Mensuration

• see also:
  • Algebra
  • Calculus
  • Circular functions
  • Coordinate geometry, parabolae, etc
  • Finance equations
  • Logarithmic, exponential and hyperbolic functions
  • Probability theory
  • http://math.about.com/library/blmeasurement.htm 

Mensuration:

• Triangle:
  • with sides a,b,c (hypotenuse) and angles opposite sides A,B,C and height above c = h:
    • right angled triangles:
      • Sine Rule:
        • a/sinA = b/sinB = c/sinC
      • Pythagorus' Theorem:
        • c² = a² + b²
      • Height above hypotenuse (h):
        • height = aCosA = bSinA = ab/c = c(Sin2A)/2
      • Area:
        • area = ab/2 = ac(CosA)/2 = bc(SinA)/2 = a²/2TanA = c²(Sin2A)/4 = ch/2
    • any triangle:
      • Cosine Rule:
        • a² = b² + c² - 2bcCosA;
      • Area:
        • area = bc(sinA)/2 = ch/2;
      • all 3 angles must sum to 180deg.
• Circle:
  • area = (pi) * r²
  • circumference = 2 (pi) * r
  • Sector of Circle (ie. creating triangle radiating from centre with base length b & height h):
    • 1 radian = 360/2(pi) = 57deg 17' 45"
    • length of chord (ie. base of triangle (b)) = 2rsin(a/2)
    • length of arc at perimeter = r (pi) angle in degrees / 180 =
    • area of sector = ar² / 2 = (pi) * r² * angle in degrees / 360
    • area of triangle = r² * (Sina) / 2
    • angle between the 2 radii = invCos (1 - (b²/2r²)) = 180deg - 2 arcSin(h/r)
  • Area of Segment of Circle:
    • A = 0.5r²(a-sina);
• Ellipse (with min,max radii of a and b):
  • circumference ~ 2 (pi) * sqrt ((a² + b²)/2)
  • area = (pi)ab
• Rectangular parallel pipe:
  • surface area = 2 (ab +bc + ca)
  • internal diagonal = sqrt (a² + b² + c²)
  • volume = abc;
• Pyramid:
  • volume = (1/3)h*area of base;
• Spheres:
  • volume = (4/3)(pi)r³
  • surface area = 4(pi)r²
  • segment of a sphere (cut by a single plane, having base circle radius b and height h):
    • area of convex surface = pi * (b² + h²) = 2(pi)rh
    • total surface area = pi * (2b² + h²)
    • volume = (pi) * h * (3b² + h2) / 6 = (pi) * h² *(3r-h) / 3
  • segment of a sphere (cut by 2 parallel planes, having base circle radius b, top circle radius t & height h):
    • area of convex surface = 2r(pi)h
    • total surface area = (pi) * (b² + 2rh + t²)
    • volume = (pi) * h * (3b² + 3t² + h²) / 6
  • wedge segment of sphere (a = angle between the 2 planes):
    • volume = (pi) * r³ * a / 270
• Right Circular Cylinder:
  • area of convex surface = 2 r (pi) h
  • total surface area = 2 r (pi) (r+h)
  • volume = (pi)r²h
• Right Circular Cones:
  • length of slant side (l) = sqrt (r² + h²)
  • area of convex surface = (pi) r l
  • total surface area = (pi)r(r+l)
  • volume = (1/3)(pi)hr²
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