Polygon Interior Angle Sum Calculator – Find Angle Sum of Polygon
Calculate the sum of interior angles of any polygon with our free online calculator. Enter the number of sides and instantly find the total interior angle sum and each angle for regular polygons.
Enter 3 or greater. For regular polygons, each angle will also be calculated.
Understanding Polygon Interior Angles
A polygon is any closed shape with straight sides – triangles, squares, pentagons, and so on. The interior angles are the angles inside the polygon at each vertex. No matter how many sides a polygon has, there's a simple formula to find the sum of all its interior angles.
The formula (n - 2) × 180° works because any polygon can be divided into (n - 2) triangles by drawing diagonals from one vertex. Since each triangle has 180°, multiply by the number of triangles. This calculator shows you the total sum, plus each individual angle if the polygon is regular (all sides and angles equal).
The Formula Explained
Where n is the number of sides (or vertices) of the polygon.
Why This Formula Works
Pick any vertex of a polygon. Draw diagonals from that vertex to all other non-adjacent vertices. This divides the polygon into triangles. A polygon with n sides creates exactly (n - 2) triangles. Since each triangle contributes 180° to the total, the sum is (n - 2) × 180°.
Each Interior Angle (Regular Polygon)
For regular polygons (all sides equal, all angles equal), divide the total sum by the number of angles to find each one.
Each Exterior Angle (Regular Polygon)
The exterior angles of any polygon always sum to 360°. For regular polygons, divide 360° by n to find each exterior angle.
Common Polygons Reference
| Sides | Name | Interior Angle Sum | Each Angle (Regular) | 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° |
| 12 | Dodecagon | 1800° | 150.00° | 30.00° |
| 15 | Pentadecagon | 2340° | 156.00° | 24.00° |
| 20 | Icosagon | 3240° | 162.00° | 18.00° |
Polygon Properties
Regular vs Irregular Polygons
Regular polygons have all sides equal and all angles equal (equilateral triangle, square, regular pentagon). Irregular polygons have sides and/or angles of different measures.
Convex vs Concave Polygons
Convex polygons have all interior angles less than 180° – no "dents" or inward-pointing vertices. Concave polygons have at least one interior angle greater than 180°.
Number of Diagonals
A diagonal connects two non-adjacent vertices. The formula for the number of diagonals in an n-sided polygon is:
Each vertex connects to (n - 3) other vertices via diagonals (not itself or its two neighbors). Multiply by n vertices, then divide by 2 since each diagonal is counted twice.
Exterior Angle Sum
The exterior angles of any polygon (one at each vertex, measured by extending each side) always sum to exactly 360° – regardless of the number of sides.
Worked Examples
Example 1: Hexagon
Find the sum of interior angles of a hexagon (6 sides)
Example 2: Decagon
Find the sum of interior angles of a decagon (10 sides)
Example 3: Finding Sides from Angle Sum
A regular polygon has interior angles of 150°. How many sides?
Example 4: Using Exterior Angles
A regular polygon has exterior angles of 30°. How many sides?
Frequently Asked Questions
What is the sum of interior angles of a polygon?
The sum is (n - 2) × 180°, where n is the number of sides. A triangle (3 sides) has 180°, a quadrilateral (4 sides) has 360°, a pentagon has 540°, and so on. Each additional side adds 180° to the sum.
Why does the formula use (n - 2)?
Any polygon can be divided into triangles by drawing diagonals from one vertex. An n-sided polygon creates exactly (n - 2) triangles. Since each triangle has 180°, multiply (n - 2) by 180°.
Does this work for irregular polygons?
Yes. The sum depends only on the number of sides, not whether the polygon is regular or irregular. A irregular hexagon still has interior angles summing to 720° – they're just not all equal.
What about concave polygons?
The formula still works. Even if a polygon has "dents" (interior angles greater than 180°), the total sum is still (n - 2) × 180°. The formula counts the total angular measure, regardless of individual angle sizes.
How do I find the number of sides from the angle sum?
Rearrange the formula: if Sum = (n - 2) × 180°, then n = (Sum / 180°) + 2. For example, if the sum is 1080°, then n = (1080 / 180) + 2 = 6 + 2 = 8 sides.
What is the sum of exterior angles?
The exterior angles of any polygon always sum to exactly 360°, regardless of the number of sides. This is true for both regular and irregular polygons, convex and concave.
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