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

Introduction

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.

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
  - see also Power of a Point
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

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

Rectangular parallel pipe:
- surface area = 2 (ab +bc + ca)
- internal diagonal = sqrt (a² + b² + c²)
- volume = abc;

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 (π) h
- total surface area = 2 r (π) (r+h)
- volume = (π)r²h

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

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.