Angles in Polygons
The Skill
Angle Facts for Lines
- Angles on a straight line add up to $180°$
- Angles around a point add up to $360°$
- Vertically opposite angles are equal
Angle Facts for Parallel Lines
When a line crosses two parallel lines:
- Corresponding angles are equal (F-shape)
- Alternate angles are equal (Z-shape)
- Co-interior (allied) angles add up to $180°$ (C-shape or U-shape)
Interior Angles of Polygons
Sum of interior angles = $(n - 2) \times 180°$
where $n$ is the number of sides.
| Polygon | Sides | Interior Angle Sum |
|---|---|---|
| Triangle | 3 | $180°$ |
| Quadrilateral | 4 | $360°$ |
| Pentagon | 5 | $540°$ |
| Hexagon | 6 | $720°$ |
Regular Polygons
A regular polygon has all sides equal and all angles equal.
Each interior angle = $\frac{(n-2) \times 180}{n}$
Each exterior angle = $\frac{360}{n}$
Key fact: Interior angle + Exterior angle = $180°$
Exterior Angles
The sum of exterior angles of any polygon = $360°$
The Traps
Common misconceptions and how to avoid them.
Confusing interior and exterior angle formulas "The Inside-Out Error"
The Mistake in Action
Find the size of each interior angle of a regular pentagon.
Wrong: $\frac{360°}{5} = 72°$
Why It Happens
Students use $\frac{360°}{n}$, which gives the exterior angle, not the interior angle.
The Fix
$\frac{360°}{n}$ gives the EXTERIOR angle.
For the INTERIOR angle of a regular polygon, either:
Method 1: Calculate interior directly $$\text{Interior angle} = \frac{(n-2) \times 180°}{n} = \frac{3 \times 180°}{5} = \frac{540°}{5} = 108°$$
Method 2: Find exterior first, then subtract from 180° $$\text{Exterior} = \frac{360°}{5} = 72°$$ $$\text{Interior} = 180° - 72° = 108°$$
Spot the Mistake
Find each interior angle of a regular pentagon.
Interior angle $= \frac{360°}{5} = 72°$
Click on the line that contains the error.
Mixing up alternate and co-interior angles "The Z and C Confusion"
The Mistake in Action
Lines AB and CD are parallel. One angle is 70°.
Wrong: $x = 70°$ (treating co-interior as alternate)
Why It Happens
Students recognise a parallel lines setup but apply the wrong rule. Alternate angles (Z-shape) are equal, but co-interior angles (C/U-shape) add to 180°.
The Fix
Alternate angles (Z-shape): angles on opposite sides of the transversal → EQUAL
Co-interior angles (C-shape): angles on the same side of the transversal, between parallel lines → ADD TO 180°
If the angles are co-interior: $$x + 70° = 180°$$ $$x = 110°$$
Tip: Trace the shape with your finger. Z-shape = equal. C-shape = sum to 180°.
Spot the Mistake
Lines AB and CD are parallel. Angle ABC = 70°. Find angle BCD (co-interior).
ABC and BCD are co-interior angles
So angle BCD = 70°
Click on the line that contains the error.
Using 180° as the sum for all polygons "The Triangle Default"
The Mistake in Action
Find the sum of interior angles in a hexagon.
Wrong: $180°$
Why It Happens
Students remember that angles in a triangle sum to $180°$ and incorrectly apply this to all shapes.
The Fix
The sum of interior angles = $(n - 2) \times 180°$
For a hexagon, $n = 6$: $$(6 - 2) \times 180° = 4 \times 180° = 720°$$
Memory aid: A hexagon can be split into 4 triangles (draw lines from one vertex), and $4 \times 180° = 720°$.
Spot the Mistake
Find the sum of interior angles in a hexagon.
Sum = $180°$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Calculate the sum of the interior angles of an octagon.
Solution
An octagon has 8 sides, so $n = 8$.
Using the formula: $$\text{Sum} = (n - 2) \times 180°$$ $$= (8 - 2) \times 180°$$ $$= 6 \times 180°$$ $$= 1080°$$
Answer: 1080°
Question
Find the size of each interior angle of a regular nonagon (9 sides).
Solution
Method 1: Direct formula $$\text{Interior angle} = \frac{(n-2) \times 180°}{n}$$ $$= \frac{(9-2) \times 180°}{9}$$ $$= \frac{7 \times 180°}{9}$$ $$= \frac{1260°}{9}$$ $$= 140°$$
Method 2: Via exterior angle $$\text{Exterior angle} = \frac{360°}{9} = 40°$$ $$\text{Interior angle} = 180° - 40° = 140°$$
Answer: 140°
Question
The interior angle of a regular polygon is 156°. How many sides does it have?
Solution
Step 1: Find the exterior angle $$\text{Exterior angle} = 180° - 156° = 24°$$
Step 2: Use the exterior angle formula $$\frac{360°}{n} = 24°$$
Step 3: Solve for n $$n = \frac{360°}{24°} = 15$$
Answer: 15 sides (pentadecagon)
Level 2: Scaffolded
Fill in the key steps.
Question
Four angles of a pentagon are 110°, 95°, 120°, and 108°. Find the fifth angle.
Level 3: Solo
Try it yourself!
Question
Three angles of a quadrilateral are $x$, $2x$, and $3x$. The fourth angle is 120°. Find the value of $x$.
Show Solution
Sum of angles in a quadrilateral = $360°$
$$x + 2x + 3x + 120° = 360°$$ $$6x + 120° = 360°$$ $$6x = 240°$$ $$x = 40°$$
Answer: $x = 40°$
Examiner's View
Mark allocation: Basic angle calculations are 1-2 marks. Problems involving multiple steps or algebra are 3-4 marks.
Common errors examiners see:
- Using $180°$ instead of $(n-2) \times 180°$ for interior angle sums
- Confusing interior and exterior angles
- Not recognising parallel line angle relationships
- Arithmetic errors in multi-step problems
What gains marks:
- Stating the angle rule you're using
- Showing clear calculations
- Giving reasons for each step
AQA Notes
AQA often tests angle reasoning with "give a reason for your answer".