Polygon Interior Angle Calculator
Calculate polygon interior angle instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Polygon Interior Angle Calculator
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Formula: Sum = (n - 2) * 180 | Each = (n - 2) * 180 / n
Worked example — Each interior angle = 108 degrees | Sum = 540 degrees
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 Sum of interior angles = (5-2) * 180 = 3 * 180 = 540 degrees Each interior angle = 540 / 5 = 108 degrees Each exterior angle = 180 - 108 = 72 degrees Number of diagonals = 5(5-3)/2 = 5 Triangles 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 Number of sides = 360 / 30 = 12 This is a regular dodecagon Sum of interior angles = (12-2) * 180 = 1800 degrees Verification: 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.
What is the apothem and how does it relate to interior angles?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It equals half the side length divided by the tangent of pi/n radians, where n is the number of sides. The apothem is directly related to the interior angle because larger interior angles (more sides) produce larger apothems relative to the side length. The area of a regular polygon equals half the perimeter times the apothem, making the apothem essential for area calculations. As the number of sides increases and interior angles approach 180 degrees, the apothem approaches the circumradius, and the polygon approaches a circle.
Which regular polygons can tessellate (tile a plane)?
Only three regular polygons can tessellate the plane by themselves: equilateral triangles (interior angle 60 degrees, six meet at each vertex), squares (90 degrees, four meet at each vertex), and regular hexagons (120 degrees, three meet at each vertex). The key requirement is that the interior angle must divide evenly into 360 degrees, so that copies of the polygon can meet at a point without gaps or overlaps. No other regular polygon satisfies this condition. Semi-regular tessellations use combinations of different regular polygons, and the study of which combinations work involves analyzing how interior angles from different polygons can sum to exactly 360 degrees.
How do interior angles work for irregular polygons?
For irregular polygons, the individual interior angles are not all equal, but their sum still follows the formula (n-2)*180 degrees. An irregular quadrilateral might have angles of 80, 100, 70, and 110 degrees, which still sum to 360. The key difference from regular polygons is that knowing the number of sides alone does not determine each individual angle. To find unknown angles in an irregular polygon, you need additional information such as other angle measures, side lengths with angle relationships, or coordinate geometry. In practice, irregular polygons are measured directly or computed from vertex coordinates using the arctangent function.
How are polygon interior angles used in architecture and engineering?
Interior angles are crucial in architectural design for creating rooms, buildings, and structural elements with specific shapes. Floor plans use polygon angles to determine wall orientations and room layouts. In structural engineering, the angles at joints determine force distribution in trusses and frames. Regular polygon shapes like hexagons are used in honeycomb structures for optimal strength-to-weight ratios. In road design, intersection angles affect visibility and traffic flow. Tile manufacturers use polygon angle calculations to create decorative patterns. Even in computer game level design, polygon angles determine walkable areas and collision boundaries.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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