Interior Angle: Definition, Formula & Examples

Definitionnoun phrase

What is an interior angle in math?

An 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.

In Spanish: ángulo interior
Also called: internal angle, angle of a polygon
Grade level: 6-8

Grade 7Grade 8GeometryCCSS 8.G.A.5CCSS HSG-CO.C.10GEDSATACTASVABHiSETACCUPLACER

Interior Angle in a picture

Cut any polygon into triangles from one corner. You always get n − 2 triangles, which is where the (n − 2) × 180° rule comes from.

Formal definition and notation

For a polygon with vertices V₁, V₂, …, V_n, the interior angle at V_k is the angle between the two sides meeting at V_k, measured on the inside of the polygon. For a convex polygon each interior angle is less than 180°; a concave polygon has at least one interior angle greater than 180°. The interior angles of any simple polygon with n sides sum to (n − 2) · 180°.

Symbol	Read it as	Meaning
S = (n − 2) · 180°	S equals n minus 2 times 180 degrees	the sum S of the interior angles of an n-sided polygon

Key facts

Why n − 2: diagonals from one vertex split the polygon into n − 2 triangles, and each triangle holds 180°.
Common sums: triangle 180°, quadrilateral 360°, pentagon 540°, hexagon 720°, octagon 1,080°.
Each angle of a regular polygon: equilateral triangle 60°, square 90°, regular pentagon 108°, regular hexagon 120°, regular octagon 135°.
Adding one side to a polygon adds 180° to the interior angle sum.
The interior angles of a regular polygon are always less than 180° and get closer to it as n grows.

Interior angle formulas

S = (n − 2) × 180° and, for a regular polygon, each = (n − 2) × 180°n

n: number of sides (also the number of interior angles)
S: sum of all interior angles
n − 2: number of triangles made by the diagonals from one vertex

Each interior angle and its exterior angle form a straight line, so interior + exterior = 180°, and the exterior angles of any convex polygon add to 360°.

Interior Angle examples, step by step

Example 1: Sum for a hexagon

Warm-up

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

A hexagon has n = 6 sides.
S = (6 − 2) × 180° = 4 × 180° = 720°.

Answer: The interior angles of a hexagon add to 720°.

Example 2: Missing angle in a quadrilateral

Core

Three interior angles of a quadrilateral are 85°, 110° and 72°. Find the fourth.

A quadrilateral’s angles add to (4 − 2) × 180° = 360°.
Add the known angles: 85 + 110 + 72 = 267.
Fourth angle: 360° − 267° = 93°.

Answer: The fourth angle is 93°.

Example 3: Find the number of sides

Test-style

Each interior angle of a regular polygon measures 156°. How many sides does it have?

Each exterior angle is 180° − 156° = 24°.
The exterior angles of a convex polygon add to 360°.
Number of sides: 360 ÷ 24 = 15.
Check: (15 − 2) × 18015 = 234015 = 156.

Answer: The polygon has 15 sides.

Common mistakes

✗ MistakePentagon sum = 5 × 180° = 900°.

✓ InsteadPentagon sum = (5 − 2) × 180° = 540°.

The polygon splits into n minus 2 triangles, not n. Multiplying by n counts two triangles too many.

✗ MistakeEach angle of a regular octagon = 1,080°.

✓ Instead1,080° is the total. Each angle is 1,080° ÷ 8 = 135°.

The formula gives the sum. For a single angle of a regular polygon, divide by the number of angles.

✗ MistakeDividing the sum by n for a polygon whose sides are not all equal.

✓ InsteadIn an irregular pentagon the five angles total 540° but can differ, such as 100°, 120°, 95°, 125°, 100°.

Only regular polygons have all interior angles equal.

Not to be confused with

Interior Anglevs.Exterior angle

An exterior angle is outside the polygon, between one side and the extension of the next. Interior + exterior = 180° at each vertex; a regular hexagon has 120° interior and 60° exterior angles.

Interior Anglevs.Interior angles of a transversal

With a transversal, interior angles are the four angles between two crossed lines, such as alternate interior angles. That meaning is about lines, not the corners of a polygon.

Quick check: 3 questions

Question 1 of 3

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

Show answer and explanation

Answer: B. (8 − 2) × 180° = 1,080°. 1,440° multiplies 8 by 180, 135° is one angle of a regular octagon, and 1,260° is the sum for a 9-gon.

Question 2 of 3

Each interior angle of a regular decagon (10 sides) measures:

Show answer and explanation

Answer: A. Sum = 8 × 180° = 1,440°, and 1,440° ÷ 10 = 144°. 36° is the exterior angle, 180° uses n instead of n − 2 before dividing, and 1,440° is the total.

Question 3 of 3

A polygon’s interior angles add to 900°. How many sides does it have?

Show answer and explanation

Answer: D. (n − 2) × 180 = 900 gives n − 2 = 5, so n = 7. Choosing 5 forgets to add the 2 back, 9 adds 4, and 6 is the hexagon (720°).

Interior Angle: frequently asked questions

How do you find an interior angle?

For a regular polygon, compute the sum (n − 2) × 180° and divide by n: a regular hexagon has 720° ÷ 6 = 120° angles. For one missing angle of any polygon, subtract the known angles from the sum: a quadrilateral with angles 90°, 90°, 100° has a fourth angle of 80°.

Can an interior angle be more than 180 degrees?

Yes, in a concave polygon. A shape like an arrowhead or a star outline has a corner that points inward, and the interior angle there is a reflex angle, greater than 180°. The sum formula (n − 2) × 180° still holds. In a convex polygon, every interior angle is less than 180°.

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

Always 180°. This matches the formula with n = 3: (3 − 2) × 180° = 180°. If two angles are 50° and 65°, the third is 180° − 115° = 65°. Every polygon’s angle sum is built from this fact, since polygons can be cut into triangles.

Practice and tools

Effortless Math practicePractice on Effortless Math Triangle CalculatorSolve for missing interior angles of a triangle

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.A.5, CCSS HSG-CO.C.10

Cite this entry

Effortless Math Team. “Interior Angle.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/interior-angle/

Flashcard

TermInterior Angleinside each corner of a polygon

DefinitionThe angle inside a polygon at a vertex. An n-sided polygon’s interior angles sum to (n − 2) × 180°: a pentagon’s total 540°.

Study the full deck

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

VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6TriangleA 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 5PolygonA 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 2QuadrilateralA 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, GeometryEquilateral TriangleAn equilateral triangle is a triangle whose three sides all have the same length. Its three angles are also equal, so each one measures 60 degrees. With side length a, its perimeter is 3a and its area is √34a². Every equilateral triangle is also isosceles and acute.Geometry · Grade 4, Grade 5, Grade 7SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s², and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3NumberA 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 2Alternate Interior AnglesAlternate interior angles are two angles that lie between two lines and on opposite sides of a transversal that crosses them. When the two lines are parallel, alternate interior angles are equal. They often trace a Z shape: in the diagram, angles 3 and 6 form one pair and angles 4 and 5 the other.Geometry · Grade 8, GeometryTransversalA transversal is a line that crosses two or more other lines at different points. Crossing two lines makes eight angles, four at each intersection. When the two lines are parallel, those angles come in equal groups: corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to 180 degrees.Geometry · Grade 8, GeometryAngleAn 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 6Polygon Angle SumThe 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°.Geometry · Grade 8, Geometry

Interior Angle

What is an interior angle in math?

Interior Angle in a picture

Formal definition and notation

Key facts

Interior angle formulas

Interior Angle examples, step by step

Example 1: Sum for a hexagon

Example 2: Missing angle in a quadrilateral

Example 3: Find the number of sides

Common mistakes

Not to be confused with

Quick check: 3 questions

Interior Angle: frequently asked questions

How do you find an interior angle?

Can an interior angle be more than 180 degrees?

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

Related terms

Practice and tools

Sources

Flashcard

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