Mathematics • Year 7 • Unit 3 • Lesson 13
Polygon Angles — Mixed Challenge
All three formulas together: interior sum, regular polygon angle, and exterior sum 360°. Includes a classic Year 7 mistake (using (n − 2) × 180 in the wrong direction) and an open-ended polygon puzzle.
1. Mixed problems
Show working. State which formula you are using. 2 marks each
1.1 Find the interior angle sum of a 12-sided polygon (dodecagon).
1.2 Find each interior angle of a regular nonagon (9 sides).
1.3 A regular polygon has each exterior angle 24°. How many sides has it?
1.4 The interior angles of a hexagon are 120°, 110°, 130°, 140°, 100° and x. Find x.
1.5 A regular polygon has each interior angle 156°. Find the number of sides.
1.6 A pentagon has angles in the ratio 1 : 2 : 3 : 4 : 5. Find each angle.
2. Find the mistake
Exactly one step contains a mistake. Spot it, explain why it's wrong, then redo the working. 3 marks
Student's question: Find each interior angle of a regular octagon (8 sides).
Step 1: Use S = (n − 2) × 180°.
Step 2: S = (8 − 2) × 180° = 6 × 180° = 1080°.
Step 3: Each interior = S × n = 1080 × 8 = 8640°.
Step 4: Answer: each interior = 8640°.
(a) Which step contains the mistake?
(b) Explain in one or two sentences why that step is wrong.
(c) Write out the corrected working, including the corrected final answer.
Stuck? An interior angle of a polygon must be less than 180°. 8640° is impossible — what should you be doing instead of multiplying?3. Open-ended challenge — design a polygon
This question has many correct answers. Show your work. 4 marks
3.1 Design ANY non-regular hexagon (6 sides) whose interior angles you choose yourself. Your hexagon must satisfy ALL of the following:
• The six interior angles add to the correct sum for a hexagon (state the sum).
• No two interior angles are equal.
• Each interior angle is strictly between 60° and 200° (the hexagon may be slightly concave, with at most one angle greater than 180°).
• Show your six angles and prove they add to the correct sum.
Bonus: Sketch your hexagon (rough freehand is fine) and label each interior angle.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Dodecagon (n = 12) sum
S = (12 − 2) × 180° = 10 × 180° = 1800°.
1.2 — Regular nonagon (n = 9)
S = (9 − 2) × 180° = 7 × 180° = 1260°. Each interior = 1260° ÷ 9 = 140°.
1.3 — Each exterior 24°
n = 360° ÷ 24° = 15 sides (regular pentadecagon).
1.4 — Hexagon with five angles given
Sum = 720°. 120 + 110 + 130 + 140 + 100 + x = 720 → 600 + x = 720 → x = 120°.
1.5 — Regular polygon, each interior 156°
Exterior = 180° − 156° = 24°. n = 360° ÷ 24° = 15 sides.
1.6 — Pentagon ratio 1:2:3:4:5
Let angles be k, 2k, 3k, 4k, 5k. Sum = 15k = (5 − 2) × 180° = 540°.
15k = 540 → k = 36°.
Angles: 36°, 72°, 108°, 144°, 180°. Check sum: 36 + 72 + 108 + 144 + 180 = 540 ✓. (Note: 180° means three vertices are collinear — geometrically a degenerate pentagon, but the algebra is correct.)
2 — Find the mistake
(a) The mistake is on Step 3.
(b) Each interior angle of a regular polygon is the TOTAL DIVIDED by n, not multiplied. The student should have done S ÷ n, not S × n. Multiplying gives an impossibly huge number; an interior angle must be less than 180°.
(c) Corrected working:
Step 1: S = (n − 2) × 180°. ✓
Step 2: S = (8 − 2) × 180° = 1080°. ✓
Step 3 (fixed): Each interior = S ÷ n = 1080° ÷ 8 = 135°.
Step 4 (fixed): Each interior angle of a regular octagon = 135°.
3 — Design a hexagon (sample solution)
Sum for a hexagon = (6 − 2) × 180° = 720°.
Sample valid hexagon: 80°, 100°, 110°, 130°, 140°, 160°.
Check sum: 80 + 100 + 110 + 130 + 140 + 160 = 720° ✓.
All six angles different ✓. Each between 60° and 200° ✓.
Another sample (with a concave vertex): 70°, 90°, 105°, 115°, 145°, 195°.
Check: 70 + 90 + 105 + 115 + 145 + 195 = 720° ✓. All different ✓. The 195° vertex creates a "dent" — concave but still a 6-sided polygon.
Marking: 1 for stating 720°; 1 for six different angles summing to 720°; 1 for all in range; 1 for sketch/bonus. Many correct designs exist — accept any valid set.