Polygons and Regular Polygons

What a polygon is and how polygons are named, which polygons are convex, which are regular, and the lines of symmetry and the order of rotational symmetry of a regular polygon.

Before you start

A triangle has 3 sides. An equilateral triangle has all three sides equal, and each of its angles is 60° (equilateral △) (from the lesson "Isosceles and Equilateral Triangles").
A quadrilateral has 4 sides. A rhombus has four equal sides, a rectangle has four right angles, and a square has four equal sides and four right angles (from the lesson "Properties of Special Quadrilaterals").
A line of symmetry folds a figure exactly onto itself. The order of rotational symmetry is the number of times a figure fits onto itself in one complete turn about its centre, counting the complete turn; order 1 means no rotational symmetry. For example, an equilateral triangle has 3 lines of symmetry and rotational symmetry of order 3, and an isosceles triangle that is not equilateral has 1 line of symmetry and order 1 (from the lesson "Line and Rotational Symmetry").
Angles at a point add up to 360° (∠s at a pt), and angles on a straight line add up to 180° (adj. ∠s on a str. line).

What Is a Polygon?

Triangles and quadrilaterals belong to a larger family of shapes: the polygons. A floor tile, a stop sign and a cell of a honeycomb are all polygons.

Polygon

A polygon is a closed plane figure with three or more straight sides that meet only at their ends.

A polygon is named by its number of sides. A polygon with $n$ sides is also called an $n$ -sided polygon; this is how a polygon with more than 10 sides is usually named, such as a 15-sided polygon.

Number of sides	Name
3	triangle
4	quadrilateral
5	pentagon
6	hexagon
7	heptagon
8	octagon
9	nonagon
10	decagon

In pentagon $ABCDE$ below, the points $A$ , $B$ , $C$ , $D$ and $E$ are its vertices (one is a vertex), and $AB$ , $BC$ , $CD$ , $DE$ and $EA$ are its sides. A polygon is named by its vertices in order round it.

An interior angle is an angle inside the polygon at one vertex, between the two sides that meet there. The interior angle at $B$ is $\angle ABC$ (marked). A pentagon has five interior angles, one at each vertex. A polygon always has as many vertices and as many interior angles as it has sides.

A diagonal is a straight line joining two vertices that are not next to each other. $AC$ (amber) is a diagonal; $AB$ is not, because it is a side.

Example 1: The parts of a polygon

Use pentagon $ABCDE$ in the figure above.

(a) Name the two sides that meet at $D$ , and name the interior angle at $D$ .

(b) Is $BE$ a side or a diagonal? Is $DE$ ?

Solution

(a) Going round the pentagon, the vertices next to $D$ are $C$ and $E$ . So the sides that meet at $D$ are $CD$ and $DE$ , and the interior angle at $D$ is $\angle CDE$ , with $D$ , its vertex, in the middle.

(b) The vertices next to $B$ are $A$ and $C$ , so $E$ is not next to $B$ : $BE$ is a diagonal. $D$ and $E$ are next to each other, so $DE$ is a side.

(c) From each vertex there is a diagonal to each vertex that is not next to it. From $A$ : $AC$ and $AD$ . From $B$ : $BD$ and $BE$ . From $C$ : $CE$ ( $CA$ is $AC$ again). From $D$ and from $E$ , every diagonal has been named already.

(a) $CD$ and $DE$ ; the interior angle at $D$ is $\angle CDE$ . (b) $BE$ is a diagonal; $DE$ is a side. (c) 5 diagonals: $AC$ , $AD$ , $BD$ , $BE$ and $CE$ .

Each part of the definition matters. The figures below are not polygons: the first is not closed, and in the second two sides cross each other at a point that is not an end of either. A circle, such as a round clock face, is not a polygon either: it has no straight sides.

Not closed

Two sides cross

Convex Polygons

Convex polygon

A polygon is convex if every one of its interior angles is less than 180°.

A polygon with an interior angle greater than 180° (a reflex angle) is not convex: it has a dent, where one corner points inwards. In the second figure below, the interior angle marked in pink is about 252°.

You can also tell by the diagonals. In a convex polygon every diagonal lies inside the polygon. In the polygon that is not convex, the diagonal drawn in amber passes outside it, across the dent.

Convex

Not convex

Example 2: Polygon or not? Convex or not?

For each figure, say whether it is a polygon. For each polygon, name it and say whether it is convex. Give a reason each time.

Solution

P is closed, with six straight sides that meet only at their ends, so it is a polygon: a hexagon. Every interior angle is less than 180° (every corner points outwards), so it is convex.

Q is closed, with six straight sides that meet only at their ends, so it is a polygon: a hexagon. At the inner corner of the L, the angle outside the L is a right angle, so the interior angle is $360^{\circ} - 90^{\circ} = 270^{\circ}$ (∠s at a pt). This is more than 180°, so Q is not convex.

R is closed, but two of its sides cross each other at a point that is not an end of either. So R is not a polygon.

P is a convex hexagon. Q is a hexagon that is not convex (one interior angle is 270°). R is not a polygon (two sides cross).

Most of the polygons you will work with from now on are convex. The rules for the angles of a polygon, in the next lesson, "Polygons and Their Interior Angles", are for convex polygons.

Regular Polygons

Regular polygon

A regular polygon is a polygon with all its sides equal and all its interior angles equal.

An equilateral triangle is a regular triangle, and a square is a regular quadrilateral. Four more regular polygons have names you should know: the regular pentagon, hexagon, octagon and decagon, drawn below. The ticks show that the sides are equal, and in each the interior angles are all equal too.

Regular pentagon

Regular hexagon

Regular octagon

Regular decagon

Both conditions are needed. A rhombus that is not a square has four equal sides, but its angles are not all equal, so it is not regular. A rectangle that is not a square has four equal angles (each 90°), but its sides are not all equal, so it is not regular either.

Equal sides, unequal angles

Equal angles, unequal sides

Example 3: Regular or not?

Is each polygon regular? Give a reason.

(a) A triangle with all three sides equal.

(b) A quadrilateral $ABCD$ with four equal sides and $\angle DAB = 90^{\circ}$ .

Solution

(a) A triangle with three equal sides is equilateral, so each of its angles is 60° (equilateral △). Its sides are all equal and its angles are all equal, so it is regular.

(b) A quadrilateral with four equal sides is a rhombus, and a rhombus is a parallelogram.

$\angle BCD = \angle DAB = 90^{\circ}$ (opp. ∠s of //gram)

$\angle ABC = 180^{\circ} - 90^{\circ} = 90^{\circ}$ (int. ∠s, AD // BC)

$\angle CDA = \angle ABC = 90^{\circ}$ (opp. ∠s of //gram)

All four angles are 90° and all four sides are equal: $ABCD$ is a square, so it is regular.

(a) Regular (an equilateral triangle). (b) Regular: it is a square. (c) Not regular: its angles are equal but its sides are not.

The Symmetry of a Regular Polygon

A regular polygon is as symmetrical as a polygon with its number of sides can be. A regular pentagon has 5 lines of symmetry. Because 5 is odd, there is no vertex opposite a vertex: each line runs from a vertex to the midpoint of the side opposite it. A regular hexagon has 6 lines of symmetry. Because 6 is even, opposite each vertex there is another vertex, and opposite each side another side: 3 lines join opposite vertices and 3 join the midpoints of opposite sides.

Regular pentagon: 5 lines

Regular hexagon: 6 lines

Turn a regular pentagon about its centre. After $\frac{1}{5}$ of a complete turn ( $360^{\circ} \div 5 = 72^{\circ}$ ), each vertex has moved to where the next vertex was, and the pentagon fits exactly onto itself. It fits again after 144°, 216°, 288° and 360°: five times in one complete turn, so its order of rotational symmetry is 5. In the same way a regular hexagon fits onto itself after every turn of $360^{\circ} \div 6 = 60^{\circ}$ , and has rotational symmetry of order 6.

Symmetry of a regular polygon

A regular polygon with $n$ sides has $n$ lines of symmetry and rotational symmetry of order $n$ . The smallest turn about its centre that makes it fit onto itself is $360^{\circ} \div n$ .

If $n$ is odd, each line of symmetry joins a vertex to the midpoint of the opposite side.

If $n$ is even, half of the lines join opposite vertices and half join the midpoints of opposite sides.

Every line of symmetry passes through the centre of the polygon.

Regular polygon	Lines of symmetry	Order of rotational symmetry
Equilateral triangle	3	3
Square	4	4
Regular pentagon	5	5
Regular hexagon	6	6
Regular octagon	8	8
Regular decagon	10	10

Only a regular polygon is sure to have this much symmetry. A rectangle that is not a square has 4 sides but only 2 lines of symmetry and rotational symmetry of order 2.

Example 4: The symmetry of a regular octagon

The figure shows a regular octagon.

(a) How many lines of symmetry does it have? Describe where they lie.

(b) State its order of rotational symmetry, and the smallest turn about its centre that makes it fit onto itself.

Solution

(a) A regular polygon with $n$ sides has $n$ lines of symmetry, and here $n = 8$ . Since 8 is even, half of them, 4 lines, join opposite vertices, and the other 4 join the midpoints of opposite sides.

(b) The order of rotational symmetry is also 8. The octagon fits onto itself 8 times in one complete turn, so the smallest turn is $360^{\circ} \div 8 = 45^{\circ}$ .

(a) 8 lines of symmetry: 4 through opposite vertices and 4 through the midpoints of opposite sides. (b) Order 8; the smallest turn is 45°.

Example 5: Naming the lines of symmetry

Regular hexagon

ABCDEF

Regular pentagon

ABCDE

(a) Describe each line of symmetry of the regular hexagon $ABCDEF$ , using its letters.

(b) In the regular pentagon $ABCDE$ , the line of symmetry through $C$ crosses the opposite side. Which side, and at what point?

Solution

(a) The hexagon has 6 sides, an even number, so 3 lines join opposite vertices and 3 join the midpoints of opposite sides.

Going round, the vertex opposite $A$ is three places on: $D$ . In the same way $B$ is opposite $E$ , and $C$ is opposite $F$ . So three lines are $AD$ , $BE$ and $CF$ .

The side opposite $AB$ is three sides on: $DE$ . In the same way $BC$ is opposite $EF$ , and $CD$ is opposite $FA$ . So the other three lines join the midpoints of $AB$ and $DE$ , of $BC$ and $EF$ , and of $CD$ and $FA$ .

(b) The pentagon has 5 sides, an odd number, so the line of symmetry through $C$ runs to the midpoint of the side opposite $C$ . The sides that meet at $C$ are $BC$ and $CD$ ; next come $DE$ and $AB$ , and the side between them, farthest from $C$ , is $EA$ .

(a) $AD$ , $BE$ and $CF$ , and the lines through the midpoints of $AB$ and $DE$ , of $BC$ and $EF$ , and of $CD$ and $FA$ . (b) It crosses $EA$ at its midpoint.

A line of symmetry can be used to find angles. Folding along it puts one half of the polygon exactly onto the other half. So:

• where it passes through a vertex, it cuts the interior angle there into two equal angles;

• where it crosses a side at its midpoint, the two angles it makes with that side are equal, and since they add up to 180° (adj. ∠s on a str. line), each is 90°.

Example 6: Using a line of symmetry

$ABCDE$ is a regular pentagon, and each of its interior angles is 108°. $M$ is the midpoint of $CD$ , and $AM$ is drawn.

(a) Explain why $AM$ is a line of symmetry of the pentagon.

(b) Find $\angle BAM$ .

Solution

(a) A regular pentagon has 5 lines of symmetry, and since 5 is odd each joins a vertex to the midpoint of the opposite side. The side opposite $A$ is $CD$ , so the line of symmetry through $A$ is $AM$ .

(b) Folding along $AM$ puts $AB$ onto $AE$ , so $\angle BAM = \angle EAM$ . Together they make the interior angle $\angle BAE = 108^{\circ}$ .

$\angle BAM = 108^{\circ} \div 2 = 54^{\circ}$ ( $AM$ is a line of symmetry)

$\angle AMC + \angle AMD = 180^{\circ}$ (adj. ∠s on a str. line), so $\angle AMC = 180^{\circ} \div 2 = 90^{\circ}$ .

Check with quadrilateral $ABCM$ : $54^{\circ} + 108^{\circ} + 108^{\circ} + 90^{\circ} = 360^{\circ}$ (∠ sum of quad.).

(b) $\angle BAM = 54^{\circ}$ (c) $\angle AMC = 90^{\circ}$ : the line of symmetry $AM$ is perpendicular to $CD$ .

Common Mistakes

Calling any closed shape a polygon. A polygon has straight sides that meet only at their ends. A round clock face has no straight sides, and a figure whose sides cross each other is not a polygon.
Calling a polygon regular because its sides are equal, or because its angles are equal. A rhombus has equal sides and a rectangle has equal angles, but neither is regular unless it is a square. A regular polygon needs both.
Counting only half the lines of symmetry. A regular hexagon has 6 lines of symmetry, not 3: as well as the 3 lines through opposite vertices, there are 3 through the midpoints of opposite sides.
Joining two vertices when $n$ is odd. In a regular pentagon there is no vertex opposite a vertex, so a line such as $AC$ is not a line of symmetry. Each line runs from a vertex to the midpoint of the opposite side.
Thinking every polygon with $n$ sides has $n$ lines of symmetry. Only a regular polygon is sure to: a rectangle that is not a square has 4 sides but 2 lines of symmetry.

Practice

Try each question, then open its solution to check your working. The figures are drawn to scale.

Practice 1: Names

Name the polygon with (a) 6 sides, (b) 8 sides, (c) 10 sides, (d) 5 sides.

(e) How many vertices and how many interior angles does a decagon have?

ShowHide solution

(a) hexagon (b) octagon (c) decagon (d) pentagon

(e) A polygon has as many vertices and as many interior angles as sides. A decagon has 10 sides, so it has 10 vertices and 10 interior angles.

(a) Hexagon (b) octagon (c) decagon (d) pentagon (e) 10 vertices and 10 interior angles.

Practice 2: Polygon or not? Convex or not?

For each figure, say whether it is a polygon. For each polygon, name it and say whether it is convex. In (d), every corner is a right angle.

(a)

(b)

(c)

(d)

ShowHide solution

(a) Closed, with 7 straight sides that meet only at their ends: a polygon, a heptagon. Every corner points outwards, so every interior angle is less than 180°: it is convex.

(b) Each line crosses two others at points that are not ends of either, so this figure is not a polygon. (The outline round the outside of the star, with 10 sides, would be a polygon: one that is not convex.)

(c) Closed, with 6 straight sides that meet only at their ends: a polygon, a hexagon. The corner at the back of the arrow points inwards, so its interior angle is more than 180°: it is not convex.

(d) Closed, with 8 straight sides that meet only at their ends: a polygon, an octagon. At each of the two inner corners the interior angle is $360^{\circ} - 90^{\circ} = 270^{\circ}$ (∠s at a pt), more than 180°: it is not convex.

(a) A convex heptagon. (b) Not a polygon (its sides cross). (c) A hexagon that is not convex. (d) An octagon that is not convex.

Practice 3: Sides, angles and diagonals

Use hexagon $ABCDEF$ .

(a) Name the two sides that meet at $C$ , and name the interior angle at $C$ .

(b) Which of $AC$ , $CD$ , $BE$ and $FA$ are diagonals?

ShowHide solution

(a) The vertices next to $C$ are $B$ and $D$ , so the sides are $BC$ and $CD$ , and the interior angle at $C$ is $\angle BCD$ .

(b) $A$ and $C$ are not next to each other (with $B$ between them), so $AC$ is a diagonal. $B$ and $E$ are not next to each other, so $BE$ is a diagonal. $CD$ and $FA$ join vertices that are next to each other: they are sides.

(c) From $B$ there is a diagonal to every vertex except $B$ itself and the two next to it, $A$ and $C$ : so to $D$ , $E$ and $F$ .

(a) $BC$ and $CD$ ; $\angle BCD$ . (b) $AC$ and $BE$ . (c) $BD$ , $BE$ and $BF$ : 3 diagonals.

Practice 4: Convex from the angles

Ali measures the interior angles of two hexagons.

(a) 100°, 150°, 190°, 90°, 120°, 70°

(b) 110°, 125°, 130°, 115°, 120°, 120°

Which hexagon is convex? Explain.

ShowHide solution

A polygon is convex if every one of its interior angles is less than 180°.

(a) One angle is 190°, which is more than 180°, so this hexagon is not convex.

(b) Every angle is less than 180°, so this hexagon is convex.

Hexagon (b) is convex. Hexagon (a) is not, because one of its angles, 190°, is more than 180°.

Practice 5: Regular or not?

Say whether each of these is a regular polygon, and give a reason.

(a) a square; (b) a rhombus with angles of 70° and 110°; (c) a rectangle 6 cm long and 4 cm wide; (d) a triangle with angles of 60°, 60° and 60°; (e) the round face of Wei Ming's clock.

ShowHide solution

(a) Regular: its four sides are equal and its four angles are all 90°.

(b) Not regular: its four sides are equal, but its angles (70°, 110°, 70°, 110°) are not all equal.

(d) Regular: a triangle with three equal angles is equilateral (equilateral △), so its sides are equal too.

(e) Not a polygon at all: a circle has no straight sides.

Regular: (a) and (d). Not regular: (b) and (c). (e) is not a polygon.

Practice 6: Lines of symmetry

For (a) a regular nonagon (9 sides) and (b) a regular decagon, state the number of lines of symmetry and the order of rotational symmetry, and describe where the lines of symmetry lie.

ShowHide solution

(a) A regular nonagon has 9 lines of symmetry and rotational symmetry of order 9. Since 9 is odd, there is no vertex opposite a vertex: each line joins a vertex to the midpoint of the side opposite it. (The smallest turn that fits it onto itself is $360^{\circ} \div 9 = 40^{\circ}$ .)

(b) A regular decagon has 10 lines of symmetry and rotational symmetry of order 10. Since 10 is even, 5 lines join opposite vertices and 5 join the midpoints of opposite sides.

Regular nonagon: 9 lines

Regular decagon: 10 lines

(a) 9 lines, each from a vertex to the midpoint of the opposite side; order 9. (b) 10 lines, 5 through opposite vertices and 5 through the midpoints of opposite sides; order 10.

Practice 7: Naming the lines of symmetry

Regular octagon

ABCDEFGH

Regular pentagon

PQRST

(a) Name the lines of symmetry of the regular octagon $ABCDEFGH$ that join two of its vertices.

(b) Describe its other lines of symmetry.

ShowHide solution

(a) 8 is even, so 4 lines join opposite vertices. The vertex opposite $A$ is four places on, $E$ ; in the same way $B$ is opposite $F$ , $C$ is opposite $G$ and $D$ is opposite $H$ .

(b) The other 4 lines join the midpoints of opposite sides. The side opposite $AB$ is four sides on, $EF$ ; so the lines join the midpoints of $AB$ and $EF$ , of $BC$ and $FG$ , of $CD$ and $GH$ , and of $DE$ and $HA$ .

(c) 5 is odd, so the line through $S$ runs to the midpoint of the side opposite $S$ . The sides that meet at $S$ are $RS$ and $ST$ ; next come $QR$ and $TP$ , and the side between them, farthest from $S$ , is $PQ$ .

(a) $AE$ , $BF$ , $CG$ and $DH$ . (b) The lines through the midpoints of $AB$ and $EF$ , $BC$ and $FG$ , $CD$ and $GH$ , $DE$ and $HA$ . (c) It crosses $PQ$ at its midpoint.

Practice 8: From the number of lines of symmetry

A regular polygon has 12 lines of symmetry.

(a) How many sides does it have?

(b) What is its order of rotational symmetry, and what is the smallest turn about its centre that makes it fit onto itself?

ShowHide solution

(a) A regular polygon with $n$ sides has $n$ lines of symmetry, so $n = 12$ : it has 12 sides.

(b) Its order of rotational symmetry is also 12. It fits onto itself 12 times in one complete turn, so the smallest turn is $360^{\circ} \div 12 = 30^{\circ}$ .

(c) 12 is even, so half of the lines join opposite vertices: $12 \div 2 = 6$ . The other 6 join the midpoints of opposite sides.

(a) 12 sides (b) order 12; the smallest turn is 30° (c) 6 lines.

Practice 9: The floor of a pavilion

The floor of a pavilion in the park near Siti's home is a regular hexagon $PQRSTU$ .

(a) How many lines of symmetry does the floor have, and what is its order of rotational symmetry?

(b) A straight path is painted across the floor from $P$ to $S$ . Is it along a line of symmetry?

(d) Another path runs from the midpoint of $QR$ to the midpoint of $TU$ . Is it along a line of symmetry?

ShowHide solution

(a) A regular hexagon has 6 lines of symmetry and rotational symmetry of order 6.

(b) $S$ is three places on from $P$ , so $P$ and $S$ are opposite vertices. Since 6 is even, the line joining opposite vertices is a line of symmetry: yes.

(c) No. $R$ is two places on from $P$ , not opposite it, and the line $PR$ does not pass through the centre of the hexagon. Every line of symmetry of a regular polygon passes through its centre.

(d) $TU$ is three sides on from $QR$ , so they are opposite sides, and the line joining their midpoints is a line of symmetry: yes.

(a) 6 lines of symmetry; order 6. (b) Yes. (c) No: $PR$ does not pass through the centre. (d) Yes.

Practice 10: Folding a paper octagon

In an art lesson, Mei Ling cuts a regular octagon $ABCDEFGH$ out of paper. Each of its interior angles is 135°. She folds it along $AE$ , opens it out, and then folds it along $MN$ , where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $EF$ .

(a) Explain why each fold is along a line of symmetry.

(b) Find $\angle BAE$ .

ShowHide solution

(a) 8 is even. $E$ is four places on from $A$ , so $AE$ joins opposite vertices; $EF$ is four sides on from $AB$ , so $MN$ joins the midpoints of opposite sides. Both are lines of symmetry.

(b) Folding along $AE$ puts $AB$ onto $AH$ , so $AE$ cuts $\angle BAH = 135^{\circ}$ into two equal angles:

$\angle BAE = 135^{\circ} \div 2 = 67.5^{\circ}$ ( $AE$ is a line of symmetry)

$\angle AMN + \angle BMN = 180^{\circ}$ (adj. ∠s on a str. line), so $\angle AMN = 90^{\circ}$ .

(b) $\angle BAE = 67.5^{\circ}$ (c) $\angle AMN = 90^{\circ}$ .

Practice 11: Equal sides, two sizes of angle

Each side of this hexagon is 2 cm long. Its interior angles are 100° and 140° in turn.

(a) Is the hexagon regular?

(b) How many lines of symmetry does it have, and where are they?

ShowHide solution

(a) Not regular: its sides are all equal, but its angles are not.

(In the next lesson, "Polygons and Their Interior Angles", you will see that the interior angles of every hexagon add up to 720°: here $3 \times 100^{\circ} + 3 \times 140^{\circ} = 720^{\circ}$ .)

(b) Fold along the line joining a 100° vertex to the vertex opposite it, a 140° vertex. The sides are all equal, and the corners on one side of the fold land on corners of the same size on the other side (100° onto 100°, 140° onto 140°), so the two halves match exactly. There are 3 such lines, one through each 100° vertex, and each is a line of symmetry.

A line through the midpoints of two opposite sides is not: folding along it would put a 100° corner onto a 140° corner. So the hexagon has 3 lines of symmetry.

(c) A turn of a third of a complete turn, 120°, moves each 100° corner onto the next 100° corner, and the hexagon fits onto itself. A turn of 60° would put a 100° corner where a 140° corner was, so it does not fit. It fits after 120°, 240° and 360°: 3 times in one complete turn.

(a) No: its angles are not all equal. (b) 3 lines, each joining a 100° vertex to the opposite 140° vertex. (c) Order 3.

Practice 12: True or false?

Say whether each statement is true or false, and give a reason.

(a) A polygon with all its sides equal is always regular.

(b) A polygon with all its interior angles equal is always regular.

(d) A regular polygon with 20 sides has 10 lines of symmetry that join opposite vertices.

(e) In a regular hexagon $ABCDEF$ , the line $AC$ is a line of symmetry.

ShowHide solution

(a) False. A rhombus that is not a square has all its sides equal but is not regular; the hexagon of Practice 11 is another example.

(b) False. A rectangle that is not a square has all its angles equal (90°) but is not regular; the pentagon of Example 3(c) is another example.

(c) True. A regular polygon with 7 sides has 7 lines of symmetry. Since 7 is odd, each joins a vertex to the midpoint of the opposite side, so each passes through a vertex.

(d) True. It has 20 lines of symmetry. Since 20 is even, half of them, $20 \div 2 = 10$ , join opposite vertices.

(e) False. The vertex opposite $A$ is $D$ , not $C$ , and $AC$ does not pass through the centre of the hexagon.

(a) False (b) false (c) true (d) true (e) false.

Key Points

A polygon is a closed plane figure with three or more straight sides that meet only at their ends.
Triangle 3 sides, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10; a polygon with $n$ sides is an $n$ -sided polygon.
A polygon has as many vertices and interior angles as sides. A diagonal joins two vertices that are not next to each other.
A polygon is convex if every interior angle is less than 180°; a polygon with a reflex interior angle is not convex.
A regular polygon has all its sides equal and all its interior angles equal; a rhombus and a rectangle that are not squares are not regular.
A regular polygon with $n$ sides has $n$ lines of symmetry and rotational symmetry of order $n$ ; the smallest turn that makes it fit onto itself is $360^{\circ} \div n$ .
For odd $n$ each line of symmetry joins a vertex to the midpoint of the opposite side; for even $n$ half join opposite vertices and half join the midpoints of opposite sides. Every line passes through the centre.
A line of symmetry through a vertex cuts the interior angle there in half; one through the midpoint of a side meets that side at right angles.