# 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

#### 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 + √(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