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:
any triangle:
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:
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:
Ellipses
Parallel Pipes
Pyramid:
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):
Cylinders
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
-
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)
if you know v and l you can solve for w using cosh2(t) - sinh2(t) = 1:
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