Geometry/Appendix A

< Geometry

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

)

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 + √(r² + h²))
- $r\,$ 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

Geometry Main Page
Motivation
Introduction
Geometry/Chapter 1 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

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Geometry/Appendix A

Length

Perimeter and Circumference

Polygon

Circle

Triangles

Right Triangles

Area

Triangles

Equilateral Triangles

Quadrilaterals

Squares

Rectangles

Parallelograms

Trapezoids

Circles

Surface Areas

Volume

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