Polygons: Properties and Angles

TL;DR

Polygons are closed 2D shapes with straight sides, and their key properties involve the number of sides and the angles formed both inside and outside. You'll learn how to calculate the sum of interior angles, the value of a single interior angle for regular polygons, and the exterior angles. Understanding these formulas lets you work with any polygon's geometry.

1. The Mental Model

Think of polygons as a family of shapes, each defined by how many straight lines make up its boundary. The 'angles' part of the topic focuses on the corners of these shapes, both the ones inside and the ones you'd see if you extended a side.

2. The Core Material

Polygons are 2D shapes made of straight line segments joined end-to-end, forming a closed figure. They don't have curves, and the sides don't cross each other.

Types of Polygons

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Polygons are named by their number of sides:
* 3 sides: Triangle
* 4 sides: Quadrilateral
* 5 sides: Pentagon
* 6 sides: Hexagon
* 7 sides: Heptagon
* 8 sides: Octagon
* n sides: n-gon

Regular vs. Irregular Polygons

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• Regular Polygon: All sides are equal in length, and all interior angles are equal. (e.g., Square, Equilateral Triangle)
• Irregular Polygon: Sides and/or angles are not all equal. (e.g., Rectangle, Scalene Triangle)

Interior Angles

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These are the angles inside the polygon at each vertex (corner).

The sum of the interior angles of an n-sided polygon is given by the formula:
Sum of interior angles = (n - 2) * 180°

For a regular n-sided polygon, each interior angle is:
Each interior angle = [(n - 2) * 180°] / n

Exterior Angles

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These are the angles formed when you extend one side of the polygon. An interior angle and its adjacent exterior angle always add up to 180° because they form a straight line.

The sum of the exterior angles of any convex polygon (regular or irregular) is always:
Sum of exterior angles = 360°

For a regular n-sided polygon, each exterior angle is:
Each exterior angle = 360° / n

Relationship between Interior and Exterior Angles

Since an interior angle and its corresponding exterior angle form a linear pair (a straight line):
Interior Angle + Exterior Angle = 180°

This means if you know one, you can easily find the other!

Let's look at how these properties connect:

graph TD
    A["Number of Sides (n)"] --> B{"Is it a Regular Polygon?"}
    B -- Yes --> C["All Sides Equal"]
    B -- Yes --> D["All Interior Angles Equal"]
    B -- No --> E["Sides/Angles May Differ"]

    A --> F["Sum of Interior Angles: (n-2) * 180°"]
    A --> G["Sum of Exterior Angles: 360°"]

    D --> H["Each Interior Angle (Regular): [(n-2)*180°]/n"]
    G -- "For Regular" --> I["Each Exterior Angle (Regular): 360°/n"]

    H -- "Forms Linear Pair" --> I_link["Interior + Exterior = 180°"]
    I_link --> H

3. Worked Example

Let's work with a regular hexagon.

1. Number of sides (n): A hexagon has 6 sides, so n = 6.

2. Sum of interior angles:
   Using the formula: (n - 2) * 180°
   (6 - 2) * 180° = 4 * 180° = 720°

3. Each interior angle:
   Since it's a regular hexagon, all interior angles are equal.
   Using the formula: [(n - 2) * 180°] / n
   720° / 6 = 120°

4. Sum of exterior angles:
   For any polygon, the sum of exterior angles is always 360°.

5. Each exterior angle:
   Since it's a regular hexagon, all exterior angles are equal.
   Using the formula: 360° / n
   360° / 6 = 60°

6. Check relationship:
   Each Interior Angle + Each Exterior Angle = 180°
   120° + 60° = 180°. This confirms our calculations are correct!

4. Key Takeaways

• The number of sides, n, is the most crucial piece of information for polygon calculations.
• The formula (n - 2) * 180° gives you the total sum of all interior angles.
• For regular polygons, you can divide the total sum of angles by n to find each individual angle.
• The sum of exterior angles for any convex polygon is always 360°, which is a super handy shortcut.
• Remember that an interior angle and its adjacent exterior angle always add up to 180°.

Common Mistakes to Avoid

• Confusing sum of angles with individual angles: Don't use (n - 2) * 180° for a single angle without dividing by n (for regular polygons).
• Forgetting the 'n-2' in the interior angle formula: It's (n-2) triangles, not n triangles, that make up the interior.
• Applying regular polygon formulas to irregular polygons: You can't assume equal angles or sides unless specified as 'regular'.
• Incorrectly assuming exterior angle sum for irregular polygons: The 360° rule always applies for exterior angles of any convex polygon, not just regular ones.

5. Now Try It

Imagine you're designing a tile pattern using regular octagons. Your task is to figure out the angles needed.

1. Calculate the sum of the interior angles of a regular octagon.
2. Determine the measure of each interior angle of this regular octagon.
3. Calculate the measure of each exterior angle of this regular octagon.

Success looks like: You provide three correct angle values, showing your steps for each calculation using the formulas you just learned. You should get angles that make sense if added together (interior + exterior = 180°).