Geometry

Formulas

(1.) $\overleftrightarrow{AB}$
Line AB

(2.) $\overline{AB}$
Line Segment AB

(3.) $\overrightarrow{AB}$
Ray AB

(4.) Given: n collinear points:
the number of different rays that can be named is: 2(n − 1) rays.

(5.) Given: n collinear points:
the number of different lines that can be named if order is important is found from the Permutation formula.
Two points are used to name a line.
Permutation: order is important: perm 2 from n
$P(n, 2) = \dfrac{n!}{(n - 2)!}$

(6.) Given: n collinear points:
the number of different lines that can be named if order is not important is found from the Combination formula.
Two points are used to name a line.
Combination: order is not important: comb 2 from n
$C(n, 2) = \dfrac{n!}{(n - 2)! * 2!}$

(7.) Given: n coplanar points:
the number of different ways to name the plane if order is important is found from the Permutation formula.
Three points are used to name a plane.
Permutation: order is important: perm 3 from n
$P(n, 3) = \dfrac{n!}{(n - 3)!}$

(8.) Given: n coplanar points:
the number of different ways to name the plane if order is not important is found from the Combination formula.
Three points are used to name a plane.
Combination: order is not important: comb 3 from n
$C(n, 3) = \dfrac{n!}{(n - 3)! * 3!}$

(9.) Regular Polygons
Where: the number of sides is greater than 4
Let:
number of sides = n
side length = s
radius (the distance from the center to a vertex) = r
apothem (shortest distance from the center to one of the sides) = a
each interior angle = θ
perimeter = P
area = A

$ \Sigma \theta = 180^\circ(n - 2) \\[3ex] \theta = \dfrac{180^\circ(n - 2)}{n} \\[5ex] a = r\cos\left(\dfrac{180^\circ}{n}\right) \\[6ex] a = \dfrac{s}{2\tan \left(\dfrac{180^\circ}{n}\right)} \\[8ex] s = 2r\sin\left(\dfrac{180^\circ}{n}\right) \\[6ex] s = 2a\tan\left(\dfrac{180^\circ}{n}\right) \\[7ex] r = \dfrac{a}{\cos\left(\dfrac{180^\circ}{n}\right)} \\[8ex] r = \dfrac{s}{2\sin\left(\dfrac{180^\circ}{n}\right)} \\[8ex] P = ns \\[3ex] P = 2rn\sin\left(\dfrac{180^\circ}{n}\right) \\[6ex] P = 2an\tan\left(\dfrac{180^\circ}{n}\right) \\[6ex] A = \dfrac{\pi r^2}{2}\sin\left(\dfrac{360^\circ}{n}\right) \\[7ex] A = \dfrac{aP}{2} \\[5ex] A = \dfrac{ans}{2} \\[5ex] A = \dfrac{\pi a^2}{\cos^2\left(\dfrac{180^\circ}{n}\right)} \sin\left(\dfrac{360^\circ}{n}\right) \\[7ex] \text{Sum of interior angles} = 180(n - 2) \\[4ex] \text{Each interior angle} = \dfrac{180(n - 2)}{n} \\[6ex] \text{Sum of exterior angles} = 360^\circ \\[4ex] \text{Each interior angle} = \dfrac{360}{n} \\[6ex] $ (10.) Section Formula
Given two points say A(x₁, y₁) and B(x₂, y₂): if a point say C(x, y) divides the line segment |AB| in the ratio: m:n, then the coordinates of C is given by:

$ C = \dfrac{mB + nA}{m + n} \\[5ex] C(x, y) = \left(\dfrac{mx_2 + nx_1}{m + n}, \dfrac{my_2 + ny_1}{m + n}\right) $

• Symbols and Meanings for Circles
• (h, k) = coordinates of the center of a circle
• r = radius of a circle
• d = diameter of a circle
• C = circumference of a circle (also known as the perimeter)
• A = area of a circle
• π = pi = $\dfrac{22}{7}$
• θ = central angle
• ° = DEG = degrees
• RAD = radians
• L = length of arc
• A_sec = area of sector
• P_sec = perimeter of sector
• A_seg = area of segment
• P_seg = perimeter of segment
• (x₁, y₁) = first endpoint of the diameter of a circle
• (x₂, y₂) = second endpoint of the diameter of a circle

Radius, Diameter, Circumference, Area

$ \underline{Circle} \\[3ex] d = 2r \\[3ex] r = \dfrac{d}{2} \\[5ex] C = \pi d \\[3ex] d = \dfrac{C}{\pi} \\[5ex] C = 2\pi r \\[3ex] r = \dfrac{C}{2\pi} \\[5ex] A = \pi r^2 \\[3ex] r = \sqrt{\dfrac{A}{\pi}} \\[5ex] A = \dfrac{\pi d^2}{4} \\[5ex] d = \sqrt{\dfrac{4A}{\pi}} \\[5ex] A = \dfrac{C^2}{4\pi} \\[5ex] C = 2\sqrt{A\pi} \\[5ex] \underline{Semicircle} \\[3ex] d = 2r \\[3ex] r = \dfrac{d}{2} \\[5ex] C = \pi r \\[3ex] C = \dfrac{\pi d}{2} \\[5ex] r = \dfrac{C}{\pi} \\[5ex] d = \dfrac{2C}{\pi} \\[5ex] A = \dfrac{\pi r^2}{2} \\[5ex] r = \sqrt{\dfrac{2A}{\pi}} \\[5ex] A = \dfrac{\pi d^2}{8} \\[5ex] d = \sqrt{\dfrac{8A}{\pi}} \\[7ex] \underline{\theta\;\;in\;\;DEG} \\[3ex] L = \dfrac{2\pi r\theta}{360} \\[5ex] \theta = \dfrac{180L}{\pi r} \\[5ex] r = \dfrac{180L}{\pi \theta} \\[5ex] A_{sec} = \dfrac{\pi r^2\theta}{360} \\[5ex] P_{sec} = \dfrac{r(\pi\theta + 360)}{180} \\[5ex] \theta = \dfrac{360A_{sec}}{\pi r^2} \\[5ex] r = \dfrac{360A_{sec}}{\pi\theta} \\[5ex] A_{sec} = \dfrac{Lr}{2} \\[5ex] A_{sec} = \dfrac{Lr}{2} \\[5ex] r = \dfrac{2A_{sec}}{L} \\[5ex] L = \dfrac{2A_{sec}}{r} \\[5ex] \underline{\theta\;\;in\;\;RAD} \\[3ex] L = r\theta \\[5ex] \theta = \dfrac{L}{r} \\[5ex] r = \dfrac{L}{\theta} \\[5ex] A_{sec} = \dfrac{r^2\theta}{2} \\[5ex] \theta = \dfrac{2A_{sec}}{r^2} \\[5ex] r = \sqrt{\dfrac{2A_{sec}}{\theta}} \\[5ex] $ (1.) Standard Form of the Equation of a Circle
$(x - h)^2 + (y - k)^2 = r^2$
where:
$x, y$ are the variables
$(h, k)$ are the coordinates of the center of the circle
$r$ is the radius of the circle

(2.) General Form of the Equation of a Circle
$x^2 + y^2 + 2gx + 2fy + c = 0$
where:
$x, y$ are the variables
$c$ is the coefficient of $x$
$d$ is the coefficient of $y$
$c, d, e$ are values/constants

(3.) Given: The Center Coordinates of a Circle and an Endpoint on the Circumference of the Circle
The coordinates of the center of the circle = $(h, k)$
The endpoint on the circumference of the circle = $(x_1, y_1)$
The radius of the circle can be found by the Distance Formula
The radius of the circle = $r$
r = $\sqrt{(x_1 - h)^2 + (y_1 - k)^2}$
The diameter of the circle = $d$
The diameter of the circle is twice the radius.
$d = 2 * r$
The second endpoint of the diameter of the circle can also be found
The second endpoint of the diameter of the circle = $(x_2, y_2)$

$ x_2 = x_1 + r \\[3ex] y_2 = y_1 + r \\[3ex] (x_2, y_2) = (x_1 + r, y_1 + r) \\[3ex] $ (4.) Given: The Endpoints of the Diameter of the Circle
$(x_1, y_1)$ = first endpoint of the diameter of a circle
$(x_2, y_2)$ = second endpoint of the diameter of a circle
The center of the circle is found using the Midpoint Formula
$(h, k)$ are the coordinates of the center of the circle

$ h = \dfrac{x_1 + x_2}{2} \\[5ex] k = \dfrac{y_1 + y_2}{2} \\[5ex] $ Euler's Theorem: The number of faces (F ), vertices (V ), and edges (E ) of a polyhedron are related by the formula: F + V = E + 2
⇒ F + V − E = 2

In naming a line, two points are used.
In naming a plane, three points are used.

We can measure angles in:
Degrees (DEG, $^\circ$)
Radians (RAD)
Gradians (GRAD)
Degrees, Minutes, and Seconds ($^\circ \:'\:''$)

$DRG$ means $Degree-Radian-Gradian$ in some calculators
$180^\circ = \pi \:\:RAD = 200 \:\:GRAD$

To convert from:
radians to degrees, multiply by $\dfrac{180}{\pi}$

degrees to radians, multiply by $\dfrac{\pi}{180}$

$DMS$ means $Degree-Minute-Second$ in some calculators

$ 1^\circ = 60' \\[3ex] 1^\circ = 3600'' \\[3ex] 1^\circ = 60' = 3600'' \\[3ex] 1' = 60'' \\[3ex] $ To convert from:
degrees to minutes, multiply by $60$
minutes to degrees, divide by $60$
degrees to seconds, multiply by $3600$
seconds to degrees, divide by $3600$
minutes to seconds, multiply by $60$
seconds to minutes, divide by $60$

Show students these angular measures in their scientific calculators.
Show them how to convert from one angular measure to another.

Ellipses

• Symbols and Meanings for Ellipses
• (h, k) = coordinates of the center of an ellipse
• Ellipse: a = horizontal distance from the center to the boundary
• Ellipse: b = vertical distance from the center to the boundary
• Horizontal Ellipse: a = half the length of the major axis OR the length of the semi-major axis
• Horizontal Ellipse: b = half the length of the minor axis OR the length of the semi-minor axis
• Vertical Ellipse: a = half the length of the minor axis OR the length of the semi-minor axis
• Vertical Ellipse: b = half the length of the major axis OR the length of the semi-major axis
• c = linear eccentricity = distance from the center of the ellipse to the foci
• $f_1,f_2$ = foci points
• $f$ = distance between the foci
• e = eccentricity
• A = area
• P = perimeter
• L = length of the latus rectum
• $L_1,\;L_2,\;L_3,\;L_4$ = endpoints of the latus rectum
• D = distance between the two directrixes
• C = fractional factorials binomial coefficient
• λ = square of the ratio of the difference of the semi-axis to the sum of the semi-axis
• n = number of terms

(1.) Standard Form of the Equation of an Ellipse

$ \dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1 \\[5ex] $ If $a \gt b$, the ellipse is a horizontal ellipse.
If $a \lt b$, the ellipse is a vertical ellipse.

$ \underline{\text{Assume a Horizontal Ellipse}} \\[3ex] \text{Length of Major Axis} = 2a \\[3ex] \text{Length of Minor Axis} = 2b \\[3ex] \text{Distance between two Foci} = 2\sqrt{a^2 - b^2} \\[5ex] $ (2.) Eccentricity

$ e = \dfrac{\sqrt{semiMajor\;\;axis\;\;length^2 - semiMinor\;\;axis\;\;length^2}}{semiMajor\;\;axis\;\;length} \\[3ex] $ In other words:
Eccentricity: Horizontal Ellipse:
Major axis is the horizontal axis
Semi-major axis length = a

$ e = \dfrac{\sqrt{a^2 - b^2}}{a} \\[3ex] $ Eccentricity: Vertical Ellipse:
Major axis is the vertical axis
Semi-major axis length = b

$ e = \dfrac{\sqrt{b^2 - a^2}}{b} \\[5ex] $

Theorems

(1.) The sum of the interior angles of a triangle is 180°

(2.) The exterior angle of a triangle is the sum of the two interior opposite angles.

(3.) The sum of angles on a straight line is 180°

(4.) The sum of the angles at a point is 360°

(5.) Interior Angle Sum Theorem: The sum of the interior angles of a polygon (regular and irregular polygons) is $180(n - 2)$ where n is the number of sides.
This is because any polygon of n sides can be split into n − 2 triangles.

(6.) Exterior Angle Sum Theorem: The sum of the exterior angles of a polygon (regular and irregular polygons) is 360°

(7.) Angle Bisector Theorem: Any point on the bisector of an angle is equidistant from the two sides of the angle.