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maths:mensuration

mensuration - how to measure area, volume, etc of various shapes

Introduction

Triangles

  • 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.

Circles

  • area = (π) * r2
  • circumference = 2 (π) * r
  • Sector of Circle (ie. creating triangle radiating from centre with base length b & height h):
    • 1 radian = 360/2(π) = 57deg 17' 45“
    • length of chord (ie. base of triangle (b)) = 2rsin(a/2)
    • length of arc at perimeter = r (π) angle in degrees / 180 =
    • area of sector = ar2 / 2 = (π) * 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);
  • chord theorem:
    • if 2 chords in a circle intersect and sections of 1st chord are lengths a,b and sections of 2nd chord are c,d, then a x b = c x d
  • Thales theorem:
    • if ABC are points on a circle and the line AB is the diameter of the circle (ie. passes through the centre), then the angle ACB is a right angle

Ellipses

  • Ellipse (with min,max radii of a and b):
    • circumference ~ 2 (π) * sqrt ((a2 + b2)/2)
    • area = (π)ab

Parallel Pipes

  • 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

Cylinders

  • Right Circular Cylinder:
    • area of convex surface = 2 r (π) h
    • total surface area = 2 r (π) (r+h)
    • volume = (π)r2h

Cones

  • Right Circular Cones:
    • length of slant side (l) = sqrt (r2 + h2)
    • area of convex surface = (π) r l
    • total surface area = (π)r(r+l)
    • volume = (1/3)(π)hr2

Catenary: Hanging cable between two poles

  • a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends
  • if w = distance between poles, p = height of pole and c = lowest height of cable above the ground which has cartesian coords (x,y) with (0,0) being the lowest point of the cable
  • thus v = vertical drop height of the cable from its attachments at the top of each pole and v = p-c
  • general catenary equations (see also https://en.wikipedia.org/wiki/Catenary) with constant a being a scaling factor:
    • y = a * cosh(x/a) - a
    • the cable length (L) = 2a * sinh(w/2a) thus sinh(w/2a) = L/2a
    • the cartesian coordinates for the top of the right pole is (w/2,v)
      • thus v = a * cosh(w/2a) - a thus cosh(w/2a) = (v + a)/a
    • if you know v and l you can solve for w using cosh2(t) - sinh2(t) = 1:
      • ((v + a)/a)2 - (L/2a)2 = 1
      • solve for a using quadratic equations then substitute a back into earlier formulae to get w
  • obviously, for the special case when cable length is twice the vertical drop of the cable, the distance between the poles must be ZERO and the above equations will not work.

 

maths/mensuration.txt · Last modified: 2021/08/28 16:10 by gary1