Review: interior and exterior angles of polygons
key notes:
🧩 What is a Polygon?
- A polygon is a closed figure made of straight line segments.
Examples: triangle, quadrilateral, pentagon, hexagon.
🟡 Interior Angles
- Interior angles are the angles inside a polygon. 🌟
- Sum of interior angles formula:
(n−2) × 180∘
👉 Here, n = number of sides
✨ Examples:
| Polygon | n | Interior Sum |
|---|---|---|
| Triangle | 3 | (3−2)×180° = 180° |
| Quadrilateral | 4 | (4−2)×180° = 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
🔵 Exterior Angles
- Formed outside the polygon when a side is extended.
- Each exterior angle forms a linear pair with an interior angle.
🧠 Most important rule:
Sum of all exterior angles of any polygon=360∘
💡 This works for ALL polygons (regular or irregular)!
🟣 Regular Polygons
A regular polygon has all sides and angles equal.
- One interior angle (regular polygon):
(n−2)×180∘ / n
- One exterior angle (regular polygon):
360∘ / n
🌟 Example:
Regular Hexagon → n = 6
Interior angle = 720° ÷ 6 = 120°
Exterior angle = 360° ÷ 6 = 60°
🔻 Interior + Exterior Angle Relationship
For every vertex in a polygon:
Interior angle+Exterior angle=180∘
(Because they form a linear pair 🔁)
🧠 Quick Memory Tricks
| Concept | Trick |
|---|---|
| Interior Sum | “(Sides − 2) × 180” 🎯 |
| Exterior Sum | “Always 360°” 🔄 |
| One exterior (regular) | 360° ÷ sides 📏 |
| Int. + Ext. | Straight line = 180° ➡️ |
✅ Fast Practice Check
- Sum of interior angles of a 7-sided polygon?
→ (7−2)×180 = 900° - Exterior angle of a regular decagon (10 sides)?
→ 360° ÷ 10 = 36°
Learn with an example
✈️ The diagram shows a convex polygon.
✈️ What is the sum of the interior angle measures of this polygon?
_______°
Look at this convex polygon.
This polygon is a triangle, so the sum of the interior angles is 180°.
✈️ The diagram shows a convex polygon.
✈️ What is the sum of the interior angle measures of this polygon?
_______°
Look at this convex polygon.
To find out how many triangles make up this polygon, pick a vertex and draw all the diagonals from that vertex.
- This polygon is made up of 3 triangles. Since the sum of the interior angle measures of each triangle is 180°, the sum of the interior angle measures of this polygon is 3 . 180° = 540°.
- Notice that this polygon has 5 sides and is made up of 3 triangles. In general, if a convex polygon has n sides it is made up of (n – 2) triangles.
- In other words, the sum of the interior angle measures of a convex polygon with n sides is (n – 2). 180°.
✈️ The diagram shows a convex polygon.
✈️ What is the sum of the exterior angle measures, one at each vertex, of this polygon?
_______°
Look at this convex polygon.
Since this polygon is convex, the sum of its exterior angle measures, one at each vertex, is 360°. The number of sides is not relevant.
let’s practice! 🖊️
