Angles in Polygons
Understanding the angles inside shapes is a key part of geometry. This guide will show you how the properties of a simple triangle can help unlock the secrets of any polygon — a vital skill for architecture, design, and your GCSE Maths exam.
The Key Angle Facts for Polygons
1. Angle Sum of a Triangle
The interior angles in any triangle always add up to $180°$. This is the foundation for all other polygon angle rules!
2. Interior Angle Sum of Any Polygon
Split the polygon into triangles from one corner. The formula is:
where $n$ is the number of sides.
3. Exterior Angles
The sum of the exterior angles of any polygon is always $360°$. Think of walking around the shape and making one full turn.
At any vertex, the interior and exterior angles add up to $180°$.
4. Regular Polygons
In a regular polygon, all sides and angles are equal. To find one angle:
- One Interior Angle $= \frac{(n-2) \times 180°}{n}$
- One Exterior Angle $= \frac{360°}{n}$
Worked Examples
Example 1: Sum of Interior Angles
Find the sum of the interior angles of a hexagon.
- A hexagon has 6 sides, so $n = 6$.
- Sum $= (n – 2) \times 180° = (6 – 2) \times 180°$.
- $= 4 \times 180°$
Answer: $720°$
Example 2: Angles in a Regular Polygon
Find one interior angle of a regular octagon.
An octagon has 8 sides, so $n = 8$. Using the exterior angle method:
- Exterior angle $= 360° \div 8 = 45°$.
- Interior angle $= 180° – 45°$.
Answer: $135°$
Example 3: Working Backwards
The interior angle of a regular polygon is $140°$. How many sides does it have?
- Find the exterior angle first: $180° – 140° = 40°$.
- Use the exterior angle formula: $n = 360° \div \text{(one exterior angle)}$.
- $n = 360° \div 40° = 9$.
Answer: The polygon has 9 sides (a nonagon).
Tutor Insights
🤔 Common Misunderstandings
- Forgetting the $(n – 2)$: Mixing up the formula and just calculating $n \times 180°$ instead.
- Assuming all polygons are regular: The single-angle formulas only work for regular polygons where all angles are equal.
📝 Common Exam Mistakes
- Simple calculation errors when multiplying or dividing.
- Not showing your working. Write down the formula you are using to earn method marks.
- Confusing interior and exterior angle formulas. Remember: exterior angles always sum to $360°$, interior angles use $(n – 2) \times 180°$.
Practice Questions
- A triangle has angles of $45°$ and $75°$. What is the size of the third angle?
- Calculate the sum of the interior angles of a nonagon (9 sides).
- What is the size of one exterior angle of a regular octagon (8 sides)?
- An exterior angle of a regular polygon is $30°$. How many sides does it have?
Show Answers
- Working: $180° – (45° + 75°) = 180° – 120°$.
Answer: $60°$. - Working: $(9 – 2) \times 180° = 7 \times 180°$.
Answer: $1260°$. - Working: $360° \div 8$.
Answer: $45°$. - Working: $360° \div 30°$.
Answer: 12 sides.
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