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Polygons: pentagons, hexagons, octagons and regular shapes

Geometry · level 2, lesson 1 of 6

A question to start

A stop sign has 8 sides. The cells of a honeycomb have 6. A soccer ball is stitched together from shapes with 5 sides and shapes with 6 sides.

Here is a puzzle. A square has all 4 sides the same length. A diamond shape (like the one on a playing card, but with straight sides) also has all 4 sides the same length. So why do mathematicians call the square regular, but not the diamond? What is the square doing that the diamond is not?

By the end of this lesson you will know the names of shapes with up to 10 sides, and you will know the two things a shape must have to be called regular.

The idea

What a polygon is

A polygon is a flat shape that is made only of straight sides, joined end to end, that closes up. The word comes from old Greek and means "many corners".

The places where two sides meet are the corners. The math word for a corner is vertex (more than one are vertices, say "VER-ti-sees").

Three things are needed:

  1. Straight sides only. No curves.
  2. Closed. The sides join up all the way round, with no gap.
  3. Sides only meet at corners. Sides must not cross over each other.
not closed a curved side sides cross

None of these is a polygon. The first has a gap. The second has a curved side. In the third, two sides cross in the middle, like a bow tie. A circle is not a polygon either: it has no straight sides at all.

Naming polygons by counting

In level 1 you learned that a shape with straight sides has as many corners as sides. So to name a polygon, count its sides (or its corners, you get the same number).

Sides Name Say it Where you see one
3 triangle TRY-ang-gul a yield sign
4 quadrilateral kwod-ri-LAT-er-al a door, a book, a kite
5 pentagon PEN-ta-gon the pentagon patches on a soccer ball
6 hexagon HEX-a-gon a honeycomb cell, a nut for a bolt
7 heptagon HEP-ta-gon
8 octagon OCK-ta-gon a stop sign
9 nonagon NON-a-gon
10 decagon DECK-a-gon

The start of each name tells you the number. Penta means 5. Hexa means 6. Octa means 8, like an octopus with 8 arms. Deca means 10, like a decade, which is 10 years.

A quadrilateral is any polygon with 4 sides. Squares and rectangles are quadrilaterals, but so are lots of other shapes: diamonds, kites, slanted shapes, and shapes with no equal sides at all.

Regular polygons

Here are polygons with 3 to 8 sides.

triangle: 3 sides square: 4 sides pentagon: 5 sides hexagon: 6 sides heptagon: 7 sides octagon: 8 sides

These are all regular polygons. A polygon is regular when:

  1. all its sides are the same length, and
  2. all its corners are the same size (equally pointy).

Both are needed. Here is why.

equal sides equal corners regular equal sides corners not equal not regular sides not equal equal corners not regular
  • The square has equal sides and equal corners. Regular!
  • The diamond (its math name is rhombus, say "ROM-bus") has 4 equal sides. But two of its corners are sharp and two are wide. Its corners are not all the same. Not regular.
  • The rectangle has 4 equal corners (all square corners). But it has two long sides and two short ones. Not regular.

So that is the answer to the puzzle at the start. The diamond passes the first test (equal sides) but fails the second (equal corners). A square is the only quadrilateral that passes both tests.

How can you check that corners are the same size, without any measuring tool? Trace one corner on thin paper and lay the tracing on each of the other corners. If they all match, the corners are equal.

Most polygons are not regular

The polygons around us are often not regular. They still have names! A shape's name only depends on counting its sides.

5 sides: pentagon 6 sides: hexagon 10 sides: decagon
  • The house shape has 5 sides. It is a pentagon, but not a regular one.
  • The L shape has 6 sides. Count carefully: up, across, down, across, down, across. It is a hexagon!
  • The star has 10 sides: 5 going in and 5 going out. It is a decagon. A star does not look like a regular decagon at all, but it has 10 straight sides and 10 corners, so that is its name.

Some polygons, like the L shape and the star, have corners that point inwards, like a dent. These are called concave polygons ("con" sounds like "cave": they have a cave in them). Polygons with no dents are convex. Every regular polygon is convex.

A special fact about triangles

Can you draw a triangle with all 3 sides the same length but corners that are different? Try it with three equal sticks (pencils or straws). Lay them in a triangle. However you push them, if the three sides stay the same length, the triangle comes out the same shape every time, and its three corners are equal.

So for triangles, equal sides always give equal corners. A triangle with 3 equal sides is automatically regular. Its special name is equilateral ("equal sides").

But with 4 equal sticks, you can push the square over into a diamond. The sides stay equal, but the corners change. That is why a quadrilateral needs both tests. A triangle made of sticks holds its shape, and a square made of sticks wobbles. This is why builders put triangles into bridges, roof frames and cranes: triangles do not wobble.

Diagonals

A diagonal is a straight line inside a polygon from one corner to another corner that is not next to it. (A line to the corner next door would just be a side.)

pentagon: 5 diagonals hexagon: 9 diagonals
  • A triangle has 0 diagonals: every corner is next to both other corners.
  • A square has 2 diagonals, corner to corner, making an X.
  • A pentagon has 5 diagonals. In a regular pentagon they make a perfect five-pointed star!
  • A hexagon has 9 diagonals.

How do we know a hexagon has 9 without missing any? Count carefully. From each corner of the hexagon, you can go to 6 corners in total, but not to itself, and not to its 2 neighbours. That leaves 3 diagonals from each corner. There are 6 corners, so 6 × 3 = 18. But wait: each diagonal has been counted twice, once from each of its two ends. So there are 18 ÷ 2 = 9 diagonals. That kind of careful counting, and then fixing the double count, is a real mathematician's trick.

Worked examples

Example 1. Name it. A shape has 8 straight sides, all the same length, and 8 corners, all the same size. What is it?

  1. Count the sides: 8. So it is an octagon.
  2. Are all the sides equal? Yes. Are all the corners equal? Yes.
  3. So it is a regular octagon, like a stop sign.

Example 2. Regular or not? A shape has 6 sides, all 5 centimetres long. Two of its corners are very sharp and four are wide. Is it a regular hexagon?

  1. It has 6 sides, so it is a hexagon.
  2. Test 1, equal sides: yes, all 5 centimetres.
  3. Test 2, equal corners: no, two are sharp and four are wide.
  4. It fails test 2, so it is a hexagon but not regular. (It is like a squashed honeycomb cell.)

Example 3. Diagonals of a pentagon. Show that a pentagon has 5 diagonals by counting.

  1. Pick one corner. A pentagon has 5 corners. Leave out the corner itself and its 2 neighbours: 5 − 3 = 2. So 2 diagonals start at each corner.
  2. 5 corners, 2 diagonals each: 5 × 2 = 10.
  3. Each diagonal was counted twice, once from each end. 10 ÷ 2 = 5.
  4. Check with the picture: the five lines of the star. Yes, 5.

Try it

  1. How many sides does an octagon have? A hexagon? A pentagon?
  2. Is a circle a polygon? Why or why not?
  3. A rectangle has 4 equal corners. Why is it not a regular polygon?
  4. Draw a pentagon that is not regular. (Hint: a house shape.)
  5. Is every square a quadrilateral? Is every quadrilateral a square?
  6. A triangle has 3 sides of 7 centimetres each. Must it be regular?
  7. A capital letter T drawn with thick straight lines (like a block letter) is a polygon. How many sides does it have? What is its name?
  8. How many diagonals does a regular octagon have? Use the counting trick.
Answers
  1. An octagon has 8 sides, a hexagon 6 and a pentagon 5. (Octopus 8, hexa 6, penta 5.)
  2. No. A polygon must have straight sides only, and a circle is one curved line with no straight sides and no corners.
  3. A regular polygon needs equal corners and equal sides. A rectangle has two long sides and two short sides, so its sides are not all equal. (Unless it is a square: a square is a regular rectangle.)
  4. For example, a square with a triangle roof on top: 2 walls, a floor and 2 roof sides make 5 sides. The roof sides are a different length from the floor, so it is not regular.
  5. Every square is a quadrilateral, because it has 4 sides. But not every quadrilateral is a square: a rectangle, a diamond or a kite are quadrilaterals that are not squares.
  6. Yes. For a triangle, equal sides always give equal corners. Three equal sticks always make the same shape, so it is an equilateral triangle, and it is regular.
  7. Walk around the block letter T: top edge, right end going down, under the right arm, down the right side of the stem, the bottom of the stem, up the left side of the stem, under the left arm, left end going up. That is 8 sides, so it is an octagon, but not a regular one. It is concave: it has two dents under the arms.
  8. 20. From each corner: 8 corners, minus itself and its 2 neighbours, leaves 5 diagonals. 8 corners × 5 =
  9. Each diagonal is counted twice, so 40 ÷ 2 = 20.

Common mistake

Thinking equal sides is enough to be regular. A diamond (rhombus) has 4 equal sides but is not regular, because its corners are not equal. You must check the corners too. (Triangles are the one exception: for a triangle, equal sides are enough, because the corners are then equal automatically.)

Thinking a polygon must look "nice" to have a name. The L shape is a hexagon, and the star is a decagon. The name only counts sides. "Regular" is the extra word for the very even ones.

Where it leads