Geometry and Measures

Angles and Polygons – Revision Notes

Edexcel IGCSE Mathematics A | Geometry and Measures

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Key Definitions and Terminology

• **Polygon**: A closed 2D shape with straight sides. Named by the number of sides (e.g., pentagon = 5 sides, hexagon = 6 sides, octagon = 8 sides, decagon = 10 sides).
• **Regular Polygon**: A polygon where **all sides are equal in length** and **all interior angles are equal in size**.
• **Irregular Polygon**: A polygon where the sides and/or angles are not all equal.
• **Interior Angle**: The angle formed **inside** a polygon between two adjacent sides.
• **Exterior Angle**: The angle formed **outside** a polygon between one side and the extension of an adjacent side. At any vertex, the interior angle + exterior angle = 180°.
• **Convex Polygon**: A polygon where all interior angles are less than 180°. A **concave polygon** has at least one interior angle greater than 180° (a reflex angle).

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Main Concepts

1. Angle Sum of a Triangle

The sum of interior angles in any triangle is 180°.

2. Angle Sum of a Quadrilateral

The sum of interior angles in any quadrilateral is 360°.

3. Interior Angle Sum of Any Polygon

For a polygon with n sides, the sum of interior angles is:

> Sum of interior angles = (n − 2) × 180°

This works because any polygon can be divided into (n − 2) triangles by drawing diagonals from one vertex.

| Polygon | Sides (n) | Sum of Interior Angles |

|-----------|-----------|------------------------|

| Triangle | 3 | 180° |

| Quadrilateral | 4 | 360° |

| Pentagon | 5 | 540° |

| Hexagon | 6 | 720° |

| Octagon | 8 | 1080° |

| Decagon | 10 | 1440° |

4. Each Interior Angle of a Regular Polygon

Since all angles are equal in a regular polygon:

> Each interior angle = (n − 2) × 180° ÷ n

5. Exterior Angles of Any Convex Polygon

The sum of exterior angles of any convex polygon is always:

> Sum of exterior angles = 360°

For a regular polygon:

> Each exterior angle = 360° ÷ n

6. Relationship Between Interior and Exterior Angles

At each vertex:

> Interior angle + Exterior angle = 180°

This relationship is essential for switching between interior and exterior angle calculations.

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Worked Examples

Worked Example 1: Finding interior angles of a regular polygon

Question: Calculate the size of each interior angle of a regular nonagon (9 sides).

Solution:

• Method 1 (using interior angle formula):
• Sum of interior angles = (9 − 2) × 180° = 7 × 180° = **1260°**
• Each interior angle = 1260° ÷ 9 = **140°** ✓
• Method 2 (using exterior angles):
• Each exterior angle = 360° ÷ 9 = 40°
• Each interior angle = 180° − 40° = **140°** ✓

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Worked Example 2: Finding the number of sides from an angle

Question: The interior angle of a regular polygon is 156°. How many sides does the polygon have?

Solution:

• Step 1: Find the exterior angle.
• Exterior angle = 180° − 156° = **24°**
• Step 2: Use the exterior angle sum.
• Number of sides = 360° ÷ 24° = **15 sides** ✓

The polygon has 15 sides.

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Worked Example 3: Finding a missing angle in an irregular polygon

Question: A pentagon has interior angles of 108°, 120°, 95° and 117°. Find the fifth angle.

Solution:

• Sum of interior angles of a pentagon = (5 − 2) × 180° = **540°**
• Sum of four known angles = 108° + 120° + 95° + 117° = **440°**
• Missing angle = 540° − 440° = **100°** ✓

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Real-World Application

Regular polygons appear frequently in real life:

• **Stop signs** are regular octagons — each interior angle is (8 − 2) × 180° ÷ 8 = **135°**
• **Honeycomb cells** are regular hexagons — they tessellate perfectly because each interior angle is 120°, and 3 × 120° = 360°, which fills the space around a point exactly.

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Exam Technique Tips

1. **Always find the exterior angle first** when working with regular polygons. It is almost always faster to calculate 360° ÷ n for the exterior angle and then subtract from 180° to get the interior angle. Edexcel mark schemes frequently award a method mark (M1) for correctly using **360 ÷ n**, so even if your final answer has an arithmetic slip, showing this step earns credit.
2. **Check that your answer is reasonable.** Edexcel questions sometimes ask you to "show that" or "explain why." Remember: interior angles of regular polygons are always **less than 180°**, exterior angles are always **positive**, and the number of sides must be a **whole number**. If you calculate a number of sides and get a decimal (e.g., 7.3), go back and check your working — a polygon cannot have a fractional number of sides. State your reasoning clearly in words for "explain" questions, as the mark scheme requires a written conclusion.

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*Remember: Learn the key formulae — they are not given on the Edexcel IGCSE formula sheet!*