Perimeter and Circumference

Triangles

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

Triangles

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

Quadrilaterals

Circles

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

Surface Areas

• Cube: 6×(${\displaystyle s^{2}}$)
  • ${\displaystyle s\,}$ is the length of a side.
• Rectangular Prism: 2×((${\displaystyle l,}$ × ${\displaystyle w\,}$) + (${\displaystyle l\,}$ × ${\displaystyle h\,}$) + (${\displaystyle w\,}$ × ${\displaystyle h\,}$))
  • ${\displaystyle l\,}$, ${\displaystyle w\,}$, and ${\displaystyle h\,}$ are the length, width, and height of the prism
• Sphere: 4×π×(${\displaystyle r\,}$²)
  • ${\displaystyle r\,}$ is the radius of the sphere
• Cylinder: 2×π×${\displaystyle r\,}$×(${\displaystyle h\,}$ + ${\displaystyle r\,}$)
  • ${\displaystyle r\,}$ is the radius of the circular base, and ${\displaystyle h\,}$ is the height
• Pyramid: ${\displaystyle A=A_{b}+{\frac {ps}{2}}}$
  • ${\displaystyle A}$ = Surface area, ${\displaystyle A_{b}}$ = Area of the Base, ${\displaystyle p}$ = Perimeter of the base, ${\displaystyle s}$ = slant height.

The surface area of a regular pyramid can also be determined based only on the number of sides(${\displaystyle n}$), the radius(${\displaystyle r}$) or side length(${\displaystyle l}$), and the height(${\displaystyle h}$)

If ${\displaystyle r}$ is known, ${\displaystyle l}$ is defined as ${\displaystyle l={\sqrt {(rcos({\frac {360}{n}})-r)^{2}+(rsin({\frac {360}{n}}))^{2}}}={\sqrt {2}}r{\sqrt {1-cos({\frac {360}{n}})}}}$

or if ${\displaystyle l}$ is known, ${\displaystyle r}$ is defined as ${\displaystyle r={\frac {l}{{\sqrt {2}}{\sqrt {1-cos({\frac {360}{n}})}}}}}$

The slant height ${\displaystyle h_{1}}$ is given by ${\displaystyle {\sqrt {r^{2}+h^{2}+{\frac {l^{2}}{4}}}}}$

The total surface area of the pyramid is given by ${\displaystyle n{\frac {l}{2}}[h_{1}+h_{0}]}$

• Cone: π×r×(r + √(r² + h²))
  • ${\displaystyle r\,}$ is the radius of the circular base, and ${\displaystyle h\,}$ is the height.