Regular and Irregular Polygons - Sixth Grade Mathematics
Polygons are two-dimensional, closed geometric figures formed by three or more straight line segments intersecting only at their endpoints.
What Defines a Polygon?
To qualify as a polygon, a geometric shape must satisfy three non-negotiable rules:
1. It must be a two-dimensional, closed planar figure (no open ends).
2. It must be composed exclusively of straight line segments (sides). No curves allowed.
3. Each line segment must intersect exactly two other segments, one at each endpoint (vertices), without crossing over itself.
Polygon vs. Non-Polygon:
Polygon: Not a Polygon (Curved): Not a Polygon (Open):
+------+ /------\ +------+
| | ( ) | |
| | \------/ |
+------+ +------+
Polygon Classification by Number of Sides
Polygons are named according to the Greek numerical prefixes indicating their count of sides and vertices.
+---------+---------------+------------------+-----------------------------+
| Sides | Polygon Name | Number of Angles | Sum of Interior Angles |
+---------+---------------+------------------+-----------------------------+
| 3 | Triangle | 3 | 180° |
| 4 | Quadrilateral | 4 | 360° |
| 5 | Pentagon | 5 | 540° |
| 6 | Hexagon | 6 | 720° |
| 7 | Heptagon | 7 | 900° |
| 8 | Octagon | 8 | 1080° |
| 9 | Nonagon | 9 | 1260° |
| 10 | Decagon | 10 | 1440° |
| 12 | Dodecagon | 12 | 1800° |
| n | n-gon | n | (n - 2) * 180° |
+---------+---------------+------------------+-----------------------------+
Regular vs. Irregular Polygons
Polygons are categorized by the equality of their sides and interior angles.
1. Regular Polygon:
- Equilateral: All sides have the exact same length.
- Equiangular: All interior angles have the exact same degree measure.
- Symmetrical and balanced in appearance.
Regular Hexagon (All sides equal, all angles = 120°):
+----------+
/ \
/ \
\ /
\ /
+----------+
2. Irregular Polygon:
- Does NOT have all sides congruent and all angles congruent.
- Can have sides of varying lengths and different angle measurements.
Convex vs. Concave Polygons
Polygons can also be classified based on whether their interior points "cave in." A convex polygon has no interior angles greater than 180 degrees. Any line segment connecting any two interior points lies entirely inside the polygon. A concave polygon has at least one interior angle greater than 180 degrees (a reflex angle). It "caves in," meaning at least one diagonal falls outside the shape.
Convex Polygon: Concave Polygon (Caves in at C):
B -------- C B
/ \ / \
/ \ / \ C (Reflex angle > 180°)
A -------------- D / \
A ----- D
Interior Angle Sum Formula
Any convex polygon with n sides can be divided into (n - 2) triangles by drawing diagonals from a single vertex. Since each triangle contains 180 degrees, the formula for the sum of interior angles S is: S = (n - 2) * 180 degrees. For a regular polygon with n sides, every interior angle is equal. The measure of each interior angle is: Individual Interior Angle = ((n - 2) * 180 degrees) / n.
Example: For a Regular Octagon (n = 8 sides):
Total Angle Sum = (8 - 2) * 180° = 6 * 180° = 1080°.
Each Interior Angle = 1080° / 8 = 135°.
Exercises
Exercise 1. State whether each figure is a polygon. If not, explain why: (a) A circle; (b) A five-pointed star; (c) A letter U; (d) An eight-sided stop sign.
Exercise 2. Calculate the sum of the interior angles of: (a) A pentagon (5 sides); (b) A decagon (10 sides); (c) A 15-gon.
Exercise 3. Find the measure of each individual interior angle in: (a) A regular hexagon; (b) A regular octagon; (c) A regular nonagon.
Exercise 4. The interior angles of a pentagon measure 105 degrees, 115 degrees, 120 degrees, and 90 degrees. What is the measure of the fifth angle? Is this pentagon regular or irregular?
Solutions
Solution to Exercise 1. For (a), a circle is not a polygon because it is formed by a smooth curve, not straight line segments. For (b), a five-pointed star is a polygon if traced along its perimeter as a 10-sided self-enclosing boundary without crossing internal lines. For (c), the letter U is not a polygon because it is open at the top and contains curved portions. For (d), a standard stop sign is an eight-sided regular octagon, which is a polygon.
Solution to Exercise 2. Using the formula S = (n - 2) * 180 degrees: For (a), pentagon (n = 5): S = (5 - 2) * 180 = 3 * 180 = 540 degrees. For (b), decagon (n = 10): S = (10 - 2) * 180 = 8 * 180 = 1440 degrees. For (c), 15-gon (n = 15): S = (15 - 2) * 180 = 13 * 180 = 2340 degrees.
Solution to Exercise 3. For (a), regular hexagon (n = 6): Total sum = (6 - 2) * 180 = 720 degrees. Each angle = 720 / 6 = 120 degrees. For (b), regular octagon (n = 8): Total sum = (8 - 2) * 180 = 1080 degrees. Each angle = 1080 / 8 = 135 degrees. For (c), regular nonagon (n = 9): Total sum = (9 - 2) * 180 = 7 * 180 = 1260 degrees. Each angle = 1260 / 9 = 140 degrees.
Solution to Exercise 4. The total sum of interior angles in a pentagon is (5 - 2) * 180 = 540 degrees. Sum the four known angles: 105 + 115 + 120 + 90 = 430 degrees. The fifth angle is 540 - 430 = 110 degrees. Because the angles are not all equal, the pentagon is irregular.