Geometry/Appendix A

This is an incomplete list of formulas used in geometry.

Length

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

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

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

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

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

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

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: $A=A_{b}+{\frac {ps}{2}}$ $A=A_{b}+{\frac {ps}{2}}$
- $A$ = 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 + √(r² + h²))
- $r\,$ is the radius of the circular base, and $h\,$ is the height.

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