Polygons explained: what a polygon is, what each one is called and how to tell

Your second grader brings home a sheet of shapes asking "Which ones are polygons?", and the hexagon and the circle look equally friendly. The answer to what is a polygon fits in one rule: a flat, closed shape with straight sides. With that rule, naming a polygon comes down to counting sides and checking a few properties.

What is a polygon, and why it matters in grades 2 to 5

A polygon is a flat shape that is closed and made only of straight sides, and it has at least 3 of them. OpenStax Contemporary Mathematics puts it in one line: "A polygon is a closed, two-dimensional shape classified by the number of straight-line sides." The sides meet only at their ends, at points called vertices (corners).

Polygons run through the elementary geometry standards. Common Core standard 2.G.A.1 asks second graders to "recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces" and to "identify triangles, quadrilaterals, pentagons, hexagons, and cubes." Grade 3 sorts quadrilaterals, and grade 5 puts all of them in a family tree, as laid out in what math each grade should know.

The word helps: "poly" means many and "gon" means angle, so a polygon is a shape with many angles.

Polygon or not: circles, open shapes, curves and cubes

A shape fails the polygon test when it breaks any one of the four rules: flat, closed, straight sides only, at least 3 sides. Four shapes trip children up most often:

• Circle: a circle has one curved edge and no straight sides, so it is not a polygon.
• Open shape: a zigzag of straight lines whose ends do not meet is not closed, so it is not a polygon.
• Shape with a curved side: a "tombstone" shape with three straight sides and a rounded top fails, because one side is curved.
• Cube or box: a cube is a 3D solid. Its faces are squares, and each face is a polygon, but the cube itself is not.

Crossing sides are a fifth case. A figure eight drawn with straight lines, where two sides cross in the middle, is not a simple polygon, and school math leaves it out. Ask your child to trace the edge with a finger: if the finger crosses its own path, the shape does not count.

How to name a polygon, step by step

Naming a polygon takes five checks, from the polygon test to the shape of its corners. Go through them aloud with a paper shape or a pattern block.

1. Check that it is a polygon. Trace the edge with a finger. It must start and end at the same point, with straight sides and no crossings.
2. Count the sides. Put a pencil dot on the first side so you know where you started, then count around. The number of sides always equals the number of corners.
3. Match the count to a name. 3 is a triangle, 4 a quadrilateral, 5 a pentagon, 6 a hexagon, 7 a heptagon, 8 an octagon, 9 a nonagon and 10 a decagon. The prefixes tri, quad, penta, hexa, hepta, octa, nona and deca mean 3 to 10.
4. Check regular or irregular. OpenStax states that "if all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons." Measure the sides with a ruler and compare the corners by laying one shape on top of another.
5. Check convex or concave. A convex polygon has every corner pointing outward. A concave polygon has at least one corner that dents inward, like the notch of an arrow or a star.

A full name stacks the checks, such as "a regular hexagon" for a honeycomb cell or "an irregular concave heptagon" for an arrow.

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The part people get wrong: a square is a rectangle and a rhombus

A square belongs to more than one family, and most adults learned it the other way. A rectangle is any quadrilateral with four right angles, so a square counts. A rhombus is any quadrilateral with four equal sides, so a square counts there as well.

Common Core 5.G.B.3 spells out why: "all rectangles have four right angles and squares are rectangles, so all squares have four right angles." The next standard, 5.G.B.4, asks fifth graders to "classify two-dimensional figures in a hierarchy based on properties." The reverse is false: most rectangles are not squares, because their sides are not all equal.

Polygon names, sides and angles from triangle to decagon

Polygon names follow the side count, and the angle sum follows a formula. OpenStax gives the sum of the interior angles of a polygon with n sides as (n − 2) × 180°, and each angle of a regular polygon as (n − 2) × 180° ÷ n. The formula works because lines from one corner split any convex polygon into n − 2 triangles, and each triangle holds 180°, a fact covered in types of triangles and angles.

The stop sign row is a rule of the road. The federal Manual on Uniform Traffic Control Devices, section 2B.05 says "the STOP sign shall be an octagon with a white legend and border on a red background." Its eight sides are equal, so it is a regular octagon with 135° corners.

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Which shapes fit together without gaps

Regular polygons tile a floor only when one angle divides 360° exactly. Triangles (60° × 6 = 360°), squares (90° × 4 = 360°) and hexagons (120° × 3 = 360°) fit around a point with no gaps, which is why honeycombs and bathroom tiles use them. A regular pentagon's 108° does not divide 360°, so regular pentagon tiles always leave gaps.

The quadrilateral family at home

Quadrilaterals are the four-sided polygons, and third grade spends most of its shape time on them. Common Core 3.G.A.1 asks children to "recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories." The types of quadrilaterals your child meets are:

• Parallelogram: two pairs of parallel sides.
• Rectangle: a parallelogram with four right angles.
• Rhombus: a parallelogram with four equal sides.
• Square: a rectangle and a rhombus at once, with four right angles and four equal sides.
• Trapezoid: at least one pair of parallel sides (some US textbooks say exactly one pair, so check your child's book).
• Kite: two pairs of equal sides next to each other, with no parallel sides needed.

A kitchen hunt makes the family stick. A cereal box face is a rectangle, a cracker may be a square, and a cake slice cut on a slant can make a parallelogram. Ask your child to name each one twice: "a square, and also a rectangle."

Practice rule: name it by its properties, not its orientation

A shape keeps its name when you turn it. A square tilted 45° is often called a "diamond", but its sides are still equal and its corners are still 90°, so it is still a square. Turn paper shapes on the table and ask "Did it change?" until the answer is an automatic "no."

Symmetry gives a second property check. A square has 4 lines of symmetry and a non-square rectangle has 2, so folding a paper shape tells them apart; the guide to lines of symmetry shows how to fold for them. Plotting a shape's corners on a grid, as in the coordinate plane explained, is the fifth- and sixth-grade version of the same check.

How to check your child's progress with polygons

Five short questions show whether your child knows the rules or only recognizes pictures. Draw each shape on paper and ask without hints.

• Count: a shape with 6 sides is drawn lopsided. Is it still a hexagon? Yes, any 6-sided polygon is a hexagon.
• Rule out: is a semicircle a polygon? No, one side is curved.
• Regular: is a long rectangle a regular polygon? No, its sides are not all equal.
• Family: is every square a rhombus? Yes. Is every rhombus a square? No.
• Angle sum (grade 5 and up): which polygon has interior angles that add up to 1,080°? 1,080 ÷ 180 = 6, and 6 + 2 = 8, so it is an octagon.

A weekly five-minute shape hunt

A weekly shape hunt shows progress without a test. Give your child five minutes to find and name polygons around the house, then count how many full names they give, such as "irregular pentagon" or "regular hexagon."

A second grader who names triangles, quadrilaterals, pentagons and hexagons by counting sides is on track for 2.G.A.1. A fourth or fifth grader should add regular or irregular and explain why a square is a rectangle. Free math games for 3rd graders keep the number facts behind those checks quick.

Common polygon mistakes and edge cases

Four mistakes cause most wrong answers on polygon worksheets, and each has a short fix.

• Calling a tilted square a diamond: "diamond" is not a math name. Turn the shape back and check sides and angles.
• Counting corners twice: children lose count on shapes with many sides. Mark the starting side with a dot and stop when you return to it.
• Thinking "hexagon" means regular: a hexagon is any 6-sided polygon. A lopsided one still counts, and only its "regular" label changes.
• Calling a cube a square: a cube is a 3D solid with 6 square faces. Name the face a square and the whole object a cube.

Concave polygons: the interior angle formula still works

A concave polygon has at least one interior angle greater than 180°. OpenStax notes that "what makes a polygon concave is having an interior angle that measures greater than 180°." The arrow in the polygon-or-not picture has 7 sides, with two notch angles of 270°, two right angles at the tail and three angles in the arrowhead that belong to one triangle.

Those angles add up to 270° + 270° + 90° + 90° + 180° = 900°, which matches (7 − 2) × 180° = 900°. The formula holds for every simple polygon, convex or concave. The same arrow is a good next step for area: split it into a rectangle and a triangle, as shown in area and perimeter explained.

Your polygon checklist and next steps

• Flat and closed: the edge starts and ends at the same point and lies flat on paper.
• Straight sides only: any curve, as in a circle or a semicircle, rules the shape out.
• Count the sides: 3 to 10 sides give triangle through decagon.
• Regular or irregular: all sides and all angles equal makes it regular.
• Convex or concave: a corner that dents inward makes it concave.
• Name by properties: a turned square is still a square, and still a rectangle and a rhombus.

Circles are the natural next shape after polygons, and the guide to the circumference and area of a circle picks up there. For a short shared session, the math game for parents and kids keeps the arithmetic behind angle sums quick.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice, so the multiplication and division behind angle sums stay fast while your child learns the shapes.