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

Quadrilaterals

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

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