Polygon Interior

Q: Does the interior angle sum formula work for concave polygons?

Yes. The formula $S = (n - 2) \times 180°$ applies to all simple polygons (polygons whose sides do not cross each other), whether convex or concave. In a concave polygon, at least one interior angle will be greater than 180° (a reflex angle), but the total sum still follows the same formula.

Polygon Interior

The points enclosed by a polygon.

Two pentagons: an outlined one labeled "Interior of a pentagon" and a shaded one showing the filled interior region.

See also

Interior, disk

Key Formula

S = (n - 2) \times 180°

Where:

$S$ = The sum of all interior angles of the polygon, measured in degrees
$n$ = The number of sides (or vertices) of the polygon, where n ≥ 3

Worked Example

Problem: Find the sum of the interior angles of a regular hexagon, and determine the measure of each interior angle.

Step 1: Identify the number of sides. A hexagon has 6 sides.

n = 6

Step 2: Apply the interior angle sum formula.

S = (n - 2) \times 180° = (6 - 2) \times 180°

Step 3: Simplify the expression.

S = 4 \times 180° = 720°

Step 4: Since the hexagon is regular (all angles equal), divide the total by 6 to find each angle.

\text{Each angle} = \frac{720°}{6} = 120°

Answer: The sum of the interior angles of a hexagon is 720°, and each interior angle of a regular hexagon measures 120°.

Another Example

This example works backwards — given the angle sum, you solve for the number of sides. It shows how to rearrange the formula.

Problem: A polygon has an interior angle sum of 1440°. How many sides does it have? If it is regular, what is each interior angle?

Step 1: Start with the interior angle sum formula and set it equal to the given sum.

(n - 2) \times 180° = 1440°

Step 2: Divide both sides by 180° to isolate the factor.

n - 2 = \frac{1440°}{180°} = 8

Step 3: Solve for n.

n = 8 + 2 = 10

Step 4: The polygon is a decagon (10 sides). If it is regular, divide the total by 10.

\text{Each angle} = \frac{1440°}{10} = 144°

Answer: The polygon has 10 sides (a decagon), and each interior angle of a regular decagon measures 144°.

Frequently Asked Questions

What is the difference between interior angles and exterior angles of a polygon?

An interior angle is formed inside the polygon at each vertex where two sides meet. An exterior angle is the supplement of an interior angle — it is formed between one side and the extension of the adjacent side. At any vertex, the interior angle and its corresponding exterior angle always add up to 180°. The sum of all exterior angles of any convex polygon is always 360°, regardless of how many sides it has.

How do you know if a point is inside or outside a polygon?

A point lies in the polygon interior if it is completely enclosed by the polygon's sides. One common method is the ray-casting test: draw a ray from the point in any direction and count how many times it crosses the polygon's boundary. If it crosses an odd number of times, the point is inside; if even, the point is outside. On the boundary itself, the point is considered neither strictly interior nor exterior.

Does the interior angle sum formula work for concave polygons?

Yes. The formula

S = (n - 2) \times 180°

applies to all simple polygons (polygons whose sides do not cross each other), whether convex or concave. In a concave polygon, at least one interior angle will be greater than 180° (a reflex angle), but the total sum still follows the same formula.

Polygon Interior vs. Polygon Exterior

	Polygon Interior	Polygon Exterior
Definition	The set of all points enclosed by the polygon's boundary	The set of all points outside the polygon's boundary
Angle sum formula	(n − 2) × 180° for all interior angles combined	Always 360° for all exterior angles combined (convex polygon)
Single angle (regular polygon)	((n − 2) × 180°) / n	360° / n
Relationship	Each interior angle + its adjacent exterior angle = 180°	Each exterior angle + its adjacent interior angle = 180°

Why It Matters

The polygon interior and the interior angle sum formula appear constantly in geometry courses, from basic triangle problems (

n = 3

gives

180°

) to tiling and tessellation questions. Architects, engineers, and game developers use these ideas when designing floor plans, structural frames, and collision detection in software. Standardized tests like the SAT and ACT regularly ask you to find missing angles in polygons using this formula.

Common Mistakes

Mistake: Using n instead of (n − 2) in the formula — for example, calculating a pentagon's angle sum as 5 × 180° = 900° instead of the correct 540°.

Correction: Always subtract 2 from the number of sides first. The factor (n − 2) represents the number of non-overlapping triangles that the polygon can be divided into, each contributing 180°.

Mistake: Dividing the total interior angle sum by n to find each angle when the polygon is not regular.

Correction: You can only divide the sum equally among the angles when the polygon is regular (all sides and all angles are equal). For irregular polygons, individual angle measures must be determined from additional information.

Related Terms

Polygon — The closed figure whose interior is discussed
Point — A location that may lie inside, on, or outside a polygon
Interior — General concept of the region inside a figure
Disk — Interior of a circle, analogous concept for curved figures
Interior Angle — Angle formed inside a polygon at each vertex
Exterior Angle — Supplement of an interior angle at each vertex
Regular Polygon — Polygon with all equal sides and equal interior angles
Convex — Property where all interior angles are less than 180°