Areas and Volumes for Calculating Weights of Castings.

AREAS AND VOLUMES FOR CALCULATING WEIGHTS OF CASTINGS Rectangle and Parallelogram Area = ab Triangle Area = 1/2 cd. Area = SQRT(s(s-a)(s-b)(s-c)) when s= 1/2(a + b + c) Example: a = 3", b = 4", c = 5" s = (3" + 4" + 5")/2 = 6" Area = SQRT(6 (6-3) (6-4) (6-5)) = 6 sq. in. Regular Polygons n = Number of sides, s= Length of one side, r= Inside radius Area = 1/2 nsr Number of Sides Area 5 1.72047 s2 = 3.63273 r2 6 2.59809 s2 = 3.46408 r2 7 3.63395 s2 = 3.37099 r2 8 4.82847 s2 = 3.31368 r2 9 6.18181 s2 = 3.27574 r2 10 7.69416 s2 = 3.24922 r2 11 9.36570 s2 = 3.22987 r2 12 11.19616 s2 = 3.21539 r2 Trapezium Area = 1/2 [a (e + d) bd + ce] Example: a = 10", b = 3", c = 5", d = 6", e = 8" Area = 1/2 [10 (8 + 6) + (3 X 6) + (5 X 8)] = 99 sq. in. Square The diagonal of a square = A X 1.414 The side of a square inscribed in a given circle is: B X .707.
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Circle θ (the Greek letter Theta) = angle included between radii π (pi) = 3.1416, D = Diameter, R = Radius, C = Chord. h = Height of Arc, L = Length of Arc. Circumference = πD = 2πR = 2 SQRT(π X Area) Diameter = 2 R = Circumference / π = 2 SQRT(Area/π) Radius = 1/2 D = Circumference / 2 π = SQRT(Area/π) Radius = ((c/2)2 + h2)/2h Area = 1/4 π D2 = 0.7854 D2 = π R2 Chord = 2 SQRT(h (D - h)) = 2R X sine 1/2θ Height of Arc, h = R - SQRT(R2-(C/2)2) Length of Arc, L = θ/360 x 2 π R = 0.0174533 Rθ 1/2 θ (in degrees) = 28.6479 L/R Sine(1/2 θ) = (C/2) / R Sector of a Circle Area = 1/2 LR Example: L = 10.472", R = 5" Area = 10.472/2 x 5 = 26.180 sq. in. or Area = π R2 X θ/360 = 0.0087266 R2θ Example: R = 5", θ = 120° Area = 3.1416 X 52 X 120/360 = 26.180 sq. in. Segment of a Circle Area = πR2 X θ/360 - C(R - h)/2 Example: R = 5", θ = 120°, C = 8.66", h = 2.5" Area = 3.1416 X 52 X 120/360 - (8.66(5 - 2.5))/2 = 15.355 sq. in. Length of arc L = 0.0174533 R θ Area = 1/2 [LR-C (R-h)] Example: R = 5", C = 8.66", h = 2.5", θ = 120° L = 0.0174533 X 5 X 120 = 10.472" Area = 1/2[(10.472 X 5) - 8.66(5 - 2.5)] = 15.355 sq. in.
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Circular Ring Area = 0.7854 (D2-d2), or 0.7854 (D-d)(D+d) Example: D = 10", d = 3" Area = 0.7854 (102 - 32) = 71.4714 sq. in. Spandrel Area = 0.2146 R2 = 0.1073 C2 Example: R = 3 Area = 0.2146 X 32 = 1.9314 Parabolic Segment Area = 2/3 sh Example: s = 3, h = 4 Area = 2/3 X 3 X 4 = 8 Ellipse Area Tab = πab = 3.1416 ab Example: a = 3, b = 4 Area = 3.1416 X 3 X 4 = 37.6992 Irregular Figures Area may be found as follows: Divide the figure into equal spaces as shown by the lines in the figure. (1) Add lengths of dotted lines. (2) Divide sum by number of spaces. (3) Multiply result by "A."
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Ring of Circular Cross Section Area of Surface = 4 π2Rr = 39.4784 Rr Area of Surface = π2 Dd = 9.8696 Dd Volume = 2 π2 Rr2 = 19.7392 Rr2 Volume = 1/4 π2 Dd2 = 2.4674 Dd2 Sphere Surface = 4 π r2 = 12.5664 r2 = π d2 Volume = 4/3 π r3 = 4.1888 r3 Volume = 1/6 π d3 = 0.5236 d3 Segment of a Sphere Spherical Surface = 2 π rh = 1/4 π(c2 + 4h2) = 0.7854 (c2 + 4h2) Total Surface = 1/4π (c2 + 8 rh) = 0.7854 (c2 + 8 rh) Volume = 1/3 π h2 (3 r - h) = 1.0472 h2 (3 r - h) or Volume = 1/24 πh (3c2 + 4h2) = 0.1309 h (3c2 + 4h2) Sector of a Sphere Total Surface = 1/2πr (4 h + c) = 1.5708 r (4 h + c) Volume = 2/3 πr2h = 2.0944 r2h Cylinder Cylindrical Surface = π dh = 2 πrh = 6.2832 rh Total Surface = 2 π r (r + h) = 6.2832 (r + h) Volume = π r2h = 1/4 πd2h = 0 7854 d2h
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Pyramid A = area of base P = perimeter of base Lateral Area = 1/2 Ps Volume = 1/3Ah Frustum of a Pyramid A = area of base a = area of top m = area of midsection P = perimeter of base p = perimeter of top Lateral Area = 1/2s (P + p) Volume = 1/3h (a + A + SQRT(aA) Volume = 16h (A + a + 4m) Cone Conical Area = πrs = πr SQRT(r2 + h2) Volume =1/3 π r2h = 1.0472 r2h = 0.2618 d2h Frustum of a Cone A = area of base a = area of top m = area of midsection R = D / 2; r = d / 2 Area of Conical Surface = 1/2 πs (D + d) = 1.5708 s (D+d) Volume = 1/3 h (R2 + Rr + r2) = 1.0472 h (R2 + Rr + r2) Volume = 1/12 h (D2 + Dd + d2) = 0.2618 h (D2 + Dd + d2) Volume = 1/3h (a + A + SQRT(aA)) = 1/6 h (a + A + 4m)