Interior Angle Calculator
Calculate interior angle sum and individual angles for any polygon
Input Values
Results
Step-by-Step Solution
Formula: Sum = (n - 2) × 180°
Sum = (6 - 2) × 180°
Sum = 4 × 180°
Sum = 0.00°
Formula: Each angle = Sum / n
Each angle = 0.00° / 6
Each angle = 0.00°
Formula: Exterior = 360° / n
Exterior = 360° / 6
Exterior = 0.00°
Verification: Interior + Exterior = 0.00° + 0.00° = 180° ✓
Common Polygons Reference Table
| Sides | Name | Angle Sum | Each Interior Angle | Each Exterior Angle |
|---|---|---|---|---|
| 3 | Triangle | 180° | 60.00° | 120.00° |
| 4 | Quadrilateral | 360° | 90.00° | 90.00° |
| 5 | Pentagon | 540° | 108.00° | 72.00° |
| 6 | Hexagon | 720° | 120.00° | 60.00° |
| 7 | Heptagon | 900° | 128.57° | 51.43° |
| 8 | Octagon | 1080° | 135.00° | 45.00° |
| 9 | Nonagon | 1260° | 140.00° | 40.00° |
| 10 | Decagon | 1440° | 144.00° | 36.00° |
| 11 | Hendecagon | 1620° | 147.27° | 32.73° |
| 12 | Dodecagon | 1800° | 150.00° | 30.00° |
Understanding Interior Angles
What are Interior Angles?
Interior angles are the angles formed inside a polygon at each vertex where two sides meet. The sum of interior angles depends only on the number of sides, following the formula (n-2) × 180°, where n is the number of sides.
Why (n-2) × 180°?
This formula comes from dividing a polygon into triangles. Any n-sided polygon can be divided into (n-2) triangles by drawing diagonals from one vertex. Since each triangle has an angle sum of 180°, the total is (n-2) × 180°.
Example for Pentagon (5 sides):
Number of triangles: 5 - 2 = 3 triangles
Angle sum: 3 × 180° = 540°
Each angle in regular pentagon: 540° / 5 = 108°
Interior vs Exterior Angles
- Interior Angle: The angle inside the polygon at each vertex
- Exterior Angle: Formed by extending one side; supplementary to interior angle
- Relationship: Interior + Exterior = 180° (always)
- Exterior Sum: Always 360° regardless of number of sides
Regular vs Irregular Polygons
In a regular polygon, all interior angles are equal. Each angle = (n-2) × 180° / n.
In an irregular polygon, angles can vary, but their sum is still (n-2) × 180°.
Frequently Asked Questions
How do you find the interior angle sum?
Use the formula (n-2) × 180°, where n is the number of sides. This works for any polygon, regular or irregular. For example, a hexagon (6 sides) has angle sum (6-2) × 180° = 720°.
Why does the formula use (n-2)?
Any polygon can be divided into (n-2) triangles from a single vertex. Since each triangle has angles summing to 180°, the polygon's total is (n-2) × 180°. This geometric proof shows why the formula works universally.
What if the polygon is irregular?
The angle sum formula (n-2) × 180° applies to all polygons, regular or irregular. However, individual angles will vary in an irregular polygon. Only regular polygons have all angles equal to the sum divided by n.
Can an interior angle be greater than 180°?
In a convex polygon, all interior angles are less than 180°. In a concave polygon, some interior angles can exceed 180° (these are called reflex angles). The angle sum formula still applies to concave polygons.
How do interior and exterior angles relate?
At each vertex, the interior angle and exterior angle are supplementary, meaning they sum to 180°. This is because they form a linear pair on a straight line. Therefore, exterior angle = 180° - interior angle.
What polygon has interior angles of 120°?
A regular hexagon has interior angles of exactly 120°. Using the formula: (n-2) × 180° / n = 120°, solving gives n = 6. This is why hexagons tessellate perfectly (three 120° angles meet at each vertex to make 360°).
As sides increase, what happens to angles?
As the number of sides increases, each interior angle of a regular polygon approaches 180°. The polygon becomes more circular. For example, a 20-gon has angles of 162°, while a 100-gon has angles of 176.4°.
Can you have a 2-sided polygon?
No, a polygon must have at least 3 sides. A 2-sided figure would just be two line segments and couldn't enclose an area. The formula (n-2) × 180° would give 0° for n=2, which doesn't make geometric sense.