POLYGONS

Introduction

A polygon is a 2-dimensional closed figure made up of three or more straight line segments. The word comes from Greek: "poly" meaning many, and "gonia" meaning angle. They are the fundamental shapes of geometry.

1. Types of Polygons

Polygons are classified by the number of sides (or angles) they have:
* 3 sides: Triangle
* 4 sides: Quadrilateral
* 5 sides: Pentagon
* 6 sides: Hexagon
* 7 sides: Heptagon
* 8 sides: Octagon
* 9 sides: Nonagon
* 10 sides: Decagon

• Regular Polygon: A polygon where all sides are equal in length (equilateral) AND all interior angles are equal in measure (equiangular). Example: A square.
• Irregular Polygon: A polygon where sides or angles are not all equal. Example: A rectangle (sides not all equal) or a rhombus (angles not all equal).
• Convex Polygon: All interior angles are less than $180^\circ$. No part of the polygon points "inward."
• Concave Polygon: At least one interior angle is greater than $180^\circ$ (a reflex angle). It looks like it has a "cave" in it.

MCQs - Types of Polygons

1. A polygon with 8 sides is called an:
   a) Hexagon
   b) Heptagon
   c) Octagon
   d) Nonagon
   Answer: c
2. Which of the following defines a regular polygon?
   a) Only all sides are equal
   b) Only all angles are equal
   c) All sides and all angles are equal
   d) It has an even number of sides
   Answer: c
3. A polygon with at least one interior angle greater than $180^\circ$ is a:
   a) Convex polygon
   b) Concave polygon
   c) Regular polygon
   d) Triangle
   Answer: b
4. Is a standard rectangle a regular polygon?
   a) Yes, because all angles are $90^\circ$
   b) Yes, because opposite sides are equal
   c) No, because not all sides are necessarily equal
   d) No, because its angles are not equal
   Answer: c
5. A polygon with 5 sides is a:
   a) Quadrilateral
   b) Pentagon
   c) Hexagon
   d) Decagon
   Answer: b
6. What is the name of a 7-sided polygon?
   a) Hexagon
   b) Heptagon
   c) Octagon
   d) Nonagon
   Answer: b
7. Which of the following shapes is typically an irregular polygon?
   a) Square
   b) Equilateral triangle
   c) Rectangle (where length != width)
   d) Regular hexagon
   Answer: c
8. A polygon with 4 sides is known as a:
   a) Triangle
   b) Quadrilateral
   c) Pentagon
   d) Hexagon
   Answer: b
9. What kind of polygon has no interior angles greater than $180^\circ$?
   a) Convex
   b) Concave
   c) Reflex
   d) Star
   Answer: a
10. A regular polygon must have:
    a) Equal sides but not necessarily equal angles
    b) Equal angles but not necessarily equal sides
    c) Both equal sides and equal angles
    d) An odd number of sides
    Answer: c

2. Properties of Polygons (Interior and Exterior Angles)

For any polygon with $n$ sides:
* Sum of Interior Angles: $S = (n - 2) \times 180^\circ$
* Example (Hexagon, $n=6$): $(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$.
* Each Interior Angle (for Regular Polygons only): $\frac{(n - 2) \times 180^\circ}{n}$
* Example (Regular Hexagon): $720^\circ / 6 = 120^\circ$.
* Sum of Exterior Angles: The sum of the exterior angles (one at each vertex) of ANY convex polygon is ALWAYS $360^\circ$, regardless of the number of sides.
* Each Exterior Angle (for Regular Polygons only): $\frac{360^\circ}{n}$
* Example (Regular Hexagon): $360^\circ / 6 = 60^\circ$.
* Note: At any vertex, Interior Angle + Exterior Angle = $180^\circ$ (they form a linear pair).

MCQs - Angles of Polygons

1. What is the sum of the interior angles of a pentagon ($n=5$)?
   a) $360^\circ$
   b) $540^\circ$
   c) $720^\circ$
   d) $900^\circ$
   Answer: b ($(5-2) \times 180 = 540$)
2. What is the sum of the exterior angles of a decagon ($n=10$)?
   a) $360^\circ$
   b) $1440^\circ$
   c) $1800^\circ$
   d) $3600^\circ$
   Answer: a (The sum of exterior angles is always $360^\circ$)
3. What is the measure of each interior angle in a regular octagon ($n=8$)?
   a) $120^\circ$
   b) $135^\circ$
   c) $144^\circ$
   d) $150^\circ$
   Answer: b ($(8-2) \times 180 / 8 = 1080 / 8 = 135$)
4. If each exterior angle of a regular polygon is $40^\circ$, how many sides does it have?
   a) 8
   b) 9
   c) 10
   d) 12
   Answer: b ($n = 360 / 40 = 9$)
5. At any vertex of a polygon, the interior angle and the exterior angle always add up to:
   a) $90^\circ$
   b) $180^\circ$
   c) $270^\circ$
   d) $360^\circ$
   Answer: b (They form a straight line)
6. What is the sum of the interior angles of a quadrilateral ($n=4$)?
   a) $180^\circ$
   b) $270^\circ$
   c) $360^\circ$
   d) $540^\circ$
   Answer: c ($(4-2) \times 180 = 360$)
7. What is the measure of each exterior angle of a regular pentagon ($n=5$)?
   a) $72^\circ$
   b) $108^\circ$
   c) $120^\circ$
   d) $144^\circ$
   Answer: a ($360 / 5 = 72$)
8. If the sum of the interior angles of a polygon is $900^\circ$, how many sides does it have?
   a) 5
   b) 6
   c) 7
   d) 8
   Answer: c ($(n-2) \times 180 = 900 \implies n=7$)
9. What is the measure of each interior angle in a regular hexagon ($n=6$)?
   a) $60^\circ$
   b) $108^\circ$
   c) $120^\circ$
   d) $135^\circ$
   Answer: c ($(6-2) \times 180 / 6 = 120$)
10. The sum of the exterior angles of a convex polygon:
    a) Increases as the number of sides increases
    b) Decreases as the number of sides increases
    c) Remains constant at $360^\circ$
    d) Remains constant at $180^\circ$
    Answer: c (Always $360^\circ$)

Fun Facts about Polygons!

• Bees are Geometers: Honeycombs built by bees are perfectly tessellated regular hexagons. Bees evolved to use hexagons because it is the most mathematically efficient shape to pack together tightly using the absolute minimum amount of wax!
• The Pentagon: The headquarters of the United States Department of Defense is famously shaped like a giant, regular five-sided polygon.
• Tessellation Limits: You can perfectly tile a floor (without leaving any gaps) using regular triangles, squares, and hexagons. But you cannot tile a floor using only regular pentagons or regular octagons!