Mensuration

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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