Polygon Angle Sums

Interior and exterior angle formulas

Polygon Angle Sums

Interior Angle Sum

For a polygon with nn sides: Sum of interior angles=(n−2)×180°\text{Sum of interior angles} = (n - 2) \times 180°

Examples:

  • Triangle (n=3n = 3): (3−2)×180°=180°(3 - 2) \times 180° = 180°
  • Quadrilateral (n=4n = 4): (4−2)×180°=360°(4 - 2) \times 180° = 360°
  • Pentagon (n=5n = 5): (5−2)×180°=540°(5 - 2) \times 180° = 540°
  • Hexagon (n=6n = 6): (6−2)×180°=720°(6 - 2) \times 180° = 720°

Regular Polygon

A polygon with all sides congruent and all angles congruent.

Measure of each interior angle: (n−2)×180°n\frac{(n - 2) \times 180°}{n}

Exterior Angle Sum

The sum of exterior angles (one at each vertex) of ANY polygon is always 360°360°.

Sum of exterior angles=360°\text{Sum of exterior angles} = 360°

Regular Polygon Exterior Angle

For a regular polygon: Each exterior angle=360°n\text{Each exterior angle} = \frac{360°}{n}

Finding Number of Sides

If you know the interior angle measure of a regular polygon: n=360°180°−interior anglen = \frac{360°}{180° - \text{interior angle}}

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the sum of the interior angles of a hexagon (6-sided polygon).

💡 Show Solution

Step 1: Recall the formula for sum of interior angles: Sum = (n - 2) × 180° where n = number of sides

Step 2: Identify n: Hexagon has 6 sides, so n = 6

Step 3: Substitute into formula: Sum = (6 - 2) × 180° Sum = 4 × 180° Sum = 720°

Step 4: Conceptual understanding: A hexagon can be divided into 4 triangles Each triangle has angles summing to 180° Total: 4 × 180° = 720° ✓

Answer: The sum of interior angles is 720°

2Problem 2easy

❓ Question:

Find the sum of the interior angles of an octagon.

💡 Show Solution

An octagon has n=8n = 8 sides.

Use the formula: Sum=(n−2)×180°\text{Sum} = (n - 2) \times 180°

=(8−2)×180°= (8 - 2) \times 180° =6×180°= 6 \times 180° =1080°= 1080°

Answer: 1080°1080°

3Problem 3easy

❓ Question:

Each interior angle of a regular pentagon measures the same. Find the measure of one interior angle.

💡 Show Solution

Step 1: Find the sum of all interior angles: Pentagon has n = 5 sides Sum = (n - 2) × 180° Sum = (5 - 2) × 180° Sum = 3 × 180° Sum = 540°

Step 2: Recall that a regular polygon has all angles equal

Step 3: Find one angle: One angle = Total sum / Number of angles One angle = 540° / 5 One angle = 108°

Step 4: Verify: 5 × 108° = 540° ✓

Answer: Each interior angle is 108°

4Problem 4medium

❓ Question:

Find the measure of each interior angle of a regular hexagon.

💡 Show Solution

A hexagon has n=6n = 6 sides.

Step 1: Find the sum of interior angles Sum=(6−2)×180°=720°\text{Sum} = (6 - 2) \times 180° = 720°

Step 2: Divide by number of angles (regular polygon) Each angle=720°6=120°\text{Each angle} = \frac{720°}{6} = 120°

Answer: Each interior angle is 120°120°

5Problem 5medium

❓ Question:

Find the measure of each exterior angle of a regular octagon.

💡 Show Solution

Step 1: Recall the exterior angle sum theorem: The sum of exterior angles of ANY polygon is always 360°

Step 2: For a regular polygon: All exterior angles are equal

Step 3: Find one exterior angle: Octagon has 8 sides, so 8 exterior angles One exterior angle = 360° / 8 One exterior angle = 45°

Step 4: Verify using interior angles (optional): Sum of interior angles = (8 - 2) × 180° = 1080° One interior angle = 1080° / 8 = 135° Exterior angle = 180° - 135° = 45° ✓

Answer: Each exterior angle is 45°

6Problem 6medium

❓ Question:

A polygon has an interior angle sum of 1800°. How many sides does it have?

💡 Show Solution

Step 1: Use the interior angle sum formula: Sum = (n - 2) × 180°

Step 2: Substitute the known sum: 1800 = (n - 2) × 180

Step 3: Solve for n: 1800/180 = n - 2 10 = n - 2 n = 12

Step 4: Verify: (12 - 2) × 180° = 10 × 180° = 1800° ✓

Step 5: Name the polygon: A 12-sided polygon is called a dodecagon

Answer: The polygon has 12 sides (dodecagon)

7Problem 7hard

❓ Question:

Each interior angle of a regular polygon measures 140°140°. How many sides does the polygon have?

💡 Show Solution

Method 1: Using exterior angles

Each exterior angle = 180°−140°=40°180° - 140° = 40°

Since exterior angles sum to 360°360°: n=360°40°=9n = \frac{360°}{40°} = 9

Method 2: Using interior angle formula (n−2)×180n=140\frac{(n-2) \times 180}{n} = 140

180n−360=140n180n - 360 = 140n 40n=36040n = 360 n=9n = 9

Answer: The polygon has 9 sides (nonagon)

8Problem 8hard

❓ Question:

Each exterior angle of a regular polygon measures 24°. Find: (a) the number of sides, (b) the sum of interior angles, and (c) each interior angle.

💡 Show Solution

Step 1: Find the number of sides using exterior angles: Sum of exterior angles = 360° For regular polygon: 360° / n = one exterior angle 360° / n = 24° n = 360° / 24° n = 15

The polygon has 15 sides

Step 2: Find sum of interior angles: Sum = (n - 2) × 180° Sum = (15 - 2) × 180° Sum = 13 × 180° Sum = 2340°

Step 3: Find each interior angle (Method 1): One interior angle = Sum / n One interior angle = 2340° / 15 One interior angle = 156°

Step 4: Verify using supplementary property: Interior + Exterior = 180° Interior = 180° - 24° = 156° ✓

Step 5: Additional verification: 15 × 156° = 2340° ✓ 360° / 15 = 24° ✓

Answer: (a) 15 sides, (b) sum = 2340°, (c) each interior angle = 156°

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