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) S^{2}
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
Sides |
Name |
n |
N-gon |
3 |
Triangle |
4 |
Quadrilateral |
5 |
Pentagon |
6 |
Hexagon |
7 |
Heptagon |
8 |
Octagon |
10 |
Decagon |
12 |
Dodecagon |
(π = 3.141592...)
Area Formulas
square = a^{ 2}
rectangle = ab
parallelogram = bh
trapezoid = h/2 (b_{1} + b_{2})
circle = πr^{ 2}
ellipse = π r_{1} r_{2}
triangle = ½ bh
equilateral triangle =√3/4 a^{2}
^{ }
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) S^{2}
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 r_{1} r_{2} r_{3}
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 r^{2}
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 r^{2}
if the angle θis in radians, then area = ((θ/(2 π))x π r^{2}
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:
r^{2} - 2cr cos(θ - α) + c^{2} = a^{2}
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.