# 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

• ${\displaystyle \pi d\ =2\pi r\,}$
• ${\displaystyle d\,}$  is the diameter
• ${\displaystyle r\,}$  is the radius

### 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.

#### Right Triangles

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

## Area

### 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

#### Equilateral Triangles

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

#### Squares

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

#### Rectangles

• ${\displaystyle ab\,}$
• ${\displaystyle a\,}$  and ${\displaystyle b\,}$  are the sides of the rectangle

#### Parallelograms

• ${\displaystyle bh\,}$
• ${\displaystyle b\,}$  is the base, ${\displaystyle h\,}$  is the height

#### Trapezoids

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

### 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.

## Volume

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