What is a Polygon?
A closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex.

Types of Polygons
Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral.
Equiangular - all angles are equal.
Equilateral - all sides are the same length.

Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180°.

Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180°.

Polygon Formulas
(N = # of sides and S = length from center to a corner)

Area of a regular polygon = (1/2) N sin(360°/N) S²

Sum of the interior angles of a polygon = (N - 2) x 180°

The number of diagonals in a polygon = 1/2 N(N-3)
The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)

Polygon Parts

Side - one of the line segments that make up the polygon.

Vertex - point where two sides meet. Two or more of these points are called vertices.

Diagonal - a line connecting two vertices that isn't a side.

Interior Angle - Angle formed by two adjacent sides inside the polygon.

Exterior Angle - Angle formed by two adjacent sides outside the polygon.

Polygon Names
Generally accepted names

 (π  = 3.141592...)

Area Formulas

square = a ²

rectangle = ab

parallelogram = bh

trapezoid = h/2 (b₁ + b₂)

circle = πr ²

ellipse = π r₁ r₂

triangle = ½ bh

equilateral triangle =√3/4 a²

triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C

triangle given a,b,c =  √[s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)

regular polygon = (1/2) n sin(360°/n) S²
   when n = # of sides and S = length from center to a corner .

Area is measured in "square" units.

Volume Formulas

cube = a ³

rectangular prism = a b c

irregular prism = b h

cylinder = b h = π r ² h

pyramid = (1/3) b h

 cone = (1/3) b h = 1/3 π r ² h

sphere = (4/3) π r ³

ellipsoid = (4/3) π r₁ r₂ r₃

Volume is measured in "cubic" units.

Surface Area Formulas
In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Surface Area of a Cube = 6 a ²

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

Surface Area of Any Prism = Surface Area = Lateral area + Area of two ends

(Lateral area) = (perimeter of shape b) * L

Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

Surface Area of a Sphere = 4 π r ²

Surface Area of a Cylinder = 2 π r ² + 2 π r h

Circles

Definition: A circle is the locus of all points equidistant from a central point.

Definitions Related to Circles

arc: a curved line that is part of the circumference of a circle

chord: a line segment within a circle that touches 2 points on the circle.

circumference: the distance around the circle.

diameter: the longest distance from one end of a circle to the other

origin: the center of the circle

π (pi): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle.

radius: distance from center of circle to any point on it.

sector: is like a slice of pie (a circle wedge).

tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.

Diameter = 2 x radius of circle

Circumference of Circle = π x diameter = 2 π x radius

Area of Circle:
    area = π r²

Length of a Circular Arc: (with central angle θ)
    if the angle θis in degrees, then length = θx (π /180) x r
    if the angle θis in radians, then length = r x θ

Area of Circle Sector: (with central angle θ)
    if the angle θis in degrees, then area = (θ/360)x π r²
    if the angle θis in radians, then area = ((θ/(2 π))x π r²

Equation of Circle: (Cartesian coordinates)

  for a circle with center (j, k) and radius (r):
    (x-j)^{^2} + (y-k)^{^2} = r^{^2}

Equation of Circle: (polar coordinates)
    for a circle with center (0, 0):   r(θ) = radius

    for a circle with center with polar coordinates: (c, α) and radius a:
      r² - 2cr cos(θ - α) + c² = a²

Perimeter Formulas
The perimeter of any polygon is the sum of the lengths of all the sides.

square = 4a

rectangle = 2a + 2b

circle = 2π r  

circle = π d (where d is the diameter)

The perimeter of a circle is more commonly known as the circumference.