Calculate a polygon’s interior angle from its number of sides, or work backward from a single interior angle or interior angle sum to find the number of sides, exterior angle, and diagonals.
Formulas & Definitions
Our calculator finds exact inside angles of regular polygons based on side counts. You can also work backward to find side numbers from a single interior angle or a total degree sum.
Valid calculations return polygon names, single exterior angles, and total diagonals. Calculating single interior angles requires regular polygons, meaning all internal angles must remain exactly equal.
Interior Angle Calculator Inputs and Outputs
Our tool handles three primary input methods, generating distinct mathematical outputs based on strict polygon rules.
| Calculation method | Input | Main result | Extra outputs | Important rule |
|---|---|---|---|---|
| Find angles from number of sides | Number of sides $n$ | Single interior angle | Polygon name, sum of interior angles, exterior angle, diagonals | $n$ must be a whole number and at least $3$ |
| Find sides from sum of interior angles | Sum of interior angles in degrees | Number of sides $n$ | Polygon name, regular interior angle, exterior angle, diagonals when valid | Sum must be at least $180^\circ$ |
| Find sides from single interior angle | Single interior angle in degrees | Number of sides $n$ | Polygon name, sum of interior angles, exterior angle, diagonals when valid | Angle must be at least $60^\circ$ and less than $180^\circ$ |
Interior Angle Formula for a Regular Polygon
Our main formula for calculating single inside angles is: $\text{Interior angle} = \frac{(n – 2) \times 180^\circ}{n}$. Here, $n$ represents a shape’s total side count.
Such formulas for single angles apply strictly to regular polygons, requiring equal internal angles and side lengths. Conversely, interior angle sum formulas apply to any simple polygon.
| Quantity | Formula |
|---|---|
| Sum of interior angles | $(n – 2) \times 180^\circ$ |
| Single interior angle | $\frac{(n – 2) \times 180^\circ}{n}$ |
| Single exterior angle | $\frac{360^\circ}{n}$ |
| Number of diagonals | $\frac{n(n – 3)}{2}$ |
How to Find Interior Angle from Number of Sides
To find an interior angle from a side count, input your number. We first compute a shape’s total internal degree sum, then divide total sums by side count, returning an exact single angle.
Alongside your main result, you instantly see matching polygon names, corresponding exterior angles, and total diagonals.
| Polygon | Sides $n$ | Sum of interior angles | Single interior angle | Exterior angle | Diagonals |
|---|---|---|---|---|---|
| Triangle | $3$ | $180^\circ$ | $60^\circ$ | $120^\circ$ | $0$ |
| Square | $4$ | $360^\circ$ | $90^\circ$ | $90^\circ$ | $2$ |
| Pentagon | $5$ | $540^\circ$ | $108^\circ$ | $72^\circ$ | $5$ |
| Hexagon | $6$ | $720^\circ$ | $120^\circ$ | $60^\circ$ | $9$ |
| Octagon | $8$ | $1080^\circ$ | $135^\circ$ | $45^\circ$ | $20$ |
| Decagon | $10$ | $1440^\circ$ | $144^\circ$ | $36^\circ$ | $35$ |
Find Number of Sides from Sum of Interior Angles
Working backward from total degree sums requires our reverse formula: $n = \frac{\text{Sum of interior angles}}{180^\circ} + 2$.
Your resulting $n$ value must be a whole number. Fractional side counts do not correspond to real polygons having whole number sides and will trigger an invalid warning.
| Sum of interior angles | Calculated sides $n$ | Valid polygon? | Polygon |
|---|---|---|---|
| $180^\circ$ | $3$ | Yes | Triangle |
| $360^\circ$ | $4$ | Yes | Quadrilateral |
| $540^\circ$ | $5$ | Yes | Pentagon |
| $720^\circ$ | $6$ | Yes | Hexagon |
| $900^\circ$ | $7$ | Yes | Heptagon |
| $500^\circ$ | $4.78$ | No | Not a real polygon |
Find Number of Sides from a Single Interior Angle
Reversing our math to find side counts from one angle uses: $n = \frac{360^\circ}{180^\circ – \text{interior angle}}$. Such reverse calculations apply exclusively to regular polygons possessing identical internal angles.
Valid shapes require angle inputs within a strict mathematical range: $60^\circ \le \text{angle} < 180^\circ$. Angles outside such bounds do not correspond to valid regular polygons here.
| Single interior angle | Calculated sides $n$ | Valid polygon? | Polygon |
|---|---|---|---|
| $60^\circ$ | $3$ | Yes | Triangle |
| $90^\circ$ | $4$ | Yes | Quadrilateral |
| $108^\circ$ | $5$ | Yes | Pentagon |
| $120^\circ$ | $6$ | Yes | Hexagon |
| $135^\circ$ | $8$ | Yes | Octagon |
| $140^\circ$ | $9$ | Yes | Nonagon |
| $100^\circ$ | $4.5$ | No | Not a real polygon |
Polygon Names and Interior Angles Reference Table
Our quick-reference guide outlines standard properties for common regular polygons, matching exact calculator outputs.
| Sides | Polygon name | Sum of interior angles | Single interior angle | Single exterior angle | Diagonals |
|---|---|---|---|---|---|
| $3$ | Triangle | $180^\circ$ | $60^\circ$ | $120^\circ$ | $0$ |
| $4$ | Quadrilateral | $360^\circ$ | $90^\circ$ | $90^\circ$ | $2$ |
| $5$ | Pentagon | $540^\circ$ | $108^\circ$ | $72^\circ$ | $5$ |
| $6$ | Hexagon | $720^\circ$ | $120^\circ$ | $60^\circ$ | $9$ |
| $7$ | Heptagon | $900^\circ$ | $128.5714^\circ$ | $51.4286^\circ$ | $14$ |
| $8$ | Octagon | $1080^\circ$ | $135^\circ$ | $45^\circ$ | $20$ |
| $9$ | Nonagon | $1260^\circ$ | $140^\circ$ | $40^\circ$ | $27$ |
| $10$ | Decagon | $1440^\circ$ | $144^\circ$ | $36^\circ$ | $35$ |
| $11$ | Hendecagon | $1620^\circ$ | $147.2727^\circ$ | $32.7273^\circ$ | $44$ |
| $12$ | Dodecagon | $1800^\circ$ | $150^\circ$ | $30^\circ$ | $54$ |
Interior and Exterior Angle Relationships
Interior angle sums always equal $(n – 2) \times 180^\circ$, while single interior angles are determined by dividing total sums by $n$.
Furthermore, single exterior angles calculate simply as $\frac{360^\circ}{n}$. Due to polygon structure, an interior angle plus its exterior angle at any individual vertex always equals exactly $180^\circ$.
| Value | What it means | Formula | When used here |
|---|---|---|---|
| Sum of interior angles | Total of all inside angles | $(n – 2) \times 180^\circ$ | From sides, and to reverse-calculate sides |
| Single interior angle | One inside angle of a regular polygon | $\frac{(n – 2) \times 180^\circ}{n}$ | Main target output |
| Single exterior angle | One outside angle of a regular polygon | $\frac{360^\circ}{n}$ | Supporting output |
Number of Diagonals by Polygon
Total diagonals connecting non-adjacent vertices are calculated using: $\text{Diagonals} = \frac{n(n – 3)}{2}$.
| Sides | Polygon | Diagonals |
|---|---|---|
| $3$ | Triangle | $0$ |
| $4$ | Quadrilateral | $2$ |
| $5$ | Pentagon | $5$ |
| $6$ | Hexagon | $9$ |
| $7$ | Heptagon | $14$ |
| $8$ | Octagon | $20$ |
| $10$ | Decagon | $35$ |
| $12$ | Dodecagon | $54$ |
Input Rules and Result Limits
Inputs must follow strict boundaries to generate valid geometric shapes.
| Input mode | Valid input rule | Invalid result behavior |
|---|---|---|
| Number of sides | Whole number, minimum $3$ | Error shown |
| Sum of interior angles | At least $180^\circ$ | Error shown below $180^\circ$ |
| Single interior angle | At least $60^\circ$ and less than $180^\circ$ | Error shown outside range |
| Reverse side calculation | Result must be a whole number | Tool marks it as not a real polygon and suppresses unsupported derived outputs |
Calculator Limitations
Our calculator determines single interior angles exclusively for regular polygons, requiring identical sides and internal angles. Interior angle sums apply universally to all simple polygons, regular or irregular.
Single interior and exterior angle outputs assume equal angles across your shape. Furthermore, fractional side counts generated during reverse calculations never represent real polygons. Under such conditions, we mark shapes invalid and halt derived value calculations.
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