Mathematics • Year 7 • Unit 3 • Lesson 13
Angles in Polygons
Master the three polygon-angle formulas: interior sum S = (n − 2) × 180°, each regular interior = S ÷ n, and exterior angle sum always 360°. Spot which formula a question needs, substitute, and solve.
1. I do — fully worked example
The "find the missing angle in a polygon" question. Read each line and write the formula at the top of your page.
Problem. A pentagon (5-sided polygon) has interior angles 110°, 95°, 120°, 100° and x. Find x.
Step 1 — Write the formula for the interior angle sum.
S = (n − 2) × 180°
Reason: this works for any n-sided polygon.
Step 2 — Substitute n = 5.
S = (5 − 2) × 180° = 3 × 180° = 540°.
Reason: a pentagon splits into 3 triangles.
Step 3 — Equation: known angles + unknown = total.
110 + 95 + 120 + 100 + x = 540
425 + x = 540.
Step 4 — Solve.
x = 540 − 425 = 115°.
Answer: x = 115°.
2. We do — fill in the missing steps
A hexagon question — fill in the blanks. 4 marks
Problem. A regular hexagon (6 sides, all interior angles equal) — find each interior angle.
Step 1 — Find the total interior sum:
S = (n − 2) × 180° = (____ − 2) × 180° = ____ × 180° = ____°.
Step 2 — Each interior angle of a regular hexagon = total ÷ n:
each interior = ____ ÷ ____ = ____°.
Step 3 — Check using the exterior angle. Each exterior angle of a regular n-gon = 360° ÷ n = 360° ÷ ____ = ____°. Interior + exterior should be 180°: ____° + ____° = ____° ✓.
3. You do — independent practice
Show working under each problem.
Foundation — apply the formulas
3.1 Find the interior angle sum of a quadrilateral (4 sides). 1 mark
3.2 Find the interior angle sum of an octagon (8 sides). 1 mark
3.3 Each exterior angle of a regular polygon is 30°. The sum of exterior angles of any convex polygon is 360°. How many sides has it? 1 mark
3.4 Find each interior angle of a regular pentagon. 1 mark
Standard — find a missing angle
3.5 A quadrilateral has angles 85°, 100°, 95° and x. Find x. 2 marks
3.6 A hexagon has five angles 105°, 130°, 110°, 115° and 140°. Find the sixth angle. 2 marks
Extension — push your thinking
3.7 A regular polygon has each interior angle equal to 144°. (a) Find each exterior angle. (b) Find the number of sides. 3 marks
3.8 A regular decagon has 10 sides. (a) Find the sum of interior angles. (b) Find each interior angle. (c) Find each exterior angle. (d) Confirm interior + exterior = 180° at each vertex. 3 marks
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What I'll revisit before next class:
Section 2 — Regular hexagon
Step 1: S = (6 − 2) × 180° = 4 × 180° = 720°.
Step 2: each interior = 720° ÷ 6 = 120°.
Step 3: each exterior = 360° ÷ 6 = 60°. Check: 120° + 60° = 180° ✓.
3.1 — Quadrilateral interior sum
S = (4 − 2) × 180° = 2 × 180° = 360°.
3.2 — Octagon interior sum
S = (8 − 2) × 180° = 6 × 180° = 1080°.
3.3 — Each exterior = 30°
n = 360° ÷ 30° = 12 sides (regular dodecagon).
3.4 — Each interior of a regular pentagon
S = (5 − 2) × 180° = 540°. Each interior = 540° ÷ 5 = 108°.
3.5 — Quadrilateral with 85°, 100°, 95°, x
Sum = 360°. 85 + 100 + 95 + x = 360 → 280 + x = 360 → x = 80°.
3.6 — Hexagon 105°, 130°, 110°, 115°, 140°, x
Sum = 720°. 105 + 130 + 110 + 115 + 140 + x = 720 → 600 + x = 720 → x = 120°.
3.7 — Regular polygon, each interior 144°
(a) Each exterior = 180° − 144° = 36°.
(b) n = 360° ÷ 36° = 10 sides (regular decagon).
3.8 — Regular decagon (n = 10)
(a) S = (10 − 2) × 180° = 8 × 180° = 1440°.
(b) Each interior = 1440° ÷ 10 = 144°.
(c) Each exterior = 360° ÷ 10 = 36°.
(d) 144° + 36° = 180° ✓.