Key takeaways
- The sum of a polygon's interior angles is (n − 2) × 180°.
- n is the number of sides (and corners) the polygon has.
- A triangle's angles always add to 180°; a hexagon's always add to 720°.
- For a regular polygon, divide the sum by n to get each individual angle.
In this guide
What are interior angles of a polygon?
A polygon is any closed shape with straight sides, like a triangle, square, or hexagon. Its interior angles are the angles inside the shape, at each corner where two sides meet.
Why does the sum of interior angles matter?
It lets you predict a shape's total angle before you even measure a single corner — useful for design, construction, and checking your work.
A stop sign is a regular octagon (n = 8). Its interior angles always sum to(8 − 2) × 180° = 1,080°, and since it's regular, each individual corner is exactly1,080° ÷ 8 = 135°— every stop sign in the world, guaranteed.
The formula
| Symbol | What it means | Example |
|---|---|---|
| Sum | The total of all interior angles, in degrees | 1,080° |
| n | The number of sides (and corners) | 8 |
How to find the sum step by step
- 1Count the number of sides. This is n.
- 2Subtract 2 from n.
- 3Multiply the result by 180°.
- 4If the polygon is regular, divide the sum by n to find each individual angle.
Now try it yourself. Change the number of sides in the calculator below and watch the total (and each angle) update, with every step shown.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting this only gives the total, not each angle. For an irregular polygon, individual angles can differ even though they still add up to the same total.
- Confusing interior angles with exterior angles. Exterior angles of any polygon always add up to 360°, which is a completely different formula.
- Using this formula on a curved or open shape. It only applies to closed, straight-sided polygons.
Tips and tricks
- For a regular polygon, always divide the sum by n at the end to get one clean interior angle.
- Working backwards from one known regular angle? Multiply by n to get the sum, then solve
(n − 2) × 180° = sumfor n.
Where you'll use it in real life
- Design: creating tiling patterns with regular polygons that fit together perfectly.
- Quality checking: verifying a drawn or built shape's angles are consistent.
- Signage: confirming the angles for a stop sign (octagon) or similar regular shape.
- Architecture: checking angles for honeycomb-style hexagonal panel layouts.
Quick summary
To find the sum of a polygon's interior angles, subtract 2 from the number of sides and multiply by 180°. For a regular polygon, divide that total by the number of sides to get each individual angle. Use the calculator above to check your own polygon instantly.
Worked example
You're designing a stop-sign shape, which is a regular octagon with 8 equal sides. What is the total of all its interior angles, and how big is each one?
- 01Sum = (n - 2) × 180°
- 02Sum = (8 - 2) × 180°
- 03Sum = 6 × 180°
- 04Sum = 1080°
- 05Each angle (if regular) = 1080 ÷ 8 = 135°
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the sum of interior angles of a triangle?
For a triangle, n = 3, so the sum is (3 - 2) × 180° = 180°, the well-known rule that a triangle's angles always add up to 180°.
How do I find each individual angle of a regular polygon?
First find the total sum using (n - 2) × 180°, then divide that total by n, since a regular polygon has all equal angles.
Does this formula work for irregular polygons too?
Yes, the total sum formula works for any polygon with n straight sides, regular or irregular. Only dividing by n to get one angle requires the polygon to be regular.
What's the sum of interior angles of a hexagon?
For a hexagon, n = 6, so the sum is (6 - 2) × 180° = 720°, and each angle in a regular hexagon is 720° ÷ 6 = 120°.
