Polygon

• If the order of the polygon is Impair, the sides “are opposed” to the tops and the angles (and vice versa ); more precisely, each top (or each angle) “is opposed” to the side located ( N - 1) /2 tops further.

Sides prolonged and diagonal

The lines which carry the sides of a polygon are called the prolonged sides of this polygon.

The sides of a polygon are not the only segments which can connect the tops between them. Any segment connecting two tops of a polygon and other that an east coast called diagonal of this polygon.

A polygon with N sides has thus $\backslash \; frac\; \{N\; (n-3)\}\{2\}\; \backslash ,$ diagonal.

Typology of the polygons

There exist many manners of classifying the polygons: according to their convexity , of their symmetries , their angles … But one initially classifies them according to their number on sides.

Classification following the number on sides

The polygons can be classified between them according to their number on sides, i.e. their order.

The most elementary polygon is the triangle: a polygon has at least three tops and three sides.

Then the quadrilateral comes, with four sides and four tops.

Starting from the order five, each name of polygon is made of a Greek Racine corresponding to the order of the polygon followed by the suffix - gone .

To find itself there in the denomination of the polygons, it should simply be retained that - kai- means " plus" in Greek, and that - told means " dizaine". For example, the word triacontakaiheptagone means three ( sorted ) tens ( - told ) more ( - kai- ) seven ( - hepta- ) units, and thus corresponds to a polygon on thirty-seven sides.

Beyond twelve sides, one has as a habit to speak all the same about polygon to N sides where N is replaced by the desired number, this in order to simplify the things.

• Note:: there exist however several old denominations for " numbers; ronds" as for a polygon with twenty sides (icosa-), with hundred sides (hecto-) and with ten thousand sides (myria-).

To note that the same principles apply to the Polyèdre S, where it is enough to replace the suffix - gone by the suffix - èdre .

Classification by convexity

It is pointed out that a diagonal of a polygon is a segment which joint two tops nonconsecutive, i.e. a segment which joint two tops and which is not a side of the polygon.

Example: the segments $$, $$, $$, $$, $$ are the 5 diagonals of the pentagon $ABCDE$ opposite.

Cross polygon

The envelope of a polygon is the polygon obtained while following contour external of this one. For example, the envelope of the preceding pentagon is a decagon whose tops are the five tops of the pentagon and the five intersections on its sides.

Concave polygon

A polygon is known as concave if it is not crossed and if one of its diagonals is not entirely inside the surface delimited by the polygon.

For example, the pentagon ACDBE opposite (on the right) is known as concave because the diagonals $$ and $$ are outside the surface delimited by the polygon.

Convex polygon

A polygon is known as convex if it is not crossed and if all its diagonals are entirely inside the surface delimited by the polygon. Thus, the Hexagone MNOPQR opposite (on the right) is known as convex .

Spangled polygon

The convex envelope of a polygon is the smallest convex polygon the container. Caution: the envelope and the convex envelope of a polygon merge only if this one is convex!

A polygon is then known as spangled if (and only if) none on its sides belongs to its convex envelope.

For example, the preceding cross pentagon and its envelope are spangled both.

Classification by symmetry

Concept of element of symmetry

A polygon can have regularities (called symmetries) which return it overall invariant by certain transformations such as rotation S or reflections. The element of symmetry of a transformation is the whole of the points invariants by this transformation:

• for a central Symmetry, the element of symmetry is the center of symmetry ;
• for a axial Symmetry, the element of symmetry is precisely this axis, known as axis-mirror because it cuts any overall invariant figure by this transformation into two images parts out of mirror one of the other;
• for a rotation, the element of symmetry is the axis of rotation (in fact, to be exact, it is rather about the intersection, called center of rotation , of this axis with the plan where the polygon is). In the case of a rotation, it is necessary to specify the angle of this rotation or its order , knowing that the product of the swing angle by its order is always equal to 2 π radians (or 1 turn or 360°…).

One can notice that, in the plan, central symmetry merges with the rotation of order two.

It is said that a polygon (or more generally any figure of geometry) presents an element of symmetry when it is overall invariant by the corresponding transformation.

In the case of a polygon, all the elements of symmetry pass by the same point. This point, if there exists, is called center polygon.

Concept of regular polygon

A polygon is known as regular if it is convex and present an axis of rotation of a nature equal to its number on sides.

That means that it is superimposed on itself when one turns it of an angle of $\{2\; \backslash \; pi\; \backslash \; over\; N\}$, where N is the order of the polygon.

The polygon thus presents the same configuration of each one of its tops which are thus laid out regularly on a circle centered on the axis of rotation.

A regular polygon is thus a convex polygon registered in a circle and of which all the sides have the same length (and angles same measurement).

Conversely, if a convex polygon is inscribable in a circle and if its sides are equal (or its equal angles), then it is regular.

Some examples and counterexamples :

• the equilateral Triangle is a regular polygon;

• the square is a regular polygon;
• the Losange (not square) is not regular (it is not inscribable in a circle).

Isosceles polygon

A polygon is known as isosceles when it presents at least a axis-mirror.

The axis-mirrors necessarily pass by tops or mediums on the sides of the polygon.

More precisely:

• if the order of the polygon is odd, any axis-mirror passes by a top and the medium on the opposite side (median, to see low);
• if the order of the polygon is even, any axis-mirror master key either by two opposite tops (principal diagonal, to see low), or by the mediums on two opposite sides (median, to see low).

An isosceles polygon which presents several axis-mirror has necessarily a center, the point of intersection of the axis-mirror.

Some examples and counterexamples :

• the isosceles Triangle, which has two equal sides, presents a axis-mirror passing by the top common to the both equal sides and the medium on the opposite side;
• the convex isosceles quadrilaterals are:

• any polygon regular is isosceles and present as much of axis-mirror than on sides;
• the parallelogram is not isosceles (except if it is about a rhombus).

Polygon centrosymetric

A polygon is known as centrosymetric when it presents a center of symmetry.

Any polygon centrosymetric has necessarily an even number of tops, and conversely, only the polygons of an even nature can be centrosymetric.

The sides opposite of a polygon centrosymetric are parallel and of the same length (order of the even polygon).

Some examples and counterexamples :

• the triangles cannot have of center of symmetry;
• the quadrilaterals centrosymetric are the parallelograms (parallel opposite sides and of the same length);
• the only quadrilaterals presenting at the same time a center of symmetry and a axis-mirror are the rectangles and the rhombuses;
• any polygon regular of an even nature has a center of symmetry.

Polygon rotosymetric

A polygon is more briefly known as rotosymetric of order N or N - rotosymetric when it presents an axis of rotation of order N .

A polygon rotosymetric of order N has a multiple number on sides of N . Conversely, a polygon can present of axis of rotation only if the order of this last divides its number on sides.

The polygons regular and centrosymetric are particular cases of polygons rotosymetric.

Some examples and counterexamples :

• a triangle cannot present of axis of rotation that if it is of order 3; it is then regular, therefore equilateral;
• any quadrilateral rotosymetric is centrosymetric;
• the simplest of polygon rotosymetric without being centrosymetric or regular case is that of the hexagon 3-rotosymetric;
• any polygon regular presents by definition an axis of rotation of the same order as the polygon;
• any polygon convex of a nature first presenting an axis of rotation is regular.

Scalene polygon

A scalene polygon is a polygon which does not present any element of symmetry. A scalene polygon thus does not have a center.

Classification by the angles

A convex polygon cannot present more than four right angles.

Right-angled polygon

A polygon is known as right-angled when it comprises at least a right angle.

Some examples and counterexamples :

• a right-angled triangle comprises a right angle and two acute angles;
• a right-angled quadrilateral comprises at least a right angle; it is however not inevitably a rectangle, which comprises four of them;
• as soon as a trapezoid comprises a right angle, it is a right-angled trapezoid; but any right-angled trapezoid comprises inevitably at least two adjacent right angles;
• the only right-angled regular polygon is the square. It is a particular case of rectangle besides, with four right angles.

Polygon birectangle

A polygon is known as birectangle when it comprises at least two right angles, consecutive or not.

Some examples and counterexamples :

• no triangle is birectangle, at least in Euclidean geometry (there exist triangles birectangles, and even trirectangular, on a sphere);
• the convex quadrilaterals birectangles are:

• the only trapezoid semi-rectangle is the rectangle;
• the only regular polygon birectangle is the square.

A polygon with two consecutive right angles presents two parallel sides.

Trirectangular polygon

A polygon is known as trirectangular when it comprises at least three right angles, consecutive or not.

Some examples and counterexamples :

• no triangle is trirectangular;
• the only trirectangular convex quadrilaterals are the rectangles, which count four right angles besides;
• the only trirectangular regular polygon is the square.

A convex polygon with three consecutive right angles presents two parallel sides twice. It resembles in fact a rectangle with a cut out corner.

Polygon équiangle

A polygon is known as équiangle when all its angles are equal.

Some examples and counterexamples :

• the only triangle équiangle is the equilateral triangle;
• the convex quadrilaterals équiangles are the rectangles;
• all the regular polygons are équiangles.

Other classifications

Equilateral polygon

A polygon is known as equilateral when all its sides have the same length.

Some examples and counterexamples :

• the equilateral convex quadrilaterals are the rhombuses;
• all the regular polygons are equilateral.

Inscribable polygon (in a circle)

A polygon is known as inscribable when all its tops are on the same circle, known as circumscribes with the polygon . Its sides are then cords of this circle, from where the name of polygon of cords given by the english-speaking to the inscribable polygons.

Some examples and counterexamples :

• any triangle is inscribable;
• a trapezoid is inscribable only if it is isosceles;
• any semi-rectangle is inscribable;
• the only inscribable parallelogram is the rectangle;
• any polygon regular is inscribable.

Circumscribable polygon (with a circle)

A polygon is known as circumscribable when all its sides are tangent with the same circle, known as registers in the polygon . The english-speaking baptized polygon of tangents this type of polygon.

Some examples and counterexamples :

• any triangle is circumscribable;
• the only circumscribable parallelograms are the rhombuses;
• any polygon regular is circumscribable.

Other definitions and properties

Mediating of a polygon

They are the Médiatrice S on its sides.

Bisectrices of a polygon

They are the Bissectrice S of its angles.

Medians and principal diagonals of a polygon

• If the order N of the polygon is even :

• If the order of the polygon is odd , it does not have there principal diagonals, only median . Each median then connects a top in the middle of the opposite side. They are then with the number of $N\; \backslash ,$.

Apothems and rays of a polygon in center

The apothems of a polygon in center connect the mediums on its sides to its center.

If the polygon is regular, it is also:

• half-medians of the polygon, if it is of an even nature;
• lines of construction defining mediating its sides;
• of the radii of the circle inscribed in the polygon.

The rays of a polygon in center connect its tops to its center.

If the polygon is regular, it is also:

• semi-diameters of the polygon, if it is of an even nature;
• of the radii of the circle circumscribed with the polygon.

Concept of angle in the center

Either has ₁ has ₂ has ₃… has _N a polygon with N sides provided with a center O .

One calls angle in the center polygon the angle $\backslash \; widehat\; \{A\_i\; O\; A\_i\}\; \_\; \{+1\}\; \backslash ,$ formed by two consecutive rays of this polygon.

If the polygon considered is regular, N angles in the center have all same measurement, 2π/ N radians, and it is also the measurement of the angle between two consecutive apothems.

Summon angles

The sum of the angles of a polygon does not bear a particular name, but is worth ( only in the case of a convex polygon ):

$S\; =\; (N\; -\; 2)\; \backslash \; times180^\; \backslash \; circ$ or $S\; =\; (N\; -\; 2)\; \backslash \; times\; \backslash \; pi$ Radian S, where N is the order of the polygon.

To note that when the order of a polygon increases by a unit, the sum of its angles increases 180° or π radians: it is the Supplément of angle.

Perimeter of a polygon

The Périmètre of a polygon is the sum lengths on its sides.

If the polygon is regular, its perimeter P is worth:

$P\; =\; 2\; N\; R\; \backslash \; sin\; (\backslash \; alpha/2)\; \backslash ,$

• N is the order of the polygon;
• $\backslash \; alpha\; \backslash ,$ is its angle in the center;
• and R the radius of the circle which is circumscribed to him.

Like $\backslash \; alpha\; \backslash ,$ is worth 2π/N radians, and that sin X ≈ X when X is close to 0, the perimeter tends towards 2 π R when N tends towards the infinite one. The perimeter of the circle well is found.

Surface of a polygon

The surface of a noncross polygon is the surface of surface encloses by the polygon.

If the polygon is regular, its surface has is worth:

$has\; =\; N\; R^2\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; alpha\}\; \{2\}\; \backslash \; right)\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; alpha\}\; \{2\}\; \backslash \; right)\; \backslash ,$

• N is the order of the polygon;
• $\backslash \; alpha\; \backslash ,$ is its angle in the center;
• and R the radius of the circle which is circumscribed to him.

As the angle in the center is worth 2 π/N radians, and that sin X ≈ X and cos X ≈ 1 when X is close to 0, the surface tends towards π R ² when N tends towards the infinite one. The surface of the disc well is found.

There exists one second possible formula to calculate the surface of a regular polygon: $A\; =\; \backslash \; frac\; \{has\; \backslash \; times\; P\}\; \{2\}$

where has is the apothem of the polygon and P its perimeter.

When the polygon is irregular, it is easy of the partitionner in triangles starting from the diagonals. To calculate its surface, it is then enough to make the sum of the surfaces of the triangles obtained.

See too

polygon|polygon

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External bonds

• Some information on the polygons

• Of many additional details on the polygons

Simple: Polygon Zh-yue: 多邊形