: Number of diagonals = n(n-3)/2 = 11(11-3)/2 = 11×8/2 = 44

Q44. Find the number of diagonals in a dodecagon.

Solution: Number of diagonals = n(n-3)/2 = 12(12-3)/2 = 12×9/2 = 54

Q45. Find the number of diagonals in a triangle.

Solution: Number of diagonals = n(n-3)/2 = 3(3-3)/2 = 3×0/2 = 0

Section F: Advanced Problems (Questions 46-55)

Q46. If each exterior angle of a regular polygon is 45°, find the number of sides.

Solution: Number of sides = 360°/exterior angle = 360°/45° = 8 sides

Q47. If each interior angle of a regular polygon is 120°, find the number of sides.

Solution: Each exterior angle = 180° - 120° = 60°
Number of sides = 360°/60° = 6 sides

Q48. A polygon has 20 diagonals. Find the number of sides.

Solution: \( \frac{n(n-3)}{2} = 20 \)
\( n(n-3) = 40 \)
\( n^2 - 3n - 40 = 0 \)
\( (n-8)(n+5) = 0 \)
n = 8 (taking positive value)
So the polygon has 8 sides.

Q49. If the sum of interior angles of a polygon is 1440°, find the number of sides.

Solution: \( (n-2) \times 180^\circ = 1440^\circ \)
\( n-2 = \frac{1440^\circ}{180^\circ} = 8 \)
n = 10
So the polygon has 10 sides.

Q50. Can a regular polygon have each exterior angle equal to 50°?

Solution: For a regular polygon to exist, 360° must be divisible by the exterior angle.
\( 360^\circ \div 50^\circ = 7.2 \)
Since this is not a whole number, such a polygon cannot exist.

Q51. The ratio of interior angle to exterior angle of a regular polygon is 3:1. Find the number of sides.

Solution: Let exterior angle = x, then interior angle = 3x
Interior angle + Exterior angle = 180°
3x + x = 180°
4x = 180°
x = 45°
Number of sides = 360°/45° = 8 sides

Q52. If each interior angle of a regular polygon exceeds each exterior angle by 60°, find the number of sides.

Solution: Let exterior angle = x
Interior angle = x + 60°
Interior angle + Exterior angle = 180°
(x + 60°) + x = 180°
2x = 120°
x = 60°
Number of sides = 360°/60° = 6 sides

Q53. What is the minimum number of sides a polygon can have?

Solution: A polygon must have at least 3 sides. A polygon with 3 sides is called a triangle.

Q54. What is the maximum exterior angle possible for a regular polygon?

Solution: The maximum exterior angle occurs in a triangle (3-sided polygon).
Maximum exterior angle = 360°/3 = 120°

Q55. What is the minimum interior angle possible for a regular polygon?

Solution: The minimum interior angle occurs in a triangle (3-sided polygon).
Minimum interior angle = 180° - 120° = 60°

Summary Table

Polygon	Sides (n)	Sum of Interior Angles	Each Interior Angle (Regular)	Each Exterior Angle (Regular)	Number of Diagonals
Triangle	3	180°	60°	120°	0
Quadrilateral	4	360°	90°	90°	2
Pentagon	5	540°	108°	72°	5
Hexagon	6	720°	120°	60°	9
Heptagon	7	900°	≈128.57°	≈51.43°	14
Octagon	8	1080°	135°	45°	20
Nonagon	9	1260°	140°	40°	27
Decagon	10	1440°	144°	36°	35
Hendecagon	11	1620°	≈147.27°	≈32.73°	44
Dodecagon	12	1800°	150°	30°	54

Key Points to Remember

The sum of exterior angles of any polygon is always 360°
Interior angle + Exterior angle = 180° (for any polygon)
Only regular polygons with exterior angles that divide 360° evenly can exist
The minimum number of sides for any polygon is 3
Diagonals connect non-adjacent vertices only

This comprehensive guide covers all essential concepts about polygons for Class 8 CBSE students, with detailed explanations, 55 practice questions, and solutions to help master the topic.

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