Polygon Interior Angles Calculator

Find the Interior Angle Sum of a Polygon

Polygon Interior Angles Formula

Formula

\[ \Sigma\alpha = (n - 2) \cdot 180^\circ \qquad \alpha = \frac{(n - 2) \cdot 180^\circ}{n} \qquad \beta = \frac{360^\circ}{n} \]

Where:

\(n\) — number of sides of the polygon (\(n \ge 3\))
\(\Sigma\alpha\) — sum of all interior angles (degrees)
\(\alpha\) — each interior angle of a regular polygon (degrees)
\(\beta\) — each exterior angle of a regular polygon (degrees)

The first formula gives the total of all interior angles for any polygon, regular or irregular, based only on how many sides it has. The other two formulas apply to regular polygons, where all angles are equal, so the total is simply divided among the \(n\) angles.

Polygon Interior Angles – Calculation Example

Example: a regular hexagon (the calculator's default), which has \(n = 6\) equal sides and equal angles.

\[ \Sigma\alpha = (6 - 2) \cdot 180^\circ = 4 \cdot 180^\circ = 720^\circ \]

\[ \alpha = \frac{720^\circ}{6} = 120^\circ \]

\[ \beta = \frac{360^\circ}{6} = 60^\circ \]

Result: the interior angles of a hexagon add up to 720°; each interior angle of a regular hexagon is 120°, and each exterior angle is 60°. Note that \(120^\circ + 60^\circ = 180^\circ\), as always for an interior/exterior angle pair.

How to use the calculator

Enter the Number of sides (n) — any whole number from 3 up, for example 6 for a hexagon.
Leave Regular polygon ticked if all sides and all angles are equal; untick it for an irregular polygon.
Read the Sum of interior angles in degrees — this is valid for every polygon with that number of sides.
When "Regular polygon" is ticked, also read Each interior angle and Each exterior angle.
Check the Calculation steps to see the arithmetic written out, with the polygon's name when it has 12 sides or fewer.

What are interior angles of a polygon?

An interior angle is the angle formed inside the polygon at a vertex, between the two sides that meet there. Every polygon with \(n\) sides has exactly \(n\) interior angles — a triangle has 3, a hexagon has 6, and so on.

The key fact is that the sum of these angles depends only on the number of sides, not on the polygon's shape. A triangle's angles always total 180°, whether it is a skinny sliver or perfectly equilateral. Each extra side adds another 180° to the total: 180° for a triangle, 360° for a quadrilateral, 540° for a pentagon, 720° for a hexagon. That is exactly what the formula \((n - 2) \cdot 180^\circ\) encodes — a polygon can be divided into \(n - 2\) triangles, each contributing 180°.

For an irregular polygon, only the total is fixed; the individual angles can be anything as long as they add up to it. For a regular polygon — equal sides and equal angles — each angle is simply the sum divided by \(n\), and each exterior angle is \(360^\circ / n\), because the exterior angles must complete a full turn around the polygon.

The result is useful in geometry homework, tiling and design work, and construction. For example, knowing that each interior angle of a regular hexagon is 120° tells you why three hexagons meet perfectly around a point (3 × 120° = 360°), which is the basis of honeycomb patterns.

Assumptions, limitations and tips

Convex polygons: the formulas assume a simple polygon. For concave (non-convex) polygons the sum formula still holds, but one or more interior angles exceed 180°.
Self-intersecting polygons (like a pentagram traced as one path) do not follow these formulas.
Whole numbers only: a polygon cannot have a fractional number of sides, so the calculator asks for a whole number of at least 3.
Exterior angles: the exterior angle shown is for a regular polygon. For any convex polygon, the exterior angles still sum to 360°, but individually they differ if the polygon is irregular.
Large values: as \(n\) grows, each interior angle of a regular polygon approaches 180°, and each exterior angle approaches 0° — a regular polygon with many sides looks almost like a circle.

A handy related fact: the number of diagonals of an \(n\)-gon is \(\frac{n(n-3)}{2}\), which comes from choosing 2 vertices out of \(n\) — you can explore that kind of counting with our Binomial Coefficient calculator. And if you enjoy number-pattern tools, try the Magic Square generator as well.

Frequently Asked Questions

How do I calculate the sum of interior angles without a calculator?

Take the number of sides, subtract 2, and multiply by 180°. For an octagon: \((8 - 2) \cdot 180° = 1080°\). The trick works because any polygon with \(n\) sides can be cut into \(n - 2\) triangles from one vertex, and each triangle contributes 180°. For a regular polygon, divide the total by \(n\) to get each individual angle: \(1080° / 8 = 135°\).

What is each angle of a regular polygon in radians?

The formulas convert easily: replace 180° with \(\pi\) and 360° with \(2\pi\). The sum becomes \((n - 2)\pi\) radians, each interior angle of a regular polygon is \(\frac{(n-2)\pi}{n}\), and each exterior angle is \(\frac{2\pi}{n}\). To convert any result manually, multiply degrees by \(\pi / 180\). For example, 120° equals \(2\pi/3 \approx 2.094\) radians.

How can I find a missing interior angle of an irregular polygon?

First compute the fixed total from the side count: \((n - 2) \cdot 180°\). Then subtract all the known angles from that total — the remainder is your missing angle. For instance, if a quadrilateral has angles 90°, 85° and 100°, the fourth angle is \(360° - 275° = 85°\). This works for convex polygons and for concave ones, where the missing angle may exceed 180° (a reflex angle).

Can an interior angle be more than 180°?

Yes, in a concave polygon. At a "dent" vertex, the interior angle is reflex — greater than 180° and less than 360°. Importantly, the sum formula still holds: a concave quadrilateral's angles still total 360°, they just are not all under 180°. The formula fails only for self-intersecting shapes, such as a pentagram traced as a single path, which follow different rules.

Why do the exterior angles of any polygon add up to 360°?

Walk once around the polygon and you end up facing your original direction, meaning you turned a full 360° in total. Each turn equals the exterior angle at that vertex, so the exterior angles — whether equal or not — must sum to 360° for any convex polygon. For a regular polygon each turn is the same, so each exterior angle is simply \(360° / n\), and it always pairs with its interior angle to make 180°.

What is the most common mistake with interior angle problems?

The top errors are: using \(n - 1\) or \(n\) instead of \(n - 2\) in the sum formula; dividing the total by the wrong number when finding each angle; and applying the regular-polygon formula \(\frac{(n-2) \cdot 180°}{n}\) to an irregular polygon, where individual angles differ. Also remember a polygon needs at least 3 sides — there is no such thing as a 2-sided polygon.

How many sides does a polygon have if I only know an interior angle?

For a regular polygon, set each interior angle equal to \(\frac{(n-2) \cdot 180°}{n}\) and solve: \(n = \frac{180°}{180° - \alpha}\). For example, an interior angle of 150° gives \(n = 180/30 = 12\), a dodecagon. A quick sanity check: a regular polygon's interior angle must be strictly between 60° (triangle) and 180°, and \(n\) must come out as a whole number.