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.
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.