Quick answer
The interior angles of an n-gon sum to (n − 2) × 180°. A pentagon has 540°, each interior angle of a regular pentagon is 108° and each exterior angle 72°.
Enter the number of sides (3 or more). The calculator returns the sum and the regular interior and exterior angles. Use this polygon interior angle calculator to check your own figures.
Related tools: Regular Polygon Calculator: Area, perimeter and angles of a regular polygon. Right Triangle Side and Angle Calculator: Complete a right triangle from any two sides. Area Calculator: Area of common two-dimensional shapes.
How to use the Polygon Interior Angle Calculator
- Enter number of sides (a whole number, between 3 and 10000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Number of sides | A whole number | ≥ 3, ≤ 10000 |
Formula and method
Formula
sum of interior angles = (n − 2) × 180°; regular interior = sum ÷ n; exterior = 360° ÷ n
IEEE-754 double precision; shown to at most 6 decimal places.
Angle-sum theorem for simple polygons.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Number of sides = 5.
| Result | Value |
|---|---|
| Sum of interior angles | 540° |
| Each interior angle (regular) | 108° |
| Each exterior angle (regular) | 72° |
Example 2
Inputs: Number of sides = 3.
| Result | Value |
|---|---|
| Sum of interior angles | 180° |
| Each interior angle (regular) | 60° |
| Each exterior angle (regular) | 120° |
The first line is the headline result (sum of interior angles). Below it the polygon Interior Angle Calculator lists each interior angle (regular) and each exterior angle (regular), which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Why (n − 2)
Any n-gon can be divided into n − 2 triangles from one vertex, each contributing 180°. A pentagon makes 3 triangles: 540°. Exterior angles of any convex polygon sum to 360°, so a regular one has 360° ÷ n each.
More worked cases
Triangle 180°, square 360°, hexagon 720° (120° each), octagon 1,080° (135° each). Interior + exterior = 180° at each vertex.
How the result changes with number of sides
Holding every other input at the first worked example, this table changes only number of sides. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Number of sides | Sum of interior angles | Each interior angle (regular) | Each exterior angle (regular) |
|---|---|---|---|
| 3 | 180° | 60° | 120° |
| 5 | 540° | 108° | 72° |
| 10 | 1,440° | 144° | 36° |
| 20 | 3,240° | 162° | 18° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Number of sides: entering "2" shows "Number of sides must be at least 3." beside that field and no result.
Choosing between the Polygon Interior Angle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Regular Polygon Calculator](/math/polygon-calculator/) | number of sides and side length | Area |
| [Right Triangle Side and Angle Calculator](/math/right-triangle-side-angle/) | known sides, leg a and leg b or hypotenuse c | Hypotenuse c |
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Pythagorean Theorem Calculator](/math/pythagorean-theorem-calculator/) | find, leg a and other side | Hypotenuse c |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- n ≥ 3
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for number of sides to confirm this.
Frequently asked questions
What is each angle of a regular decagon?
Each interior angle of a regular decagon is (10 − 2) × 180° ÷ 10 = 144°.
What is the sum for a hexagon?
A hexagon's interior angles sum to (6 − 2) × 180° = 720°.
Do irregular polygons have the same sum?
Yes, only the individual angles differ.
What does the Polygon Interior Angle Calculator calculate?
Interior and exterior angles of a polygon. It returns Sum of interior angles, Each interior angle (regular) and Each exterior angle (regular).
What formula does the Polygon Interior Angle Calculator use?
sum of interior angles = (n − 2) × 180°; regular interior = sum ÷ n; exterior = 360° ÷ n. Angle-sum theorem for simple polygons.
What do I need to enter in the polygon Interior Angle Calculator?
You enter Number of sides.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. n ≥ 3.
What is an example of a polygon Interior Angle result?
With number of sides 5, the sum of interior angles is 540°.
Related calculators
- Regular Polygon Calculator — Area, perimeter and angles of a regular polygon.
- Right Triangle Side and Angle Calculator — Complete a right triangle from any two sides.
- Area Calculator — Area of common two-dimensional shapes.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Pythagorean Theorem Calculator — Find the hypotenuse or a missing leg of a right triangle.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.