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.

Build · I Do / We Do / You Do

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.

Interior angle sum of a pentagon = (5 − 2) × 180° = 540°.

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°.

Stuck? Revisit lesson § "Interior Angle Sum", S = (n − 2) × 180° works because the polygon splits into (n − 2) triangles.

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°: ____° + ____° = ____° ✓.

Stuck? Revisit lesson § "Regular Polygons", each interior of a regular hexagon is 120°.

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

Stuck on 3.7? Interior + exterior = 180°, so exterior = 180 − 144 = 36°. Then n = 360 ÷ exterior.

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Answers, Do not peek before attempting

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° ✓.