Angles in Polygons
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Polygon Vocabulary
- Polygon: a 2D shape made entirely of straight sides.
- Regular polygon: all sides and all angles equal.
- Irregular polygon: sides and/or angles are not all equal.
- Interior angle: the angle inside the polygon at each vertex.
- Exterior angle: the angle between a side and the extension of the next side - it lies on a straight line with the interior angle.
| Polygon | Sides |
|---|---|
| Triangle | 3 |
| Quadrilateral | 4 |
| Pentagon | 5 |
| Hexagon | 6 |
| Heptagon | 7 |
| Octagon | 8 |
| Nonagon | 9 |
| Decagon | 10 |
Sum of Interior Angles
Splitting a polygon into triangles from one vertex derives the formula: a polygon with n sides splits into (n − 2) triangles, and since each triangle's angles sum to 180°, the sum of interior angles = (n − 2) × 180°.
A hexagon splits into 4 triangles from one vertex: (6 − 2) × 180° = 720°
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Question: Find the sum of the interior angles of a hexagon (6 sides).
- Step 1: Sum = (6 − 2) × 180 = 4 × 180
- Answer: 720°
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Question: Find the sum of the interior angles of a decagon (10 sides).
- Step 1: Sum = (10 − 2) × 180 = 8 × 180
- Answer: 1440°
Interior Angles of Regular Polygons
In a regular polygon, every interior angle is equal, so each interior angle = (sum of interior angles) ÷ n.
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Question: Find each interior angle of a regular pentagon.
- Step 1: Sum = (5 − 2) × 180 = 540°
- Step 2: Each angle = 540 ÷ 5
- Answer: 108°
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Question: Find each interior angle of a regular octagon.
- Step 1: Sum = (8 − 2) × 180 = 1080°
- Step 2: Each angle = 1080 ÷ 8
- Answer: 135°
Sum of Exterior Angles
The exterior angles of any polygon - regular or irregular - always add up to 360°. This makes sense because walking all the way around the polygon turns you through one complete rotation.
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Question: A quadrilateral has exterior angles of 80°, 95°, 100°, and x. Find x.
- Step 1: 80 + 95 + 100 + x = 360
- Step 2: 275 + x = 360
- Answer: x = 85°
Exterior Angles of Regular Polygons
In a regular polygon, each exterior angle = 360° ÷ n.
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Question: Find each exterior angle of a regular hexagon.
- Step 1: Each exterior angle = 360 ÷ 6
- Answer: 60°
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Question: A regular polygon has each exterior angle equal to 24°. How many sides does it have?
- Step 1: n = 360 ÷ 24
- Answer: 15 sides
Interior and Exterior Angle Relationship
At each vertex, the interior and exterior angles lie on a straight line, so interior angle + exterior angle = 180°. This gives a useful cross-check or alternative method.
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Question: A regular polygon has an exterior angle of 40°. Find its interior angle and its number of sides.
- Step 1: Interior angle = 180 − 40 = 140°
- Step 2: Number of sides = 360 ÷ 40
- Answer: Interior angle = 140°, 9 sides
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Question: A regular polygon has an interior angle of 156°. Find its number of sides.
- Step 1: Find the exterior angle first: 180 − 156 = 24°
- Step 2: Number of sides = 360 ÷ 24
- Answer: 15 sides
Real-World Applications
Polygon angles appear in many real designs:
- Floor and wall tiling: hexagonal and other regular-polygon tiles.
- Road signs: the regular octagon of a stop sign.
- Engineering: hexagonal nuts and bolts.
- Architecture: pentagon and polygon-shaped buildings and windows.
- Tessellation art: patterns built from repeating regular polygons.
Exam Tips
Common Mistakes
MistakeUsing n instead of (n − 2) when finding the sum of interior angles
Fixalways subtract 2 from the number of sides before multiplying by 180°
MistakeDividing the sum of interior angles by n for an irregular polygon
Fixdividing by n to find "each angle" only works for regular polygons, where every angle is equal
MistakeConfusing the sum of exterior angles with the sum of interior angles
Fixexterior angles of ANY polygon always sum to 360°; interior angles depend on the number of sides
MistakeForgetting that interior and exterior angles sum to 180° at each vertex
Fixinterior angle + exterior angle = 180°, since they lie on a straight line
MistakeMiscounting the number of sides when naming a polygon
Fixcount carefully, or check a reference table of polygon names
MistakeTrying to find the number of sides directly from an interior angle, without finding the exterior angle first
Fixconvert the interior angle to its exterior angle (180° minus it) first, THEN divide 360° by that
For Exams
- Learn the formula: sum of interior angles = (n − 2) × 180°.
- Remember: exterior angles of any polygon always sum to 360°.
- For regular polygons: each interior angle = sum ÷ n; each exterior angle = 360° ÷ n.
- Use interior + exterior = 180° as a quick cross-check.
- Given an interior angle, convert to the exterior angle first, then divide 360° by it to find the number of sides.
- Show your working step by step, especially when finding the number of sides from an angle.
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