Geometry/Appendix A

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This is an incomplete list of formulas used in geometry.

Length[edit]

Perimeter and Circumference[edit]

Polygon[edit]

  • Sum the lengths of the sides.

Circle[edit]

  • \pi d\ = 2\pi r\,
    • d\, is the diameter
    • r\, is the radius

Triangles[edit]

  • 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[edit]

  • Pythagorean Theorem: c^2=a^2+b^2
    • a, b, c\, are sides.

Area[edit]

Triangles[edit]

  • 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[edit]

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

Quadrilaterals[edit]

Squares[edit]

  • s^2\,
    • s\, is the length of the square's side

Rectangles[edit]

  • ab\,
    • a\, and b\, are the sides of the rectangle

Parallelograms[edit]

  • bh\,
    • b\, is the base, h\, is the height

Trapezoids[edit]

  • \frac{(b_1+b_2)h}{2}\,
    • b_1,b_2\, are the two bases, h\, is the height

Circles[edit]

  • \pi r^2\,
    • r\, is the radius

Surface Areas[edit]

  • 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[edit]

  • 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

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Chapter 21 · Appendix B