# Geometry/Appendix A

This is an incomplete list of formulas used in geometry.

## Length
### Perimeter and Circumference
#### Polygon

* Sum the lengths of the sides.

#### Circle

* $\pi d\ =2\pi r\,$
** $d\,$ is the diameter
** $r\,$ is the radius

### Triangles

* Law of Sines: ${\frac {a}{sin(A)}}={\frac {b}{sin(B)}}={\frac {c}{sin(C)}}$
** $a,b,c\,$ are sides, $A,B,C\,$ are the angles corresponding to $a,b,c\,$ respectively.
* Law of Cosines: $c^{2}=a^{2}+b^{2}-2ab\cos(C),$
** $a,b,c\,$ are sides, $A,B,C\,$ are the angles corresponding to $a,b,c\,$ respectively.

#### Right Triangles

* Pythagorean Theorem: $c^{2}=a^{2}+b^{2}$
** $a,b,c\,$ are sides where c is greater than other two.

## Area
### Triangles

* $A={\frac {bh}{2}}\,$
** $b\,$ = base, $h\,$ = height (perpendicular to base), $A\,$ = area
* Heron's Formula: $A={\sqrt {s(s-a)(s-b)(s-c)}}\,$
** $a,b,c\,$ are sides, and $s={\frac {a+b+c}{2}}\,$, $A\,$ = area

#### Equilateral Triangles

* ${\frac {{\sqrt {3}}a^{2}}{4}}\,$
** $a\,$ is a side

### Quadrilaterals
#### Squares

* $s^{2}\,$
** $s\,$ is the length of the square's side

#### Rectangles

* $ab\,$
** $a\,$ and $b\,$ are the sides of the rectangle

#### Parallelograms

* $bh\,$
** $b\,$ is the base, $h\,$ is the height

#### Trapezoids

* ${\frac {(b_{1}+b_{2})h}{2}}\,$
** $b_{1},b_{2}\,$ are the two bases, $h\,$ is the height

### Circles

* $\pi r^{2}\,$
** $r\,$ is the radius

### Surface Areas

* Cube: 6×($s^{2}$)
** $s\,$ is the length of a side.
* Rectangular Prism: 2×(($l,$ × $w\,$) + ($l\,$ × $h\,$) + ($w\,$ × $h\,$))
** $l\,$, $w\,$, and $h\,$ are the length, width, and height of the prism
* Sphere: 4×π×($r\,$$^2$)
** $r\,$ is the radius of the sphere
* Cylinder: 2×π×$r\,$×($h\,$ + $r\,$)
** $r\,$ is the radius of the circular base, and $h\,$ is the height
* Pyramid: $A=A_{b}+{\frac {ps}{2}}$
** $A$ = Surface area, $A_{b}$ = Area of the Base, $p$ = Perimeter of the base, $s$ = slant height.
       The surface area of a regular pyramid can also be determined based only on the number of sides($n$), the radius($r$) or side length($l$), and the height($h$)
       If $r$ is known, $l$ is defined as $l={\sqrt {(rcos({\frac {360}{n}})-r)^{2}+(rsin({\frac {360}{n}}))^{2}}}={\sqrt {2}}r{\sqrt {1-cos({\frac {360}{n}})}}$
       or if $l$ is known, $r$ is defined as $r={\frac {l}{{\sqrt {2}}{\sqrt {1-cos({\frac {360}{n}})}}}}$
       The slant height $h_{1}$ is given by ${\sqrt {r^{2}+h^{2}+{\frac {l^{2}}{4}}}}$
       The total surface area of the pyramid is given by $n{\frac {l}{2}}[h_{1}+h_{0}]$
* Cone: π×r×(r + √(r$^2$ + h$^2$))
** $r\,$ is the radius of the circular base, and $h\,$ is the height.

## Volume

* Cube $s^{3}=s\cdot s\cdot s$
** s = length of a side
* Rectangular Prism $l\cdot w\cdot h$
** l = length, w = width, h = height
* Cylinder(Circular Prism)$\pi r^{2}\cdot h$
** r = radius of circular face, h = distance between faces
* Any prism that has a constant cross sectional area along the height:
** $A\cdot h$
** A = area of the base, h = height
* Sphere: ${\frac {4}{3}}\pi r^{3}$
** r = radius of sphere
* Ellipsoid: ${\frac {4}{3}}\pi abc$
** a, b, c = semi-axes of ellipsoid
* Pyramid: ${\frac {1}{3}}Ah$
** A = area of base, h = height from base to apex
* Cone (circular-based pyramid):${\frac {1}{3}}\pi r^{2}h$
** r = radius of circle at base, h = distance from base to tip

### Navigation

* Geometry Main Page
* Motivation
* Introduction
* Geometry/Chapter 1 - HS Definitions and Reasoning (Introduction)
** Geometry/Chapter 1/Lesson 1 Introduction
** Geometry/Chapter 1/Lesson 2 Reasoning
** Geometry/Chapter 1/Lesson 3 Undefined Terms
** Geometry/Chapter 1/Lesson 4 Axioms/Postulates
** Geometry/Chapter 1/Lesson 5 Theorems
** Geometry/Chapter 1/Vocabulary Vocabulary
* Geometry/Chapter 2 Proofs
* Geometry/Chapter 3 Logical Arguments
* Geometry/Chapter 4 Congruence and Similarity
* Geometry/Chapter 5 Triangle: Congruence and Similiarity
* Geometry/Chapter 6 Triangle: Inequality Theorem
* Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
* Geometry/Chapter 8 Perimeters, Areas, Volumes
* Geometry/Chapter 9 Prisms, Pyramids, Spheres
* Geometry/Chapter 10 Polygons
* Geometry/Chapter 11 $\mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}$
* Geometry/Chapter 12 Angles: Interior and Exterior
* Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
* Geometry/Chapter 14 Pythagorean Theorem: Proof
* Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
* Geometry/Chapter 16 Constructions
* Geometry/Chapter 17 Coordinate Geometry
* Geometry/Chapter 18 Trigonometry
* Geometry/Chapter 19 Trigonometry: Solving Triangles
* Geometry/Chapter 20 Special Right Triangles
* Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
* Geometry/Chapter 22 Rigid Motion
* Geometry/Appendix A Formulae
* Geometry/Appendix B Answers to problems
* Appendix C. Geometry/Postulates & Definitions
* Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry
← Chapter 21 · Appendix B →