# Gary Ayton's camping and photography wiki

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maths:mensuration

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

## 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. 