Quadrilaterals, Polygons - Regular Polygons, Circle
     Quadrilaterals
      Square
      Rhombus
      Rectangle
      Parallelogram
      Kite or deltoid
      Trapezium or trapezoid
      Isosceles trapezoid
      Cyclic quadrilateral
      Tangential quadrilateral
     Polygons
     Regular polygons
Quadrilaterals
Square
A square is a plane figure with four sides of equal length and four right angles.
The two diagonals, which are of equal length, bisect each other perpendicularly. 
The square has four axes of symmetry.
 
perimeter   P = 4a
area   A = a a = a2 A = a2
Rhombus or rhomb
A rhombus is an oblique-angled parallelogram whose four sides are equal.
Rhombus diagonals bisect each other perpendicularly.
The rhombus is symmetrical about both of its diagonals.
 
perimeter   P = 4a
area    A = a h     and
Rectangle
A rectangle is a plane figure with four straight sides, two parallel pairs of equal length forming four right angles. 
The two diagonals of a rectangle have equal lengths.
The rectangle has two axes of symmetry.
 
diagonal
perimeter        P = 2(a + b)
area        A = a
Parallelogram
A parallelogram is a quadrilateral (plane figure with four sides) with opposite sides parallel and of equal length. 
The opposite angles of a parallelogram are also equal. 
In the case in which the angles are all right angles the parallelogram is a rectangle, and when all the sides are equal it is a rhombus.
 
perimeter        P = 2(a + b)
area        A = a
Kite or deltoid
A kite or deltoid is a quadrilateral with two pairs of equal adjacent sides. 
Two of the angles in a kite are opposite and equal.
The kite's diagonals cross perpendicularly, and one of them (the shorter one) is being bisected by the other.
 
perimeter      P = 2(a + b)
area   
Trapezium or trapezoid
A trapezium or trapezoid is a quadrilateral with two parallel sides of unequal length. 
The trapezium's median is a straight line joining the midpoints of nonparallel sides and parallel to the two parallel sides.
 
a + d = 180,  b + g = 180   =>   a + b + g + d = 360
median      m = 1/2(a + c),       area A = m
perimeter   P = a + b + c + d
A = ch + 1/2(a - c)h = 1/2(a + c)h   A =1/2(a + c)h
Isosceles trapezoid
A trapezoid with the two nonparallel sides of equal length and the angles opposite those sides equal, is called an isosceles trapezoid.
 
                 a + b = 180,
median      m = 1/2(a + c),       area A = m
perimeter   P = a + 2b + c
A = ch + 1/2(a - c)h = 1/2(a + c)h   A =1/2(a + c)h
Cyclic quadrilateral
A quadrilateral inscribed in a circle, so that all its vertices lie on the circumference is called a cyclic quadrilateral. 
The opposite angles of a cyclic quadrilateral are supplementary.
 
a + g = b + d = 180,
d1  d2  = a    c + b  d
perimeter    P = a + b + c + d,     s = 1/2
area
Tangential quadrilateral
A quadrilateral whose sides all lie tangent to the circle inscribed within the quadrilateral is called a tangential quadrilateral.
 
s = a + c = b + d,
perimeter    P = a + b + c + d,     s = 1/2
area A = r s
Polygons
A polygon is closed plane figure bounded by a number of straight line segments with the same number of vertices.
The sum of interior angles is  (n - 2) 180, where n is the number of sides of a polygon. 
The sum of exterior angles of a polygon is 360.
 
 dn - number of diagonals
b' - the exterior angle
dn = 1/2 n (n - 3)
Sn= (n - 2) 180 - the sum of interior angles
Regular polygons
In a regular polygon all sides are equal and all its angles are equal.
The exterior angle of a regular polygon is b' = 360/n, where n is the number of sides of a regular polygon.
 a = b' = 360/n b' - the exterior angle
b = 180 - b' = (n - 2) 180/n,     P = n a
dn = 1/2 n (n - 3) - number of diagonals
A = 1/2 n a r = 1/2P r - area of a regular polygon
Geometry and use of trigonometry contents - A
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