Comprehensive Geometry and Trigonometry Formula Guide

Circle Geometry: Angle Relationships and Measures

• Intersecting Chords or Secants (on the Interior)
    - When chords or secants intersect inside a circle, the measure of the angle formed is half the sum of the measures of the intercepted arcs.
    - Formula for angle 1: $m\angle 1 = \frac{1}{2}(m\text{arc AB} + m\text{arc CD})$
    - Formula for angle 2: $m\angle 2 = \frac{1}{2}(m\text{arc AC} + m\text{arc BD}$

• Intersecting Tangents & Chords/Secants (on the Circle)
    - When a tangent and a chord (or secant) intersect at the point of tangency on the circle, the measure of the angle is half the measure of the intercepted arc.
    - Formula for angle 1: $m\angle 1 = \frac{1}{2}(m\text{arc ABC})$
    - Formula for angle 2: $m\angle 2 = \frac{1}{2}(m\text{arc AC})$

• Intersecting Lines (on the Exterior)
    - For all cases where lines intersect outside the circle, the measure of the exterior angle is half the difference of the measures of the intercepted arcs.
    - Intersecting Secants (on the Exterior): $m\angle A = \frac{1}{2}(m\text{arc DE} - m\text{arc BC}$
    - Intersecting Tangent & Secant (on the Exterior): $m\angle A = \frac{1}{2}(m\text{arc BD} - m\text{arc BC}$
    - Intersecting Tangents (on the Exterior): $m\angle A = \frac{1}{2}(m\text{arc BDC} - m\text{arc BC}$

Circle Geometry: Segment Length Theorems

• Intersecting Chords or Secants (on the Interior)
    - When two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.
    - Formula: $a \times b = c \times d$

• Intersecting Secants (on the Exterior)
    - When two secant segments are drawn to a circle from an exterior point, the product of the external segment and the whole secant segment is the same for both secants.
    - Formula: $a \times (a + b) = c \times (c + d)$

• Intersecting Tangent & Secant (on the Exterior)
    - When a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the measure of the tangent segment is equal to the product of the measures of the external secant segment and the whole secant segment.
    - Formula: $a^2 = b \times (b + c)$

Laws of Trigonometry

• Law of Sines
    - The Law of Sines relates the side lengths of a triangle to the sines of its angles.
    - Formula: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Law of Cosines
    - The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
    - Formula for side a: $a^2 = b^2 + c^2 - 2 \times b \times c \cos(A)$
    - Formula for side b: $b^2 = a^2 + c^2 - 2 \times a \times c \cos(B)$
    - Formula for side c: $c^2 = a^2 + b^2 - 2 \times a \times b \cos(C)$

Two-Dimensional Area Formulas

• Parallelogram
    - Formula: $A = b \times h$
    - Definitions: $b = \text{base}$, $h = \text{height}$

• Trapezoid
    - Formula: $A = \frac{1}{2} \times h(b_1 + b_2)$
    - Definitions: $b_1, b_2 = \text{bases}$, $h = \text{height}$

• Rhombus
    - Formula: $A = \frac{1}{2} \times d_1 \times d_2$
    - Definitions: $d_1, d_2 = \text{diagonals}$

• Kite
    - Formula: $A = \frac{1}{2} \times d_1 \times d_2$
    - Definitions: $d_1, d_2 = \text{diagonals}$

• Circle
    - Formula: $A = \pi \times r^2$
    - Definitions: $r = \text{radius}$

• Sector
    - Formula: $A = \frac{x}{360} \times \pi \times r^2$
    - Definitions: $x = \text{degree measure of the intercepted arc}$

• Regular Polygon
    - Formula: $A = \frac{1}{2} \times a \times P$
    - Definitions: $a = \text{apothem}$, $P = \text{perimeter}$

Three-Dimensional Surface Area Formulas

• Prism
    - Formula: $SA = P \times h + 2 \times B$
    - Definitions: $P = \text{perimeter of base}$, $h = \text{height}$, $B = \text{area of base}$

• Pyramid
    - Formula: $SA = \frac{1}{2} \times P \times l + B$
    - Definitions: $l = \text{slant height}$, $B = \text{area of base}$

• Cone
    - Formula: $SA = \pi \times r \times l + \pi \times r^2$
    - Definitions: $l = \text{slant height}$, $r = \text{radius}$

• Cylinder
    - Formula: $SA = 2 \times \pi \times r^2 + 2 \times \pi \times r \times h$

• Sphere
    - Formula: $SA = 4 \times \pi \times r^2$

Three-Dimensional Volume Formulas

• Prism
    - Formula: $V = B \times h$

• Pyramid
    - Formula: $V = \frac{1}{3} \times B \times h$

• Cone
    - Formula: $V = \frac{1}{3} \times B \times h$
    - Note: This is also expressed as $V = \frac{1}{3} \times \pi \times r^2 \times h$

• Cylinder
    - Formula: $V = B \times h$
    - Note: This is also expressed as $V = \pi \times r^2 \times h$

• Sphere
    - Formula: $V = \frac{4}{3} \times \pi \times r^3$