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.

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: $\frac{1}{3} A h$
• 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