Polygon Angle Sum: Formula (n − 2) × 180 & Examples

Definitionnoun phrase

What is the polygon angle sum?

The polygon angle sum is the total of the interior angles of a polygon. For a polygon with n sides it equals (n − 2) × 180°, because diagonals from one vertex cut the polygon into n − 2 triangles. A pentagon’s angles add to 540°, and a hexagon’s add to 720°.

In Spanish: suma de los ángulos de un polígono
Also called: polygon angle sum theorem, sum of interior angles, interior angle sum
Grade level: Geometry

Grade 8GeometryCCSS 8.G.A.5SATACTGEDASVAB

Polygon Angle Sum in a picture

From one vertex, the diagonals cut an n-gon into n − 2 triangles, and each triangle adds 180°.

Formal definition and notation

For a convex polygon with n ≥ 3 sides, the sum of the interior angle measures is S = (n − 2) · 180°. The same formula holds for any simple polygon, convex or concave. The exterior angles of a convex polygon, one at each vertex, always add to 360°.

Symbol	Read it as	Meaning
n	n	the number of sides (and of angles)
S	S	the sum of the interior angles

Key facts

Triangle 180°, quadrilateral 360°, pentagon 540°, hexagon 720°, octagon 1,080°.
Each extra side adds one more triangle, so the sum goes up by 180°.
The formula works for every polygon with that number of sides, regular or irregular.
The exterior angles of a convex polygon always add to 360°, whatever the number of sides.
Only in a regular polygon can you divide the sum by n to get each angle.

Polygon angle sum formula

S = (n − 2) × 180°

n − 2: the number of triangles made by the diagonals from one vertex
180°: the angle sum of each triangle
360°: the sum of the exterior angles, one per vertex, for any convex polygon

Working backward: n = S180 + 2. An angle sum of 1,440° gives n = 8 + 2 = 10.

Polygon Angle Sum examples, step by step

Example 1: Angle sum of an octagon

Warm-up

What is the sum of the interior angles of an octagon?

An octagon has n = 8 sides.
S = (8 − 2) × 180 = 6 × 180 = 1,080.

Answer: The angles of an octagon add to 1,080°.

Example 2: A missing angle in a pentagon

Core

Four angles of a pentagon measure 100°, 110°, 95° and 120°. Find the fifth angle.

Angle sum of a pentagon: (5 − 2) × 180 = 540.
Add the known angles: 100 + 110 + 95 + 120 = 425.
Subtract: 540 − 425 = 115.

Answer: The fifth angle is 115°.

Example 3: Find the number of sides

Test-style

The interior angles of a polygon add to 1,440°. How many sides does it have?

Set up (n − 2) × 180 = 1,440.
Divide by 180: n − 2 = 8.
Add 2: n = 10. It is a decagon.

Answer: The polygon has 10 sides (a decagon).

Common mistakes

✗ MistakeAngle sum of a hexagon: 6 × 180 = 1,080°.

✓ Instead(6 − 2) × 180 = 720°.

A hexagon splits into 4 triangles, not 6. Drawing triangles from the center adds a full 360° turn at the center that is not part of the interior angles.

✗ MistakeEach angle of an irregular pentagon is 540 ÷ 5 = 108°.

✓ InsteadOnly a regular pentagon has five 108° angles. An irregular one has the same total, 540°, shared unevenly.

Dividing the sum by n assumes all angles are equal, which is true only for regular polygons.

✗ MistakeThe exterior angles of an octagon add to 1,080°.

✓ InsteadExterior angles of any convex polygon add to 360°. The 1,080° is the interior sum.

Walking around the polygon, you turn once at each corner and end up facing the way you started: one full turn.

Not to be confused with

Polygon Angle Sumvs.Exterior angle sum

Interior angles add to (n − 2) × 180°, which grows with n. Exterior angles of a convex polygon always add to 360°.

Polygon Angle Sumvs.Regular polygon

The angle sum works for every polygon. For a regular polygon only, divide by n to get each angle: 720 ÷ 6 = 120° for a regular hexagon.

Quick check: 3 questions

Question 1 of 3

What is the sum of the interior angles of a hexagon?

Show answer and explanation

Answer: D. (6 − 2) × 180 = 720. A multiplies 6 by 180, B is a pentagon, and C is a quadrilateral.

Question 2 of 3

The interior angles of a polygon add to 900°. How many sides does it have?

Show answer and explanation

Answer: B. 900 ÷ 180 = 5 triangles, and n = 5 + 2 = 7. A stops at the number of triangles; C adds 4 instead of 2.

Question 3 of 3

What is the sum of the exterior angles of a convex 12-sided polygon?

Show answer and explanation

Answer: A. Exterior angles of any convex polygon add to 360°. B is the interior sum (12 − 2) × 180, C is 12 × 180, and D is one exterior angle of a regular 12-gon.

Polygon Angle Sum: frequently asked questions

Why is the formula n minus 2 times 180?

Pick one vertex and draw every diagonal from it. The diagonals cannot go to the vertex itself or to its two neighbors, and they cut the polygon into n minus 2 triangles. The triangle angles together make up exactly the polygon angles, and each triangle holds 180 degrees, so the total is (n minus 2) times 180.

Does the angle sum formula work for concave polygons?

Yes. A concave polygon, one with a dent, still splits into n minus 2 triangles, and its angles still add to (n minus 2) times 180 degrees. One of its interior angles is simply more than 180 degrees. The exterior-angle rule of 360 degrees, though, is stated for convex polygons.

Practice and tools

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.A.5

Cite this entry

Effortless Math Team. “Polygon Angle Sum.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/polygon-angle-sum/

Flashcard

TermPolygon Angle Sum(n − 2) × 180°

DefinitionThe total of a polygon’s interior angles. A pentagon splits into 3 triangles, so its angles add to 540°.

Study the full deck

Browse all terms in the Math Dictionary · More geometry terms · Random term

PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Interior AngleAn interior angle is an angle inside a polygon, formed at a vertex by two sides that meet there. A polygon with n sides has n interior angles, and they add up to (n − 2) × 180 degrees. A triangle’s interior angles total 180 degrees, and a rectangle’s four right angles total 360 degrees.Geometry · Grade 7, Grade 8, GeometryNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5QuadrilateralA quadrilateral is a polygon with four straight sides, four vertices and four angles. Its four interior angles always add to 360 degrees. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are special quadrilaterals, each named for extra properties of its sides or angles.Geometry · Grade 2, Grade 3, Grade 5Regular PolygonA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is (n − 2) × 180°n, and each exterior angle is 360°n.Geometry · Grade 5, Grade 8, GeometryDegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6

Polygon Angle Sum

What is the polygon angle sum?

Polygon Angle Sum in a picture

Formal definition and notation

Key facts

Polygon angle sum formula

Polygon Angle Sum examples, step by step

Example 1: Angle sum of an octagon

Example 2: A missing angle in a pentagon

Example 3: Find the number of sides

Common mistakes

Not to be confused with

Quick check: 3 questions

Polygon Angle Sum: frequently asked questions

Why is the formula n minus 2 times 180?

Does the angle sum formula work for concave polygons?

Related terms

Practice and tools

Sources

Flashcard

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