Naming and classifying polygons

When you reach for it

Grainne, Ronan and Bronagh work in a shop that cuts road signs out of flat metal sheets. A sign’s shape carries its message before anyone reads a word. A stop sign has eight sides. A yield sign is a triangle standing on one point. A school zone sign has five sides and is shaped like a house. A warning sign is a square turned onto one corner, like a diamond. The round sign before a railroad crossing has no straight sides at all.

To talk about these shapes, the shop needs names, and the names come from one plain count: how many straight sides does the shape have? That count gives the name: triangle, quadrilateral, pentagon, hexagon, and on up. A second question then sorts shapes that share a name: does the shape have a dent? A shape with no dent is convex, and a shape with a dent is concave.

You reach for these names whenever a flat shape with straight sides is in front of you and you need to say what it is, sort it, or know which facts about it apply. The names also open the families inside them. Triangles are sorted further in Classifying triangles, Quadrilaterals in Parallelograms, rectangles, rhombuses and squares: what each one guarantees, and the hierarchy of shapes fits all of those families together.

The procedure

The lesson runs over two days. Part 1 says what a polygon is and names it by its sides. Part 2 sorts polygons into convex and concave, names the regular ones, and puts the names together.

Part 1: what a polygon is, and its name

A polygon is a flat, closed figure made of straight line segments. The segments are its sides. Each side meets exactly two other sides, one at each of its ends, and sides meet only at their ends. A point where two sides meet is a vertex (plural vertices). The corner at each vertex makes an angle of the polygon, measured on the inside.

Each part of that definition rules something out, and Figure 1 shows what.

• Closed. The sides join up all the way around. A path that stops short of where it began has a gap, so it is not a polygon.
• Straight. Every side is a straight segment. A shape with a curved side, such as a circle or a half circle, is not a polygon.
• Sides meet only at their ends. In the bow tie, two sides cross in the middle, at a point that is not an end of either. It is not a polygon.
• Flat. A polygon lies flat, like a drawing on paper. A box is solid, so it is not a polygon, though each flat face of a box is one.

Sides, vertices and angles come in equal numbers. Walk around a polygon in one direction. Each side ends at a vertex, and the next side starts there. So each side can be paired with the vertex at its far end, and each vertex with the one side that arrives at it. Nothing is left over, so the two counts are equal. Each vertex has one angle, so the angles match too. A polygon with 5 sides has 5 vertices and 5 angles, and any one of the three counts gives the other two.

The name counts the sides.

The names are built from old words for numbers: penta means five, hexa six, hepta seven, octa eight (an octopus has eight arms), nona nine and deca ten. The ending -gon comes from a Greek word for a corner or angle, and poly means many, so a polygon is a “many-cornered” shape. Triangle means three angles, and quadrilateral means four sides. For any other count, name the polygon by its number: a polygon with 11 sides is an 11-gon, and one with 20 sides is a 20-gon. A few of these have old names too, such as dodecagon for 12 sides, but 12-gon is just as correct.

The name says nothing else about the shape. Any polygon with six sides is a hexagon, whether its sides are equal or not, whether it is long or short, and whether a corner is pushed in. All three shapes in Figure 2 are hexagons.

Count so that each side is counted once. Pick one vertex as a starting mark and count sides going around in one direction until you are back at the mark. On a shape with many sides, a count by eye slips easily, and the mark is what stops a side being counted twice or missed. Counting vertices gives the same number, so count whichever is easier to see.

Part 2: convex or concave, and regular

Polygons with the same number of sides can still differ. The first difference is a dent.

A polygon is convex if every one of its angles is less than a straight angle, 180^\circ. Every corner points outward. A polygon is concave if at least one of its angles is more than 180^\circ. That corner points into the shape and makes a dent. No angle of a polygon is exactly 180^\circ, because two segments that meet in a straight line make one side, not two.

The dashed segments in Figure 3 show a second test. In a convex polygon, a segment joining any two of its points stays inside it. In a concave polygon, the segment across the mouth of a dent runs outside. The two tests always agree, so use whichever is easier to see.

Every triangle is convex. Its three angles add to 180^\circ (Triangle angle sum), and each angle is more than 0^\circ, so no single angle can reach 180^\circ. A quadrilateral can already be concave, as an arrowhead with one corner pushed in shows.

Regular polygons. A polygon is regular when all its sides are equal and all its angles are equal, as in Classifying shapes in a hierarchy. A stop sign is a regular octagon, and a yield sign is a regular triangle, which is the equilateral one. Both conditions are needed, and Example 6 shows a polygon with equal sides that is not regular. A regular polygon is always convex. The angles of a polygon with n sides add to (n - 2) \times 180^\circ (Angle sums of polygons), which is less than n \times 180^\circ. When all n angles are equal, each one is that total split n ways, so each is less than 180^\circ.

Putting the names together. A full description uses the count and the sort: a convex pentagon, a concave hexagon, a regular octagon. Each polygon has one count, so it gets exactly one name from the table. Each polygon is also either convex or concave and never both, because either every angle is less than 180^\circ or at least one is not.

Worked examples

Examples 1 to 4 go with Part 1. Examples 5 to 7 go with Part 2.

Boundaries

• The names apply only to polygons. A shape with a curved side, a gap, or sides that cross has no side count that names it, however many straight pieces it has.
• A vertex must be a real corner. Two segments that meet in a straight line are one side, so a dot drawn partway along a side does not add a side or a vertex.
• The count is the whole of the name. “Hexagon” says six sides and nothing about lengths or angles. To say more, add convex, concave or regular, or use the family names for Triangles and Quadrilaterals.
• Convex or concave is decided by the angle on the inside. At a dent, the angle seen from outside is less than 180^\circ; the one that counts is the angle inside the shape.
• A polygon is flat. A solid has faces that are polygons, but the solid itself is not one.

Where this goes next

Once a polygon has a name, the facts about it can be gathered. Quadrilaterals split into trapezoids, parallelograms, rectangles and more in Parallelograms, rectangles, rhombuses and squares: what each one guarantees, and Classifying shapes in a hierarchy shows how a shape can belong to several families at once, as in Shapes in more than one category, and quadrilaterals in none. The Perimeter of a polygon adds up its sides, so a regular polygon’s perimeter is its number of sides times the length of one. Tessellations asks which polygons can cover a floor with no gaps and no overlaps, and the answer depends on their angles.

Exercises

Part 1: polygon or not

Say whether each figure is a polygon. If it is, name it. If it is not, say which part of the definition it fails.

1. Six straight segments joined end to end that close up, with sides meeting only at their ends.
2. A half circle: a straight edge with a curved edge joining its two ends.
3. The letter Z, drawn with three straight strokes.
4. A solid cube.
5. Four straight segments joined end to end that close up, with two of them crossing in the middle.

Part 1: naming by sides

6. A polygon has 7 sides. Name it, and give its number of vertices.
7. A polygon has 10 vertices. How many sides does it have, and what is it called?
8. A polygon has 9 angles. Name it.
9. Name a polygon with 15 sides.
10. Name each of the shapes A to F below by its number of sides.

Part 1: counting with signs

11. How many edges do 5 yield signs, 2 stop signs and 3 school zone signs have altogether?
12. Ronan cuts one corner off a pentagon with one straight cut, from a point on one side to a point on the next. Name the shape that is left.
13. Bronagh cuts all three corners off a triangle the same way, with cuts that do not meet. Name the shape that is left.
14. A polygon has as many sides as a stop sign and a yield sign together. Name it.

Part 2: convex or concave

15. For each of the shapes A to F in Exercise 10, say whether it is convex or concave, and count its dents.
16. Each angle of a hexagon measures 120^\circ. Is it convex or concave?
17. A pentagon has angles of 60^\circ, 60^\circ, 240^\circ, 90^\circ and 90^\circ. Is it convex or concave, and how many dents does it have?
18. Grainne joins two vertices of a polygon with a straight segment, and part of the segment lies outside the polygon. What can she say about the polygon?

Part 2: regular polygons and sorting

19. A hexagon has six sides of 5 cm each, but its angles are not all equal. Is it regular?
20. A triangle has three sides of 9 cm each and three angles of 60^\circ each. Is it regular? Describe it as fully as you can.
21. Ronan wants a concave polygon with as few sides as possible. How many sides must it have? Describe one.

Reasoning

22. Explain why every polygon has as many vertices as sides.
23. Explain why no triangle can be concave.
24. Bronagh says, “A polygon whose sides are all equal must be regular.” Use a shape from this lesson to show that she is wrong.
25. Ronan says a circle is a polygon with a great many sides. Explain why a circle is not a polygon.