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Polygon Angle Sums

Interior and exterior angle formulas

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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 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°

2Problem 2medium

❓ 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°

3Problem 3hard

❓ 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)

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❓ Frequently Asked Questions

What is Polygon Angle Sums?▾
Interior and exterior angle formulas
How can I study Polygon Angle Sums effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
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What course covers Polygon Angle Sums?▾
Polygon Angle Sums is part of the Geometry course on Study Mondo, specifically in the Quadrilaterals and Polygons section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Polygon Angle Sums?▾
Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.