Polygon Interior Angle Calculator
Calculate polygon interior angle instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Formula
Sum = (n - 2) * 180 | Each = (n - 2) * 180 / n
Where n is the number of sides. The sum of interior angles equals (n-2) times 180 degrees because the polygon can be divided into (n-2) triangles. For regular polygons, divide by n to get each individual angle.
Worked Examples
Example 1: Regular Pentagon Interior Angles
Problem: Find each interior angle and total angle sum of a regular pentagon.
Solution: Number of sides n = 5\nSum of interior angles = (5-2) * 180 = 3 * 180 = 540 degrees\nEach interior angle = 540 / 5 = 108 degrees\nEach exterior angle = 180 - 108 = 72 degrees\nNumber of diagonals = 5(5-3)/2 = 5\nTriangles formed = 5 - 2 = 3
Result: Each interior angle = 108 degrees | Sum = 540 degrees
Example 2: Finding Sides from Interior Angle
Problem: A regular polygon has each interior angle measuring 150 degrees. How many sides?
Solution: Exterior angle = 180 - 150 = 30 degrees\nNumber of sides = 360 / 30 = 12\nThis is a regular dodecagon\nSum of interior angles = (12-2) * 180 = 1800 degrees\nVerification: 1800 / 12 = 150 degrees per angle
Result: 12 sides (dodecagon) | Sum of angles = 1800 degrees
Frequently Asked Questions
What is the interior angle sum formula for polygons?
The interior angle sum formula states that the sum of all interior angles of a polygon with n sides equals (n - 2) times 180 degrees. This formula works because any polygon can be divided into (n - 2) non-overlapping triangles by drawing diagonals from a single vertex, and each triangle contributes 180 degrees to the total angle sum. For a triangle (n=3), the sum is 180 degrees. For a quadrilateral (n=4), it is 360 degrees. For a pentagon (n=5), it is 540 degrees. This formula applies to both regular and irregular polygons, convex and concave, as long as the polygon is simple (non-self-intersecting).
How do you find each interior angle of a regular polygon?
For a regular polygon where all sides and angles are equal, each interior angle equals the sum of interior angles divided by the number of sides: ((n - 2) * 180) / n degrees. This simplifies to 180 - 360/n, which shows that each interior angle equals 180 degrees minus the exterior angle. A regular triangle has angles of 60 degrees, a square has 90 degrees, a regular pentagon has 108 degrees, and a regular hexagon has 120 degrees. As the number of sides increases, each interior angle approaches but never reaches 180 degrees, with the polygon increasingly resembling a circle.
What is the difference between interior and exterior angles?
Interior angles are formed inside the polygon between two adjacent sides. Exterior angles are formed between one side and the extension of the adjacent side, lying outside the polygon. At each vertex, the interior and exterior angles are supplementary, summing to 180 degrees. While the sum of interior angles increases with the number of sides ((n-2)*180), the sum of exterior angles is always 360 degrees regardless of the polygon type. This makes exterior angles simpler for certain calculations, while interior angles are more intuitive for understanding the shape and for construction purposes.
How do you find the number of sides from an interior angle?
If you know the interior angle of a regular polygon, you can find the number of sides using the formula n = 360 / (180 - interior angle). First compute the exterior angle as 180 minus the interior angle, then divide 360 by this exterior angle. For example, if the interior angle is 144 degrees, the exterior angle is 36 degrees, and n = 360/36 = 10 (decagon). The result must be a whole number of 3 or greater for a valid regular polygon. If the calculation yields a non-integer, then no regular polygon has that exact interior angle measure.
What are diagonals of a polygon and how many are there?
A diagonal is a line segment connecting two non-adjacent vertices of a polygon. The number of diagonals in an n-sided polygon is n(n-3)/2. This formula comes from choosing 2 vertices from n (which gives n(n-1)/2 line segments) and subtracting the n sides. A triangle has 0 diagonals, a quadrilateral has 2, a pentagon has 5, a hexagon has 9, and a decagon has 35. Diagonals are used to triangulate polygons, which is essential for computing areas, for rendering polygons in computer graphics, and for many computational geometry algorithms.
Why can a polygon be divided into triangles using diagonals?
Any simple polygon with n vertices can be triangulated into exactly (n-2) non-overlapping triangles by drawing (n-3) non-crossing diagonals from a single vertex. This is possible because a diagonal from one vertex to a non-adjacent vertex splits the polygon into two smaller polygons, and the process can be repeated recursively. The triangulation is not unique (there are typically many valid triangulations), but the number of resulting triangles is always n-2. This property is the foundation of the interior angle sum formula and is heavily used in computational geometry, finite element methods, and 3D rendering pipelines.