A 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.
In Spanish
polígono regular
Also called
regular n-gon, equilateral and equiangular polygon
A regular polygon needs both: equal sides (ticks) and equal angles. The rhombus has only the first and the rectangle only the second.
Formal definition and notation
A polygon is regular when it is both equilateral (all sidescongruent) and equiangular (all interior angles congruent). A regular n-gon has a center equidistant from all vertices; it has n lines of symmetry and rotational symmetry of order n. Its area is A = 12aP, where a is the apothem and P the perimeter.
A regular n-gon has n lines of symmetry and matches itself after a turn of 360°n.
Only three regular polygons tile a flat floor by themselves with no gaps: equilateral triangles, squares and regular hexagons.
Regular polygon formulas
Regular polygon formulas
each interior angle = (n − 2) × 180°n
360°n
each exterior angle (and the smallest turn that maps the polygon onto itself)
A = 12aP
area from the apothem and the perimeter
n = 360°exterior angle
number of sides from one exterior angle
Interior and exterior angles at each vertex add to 180°: for a regular hexagon, 120° + 60° = 180°.
Try it: turn a regular polygon
Interactive
This square is a regular polygon centered at the origin. Choose Rotate and try 90°, 180° and 270°: every quarter turn lands the image exactly on the original, because a regular 4-gon matches itself after each 360° ÷ 4 = 90° turn.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Regular Polygon examples, step by step
Example 1: Each angle of a regular pentagon
Warm-up
Find the measure of each interior angle of a regular pentagon.
Angle sum: (5 − 2) × 180 = 540°.
The 5 angles are equal: 540 ÷ 5 = 108.
Answer: Each interior angle is 108°.
Example 2: Sides from an exterior angle
Core
Each exterior angle of a regular polygon is 24°. How many sides does it have, and what is each interior angle?
Exterior angles add to 360°: n = 360 ÷ 24 = 15.
Interior angle: 180 − 24 = 156°.
Answer: It has 15 sides, and each interior angle is 156°.
Example 3: Area of a regular hexagon
Test-style
A regular hexagon has sides of 6 cm. Find its perimeter and its exact area.
Perimeter: 6 × 6 = 36 cm.
Lines from the center to the 6 vertices cut it into 6 equilateral triangles with sides of 6 cm.
One equilateral triangle: √34 × 62 = 9√3.
Hexagon: 6 × 9√3 = 54√3 ≈ 93.5.
Answer: The perimeter is 36 cm and the area is 54√3, about 93.5 square cm.
Common mistakes
✗ MistakeA rhombus is regular because all four sides are equal.
✓ InsteadA rhombus is regular only if its angles are also equal, which makes it a square.
Regular requires equal sides and equal angles. A leaning rhombus has angles such as 60° and 120°.
✗ MistakeA rectangle is a regular polygon because all its angles are 90°.
✓ InsteadA 6 by 4 rectangle is not regular: its sides are not all equal.
Equal angles alone are not enough; the square is the only regular quadrilateral.
✗ MistakeEach interior angle of a regular hexagon is 360 ÷ 6 = 60°.
✓ Instead360 ÷ 6 = 60° is the exterior angle. The interior angle is 180 − 60 = 120°.
Dividing 360 by n gives the exterior turn at each corner. Interior and exterior angles add to 180°.
Not to be confused with
Regular Polygonvs.Irregular polygon
An irregular polygon has at least two different sides or two different angles, like a 6 by 4 rectangle. It keeps the same name (rectangle, hexagon) but is not regular.
Regular Polygonvs.Equilateral polygon
Equilateral means equal sides only. For triangles that forces equal angles, but a rhombus is equilateral without being regular.
Quick check: 3 questions
Question 1 of 3
What is each interior angle of a regular hexagon?
Show answer and explanation
Answer: C.(6 − 2) × 1806 = 7206 = 120. A is the exterior angle, B is a regular pentagon, and D is the total.
Question 2 of 3
Which shape is always a regular polygon?
Show answer and explanation
Answer: D. A square has 4 equal sides and 4 equal angles. A rectangle can have unequal sides, a rhombus unequal angles, and an isosceles triangle can have a different third side.
Question 3 of 3
What is each exterior angle of a regular decagon (10 sides)?
Show answer and explanation
Answer: A.360 ÷ 10 = 36. B is the interior angle, C divides 180 by 10, and D is the interior angle sum.
Regular Polygon: frequently asked questions
Is a rectangle a regular polygon?
Only if it is a square. A regular polygon needs all sides equal and all angles equal. Every rectangle has four equal 90-degree angles, but its length and width usually differ, so most rectangles are not regular. The square, with equal sides as well, is the one regular rectangle.
How do you find the interior angle of a regular polygon?
Find the angle sum, (n − 2) × 180, then divide by the number of angles, n. For a regular octagon: (8 − 2) × 180 = 1,080, and 1,080 ÷ 8 = 135 degrees. A shortcut: subtract the exterior angle from 180, so 180 − 3608 = 180 − 45 = 135.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSG-CO.A.3
Cite this entry
Effortless Math Team. “Regular Polygon.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/regular-polygon/
Flashcard
TermRegular Polygonequal sides AND equal angles
DefinitionA polygon with all sides equal and all angles equal. A regular hexagon has six 120° angles; a rhombus is not regular.
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 2CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometryInterior 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, GeometrySymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometryPerimeterPerimeter 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 5NumberA 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 2MidpointA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is (x1 + x22, y1 + y22).Geometry · Grade 8, Algebra 1, GeometryLengthLength 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 3TriangleA 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 5Equilateral 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 7QuadrilateralA 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 5SquareA 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 3Polygon 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, GeometryRotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).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