A 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.
A polygon is closed, flat and made only of straight sides that meet at their ends. A concave shape like the arrowhead still counts.
Formal definition and notation
A polygon is a closed plane figure formed by n ≥ 3 line segments (sides) joined end to end, where each side meets exactly two others, one at each endpoint, and no two sides cross. A polygon is convex if every interior angle is less than 180° and concave if at least one interior angle is greater than 180°.
Symbol
Read it as
Meaning
n-gon
n gon
a polygon with n sides, such as a 12-gon
ABCDE
polygon A B C D E
a polygon named by its vertices in order around the shape
Key facts
Every polygon has the same number of sides, vertices and angles.
Every regular polygon is a polygon, but a polygon only needs straight sides and a closed shape. A regular one also has all sides equal and all angles equal.
Quick check: 3 questions
Question 1 of 3
Which shape is a polygon?
Show answer and explanation
Answer: B. A triangle is closed and made of 3 straight sides. A circle and a half circle have curved parts, and an open zigzag is not closed.
Question 2 of 3
How many sides does a heptagon have?
Show answer and explanation
Answer: C. Hepta- means 7, so a heptagon has 7 sides. A hexagon has 6, an octagon 8 and a nonagon 9.
Question 3 of 3
Each side of a regular hexagon is 4.5 inches long. What is its perimeter?
Show answer and explanation
Answer: A. A hexagon has 6 equal sides: 6 × 4.5 = 27. Choice B uses 5 sides, C uses 4, and D uses 8.
Polygon: frequently asked questions
Is a circle a polygon?
No. A polygon is made only of straight line segments that meet at corners. A circle is one smooth curve, with no straight sides and no vertices, so it is not a polygon. The same rule rules out ovals, half circles and any shape that has even one curved edge.
What is the difference between a regular and an irregular polygon?
A regular polygon has all sides the same length and all angles the same size, like a square or the octagon on a stop sign. An irregular polygon breaks at least one of those rules, like a rectangle that is not a square. Both kinds keep the same name: any 5-sided polygon is a pentagon.
Can a polygon have a dent in it?
Yes. A polygon with a dent is called concave: at least one interior angle is more than 180 degrees, so the shape caves inward, like an arrowhead. A polygon with no dents is convex: every interior angle is less than 180 degrees. Both kinds are polygons, as long as the sides are straight and do not cross.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 2.G.A.1, CCSS 6.G.A.3
Cite this entry
Effortless Math Team. “Polygon.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/polygon/
Flashcard
TermPolygonclosed, flat, straight sides
DefinitionA closed flat shape made of straight sides that meet at vertices. A pentagon has 5 sides, 5 vertices and 5 angles; a circle is not a polygon.
Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5, Grade 6VertexA 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 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 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 5CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3SquareA 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 3Regular 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 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, GeometryAreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5