### Design Equations and Formula Circular Pitches and Equivalent Diametral Pitches Table

Gear Design and Selection Application

Spur Gear design formula for geometry, pitch, tooth clearance and critical functional data.
(Inch Units Applicable for Constants) Spur Gear Design Calculator

Where:
φ = Pressure Angle
b = Dedendum
c = Clearance
C = Center Distance
D = Pitch Diameter
DG = Pitch Diameter of Gear
DP = Pitch Diameter of Pinion
DB = Base Circle Diameter
DO = Outside Diameter
DR = Root Diameter
F = Face Width
hk = Working Depth of Tooth
ht = Whole Depth of Tooth
mG = Gear Ratio
N = Number of Teeth
NG = Number of Teeth in Gear
NP = Number of Teeth in Pinion
p = Circular Pitch
P=Diametral Pitch

Equations for Standards Spur Gears

 To Find Equation Base Circle Pitch DB = D cosφ Circular Pitch p = ( π D )/ N p = π / P Center Distance C = Np (mG + 1) / 2P C = ( Dp + DG ) / 2 C = ( NG + Np ) / 2P C = (NG + Np) p / 2P C = (NG + Np) p / 6.2832 Diametral Pitch P = π / p P = N / D P = [ Np ( mG + 1) ] / 2C Gear Ratio mG = NG / Np Number of Teeth N = P D N = ( π D ) / p Outside Diameter (Full Depth Teeth) DO = ( N + 2 ) / P DO = [ ( N + 2 ) p ] / π Outside Diameter (American Standard Stub Teeth) DO = ( N + 1.6 ) / P DO = [ ( N + 1.6 ) p ] / π Outside Diameter DO = D + 2a Pitch Diameter D = N / P D = (N p ) / π Root Diameter DR = D - 2b Whole Depth a + b Working Depth aG + ap

Formulas for Tooth Parts, 20-and 25-degree Involute Full-depth Teeth
ANSI Coarse Pitch Spur Gear Tooth Forms ANSI B6.1

 To Calculate Diametral Pitch, P, Known Circular Pitch, p, Known Addendum a = 1.000 / P a = 0.3183 × p Dedendum (Preferred) b = 1.250 / P b = 0.3979 × p (Shaved or Ground Teeth)a b = 1.350 / P b = 0.4297 × p Working Depth hk = 2.000 / P hk = 0.6366 × p Whole Depth (Preferred) ht = 2.250 / P ht = 0.7162 × p (Shaved or Ground Teeth) ht = 2.350 / P ht = 0.7480 × p Clearance (Preferred)b c = 0.250 / P c = 0.0796 × p (Shaved or Ground Teeth) c = 0.350 / P c = 0.1114 × p Fillet Radius (Rack)c rf = 0.300 / P rf = 0.0955 × p Pitch Diameter D = N / P D = 0.3183 × Np Outside Diameter DO = (N + 2) / P DO = 0.3183 × (N + 2) p Root Diameter (Preferred) DR = (N − 2.5) / P DR = 0.3183 × (N − 2.5) p Root Diameter (Shaved or Ground Teeth) DR = (N − 2.7) / P DR = 0.3183 × (N − 2.7) p Circular Thickness Basic t = 1.5708 / P t = p / 2

Equations Tooth Parts, 20-and 25-degree Involute Full-depth Teeth ANSI Coarse Pitch Spur Gear Tooth Forms ANSI B6.1

Spur Gear Design Calculator

a When gears are preshave cut on a gear shaper the dedendum will usually need to be increased to 1.40/P to allow for the higher fillet trochoid produced by the shaper cutter. This is of particular importance on gears of few teeth or if the gear blank configuration requires the use of a small diameter shaper cutter, in which case the dedendum may need to be increased to as much as 1.45/P. This should be avoided on highly loaded gears where the consequently reduced J factor will increase gear tooth stress excessively.
b A minimum clearance of 0.157/P may be used for the basic 20-degree and 25-degree pressure angle rack in the case of shallow root sections and use of existing hobs or cutters.
c The fillet radius of the basic rack should not exceed 0.235/P for a 20-degree pressure angle rack or 0.270/P for a 25-degree pressure angle rack for a clearance of 0.157/P. The basic rack fillet radius must be reduced for teeth with a 25-degree pressure angle having a clearance in excess of 0.250/P.

#### Helical Gear

 Calculate When Defined Formula Normal D.P. (Pn) Transverse D.P.(P) and the Helix Angle (A) Pn= P / cos A Normal D.P. (Pn) Number of teeth (N) and the Helix Angle (A) Pn= N / (D X cos A) Pitch Diameter (D) Number of teeth (N), the Normal Diametral Pitch and the Helix Angle (A) D = N / (Pn X cos A) Outside Diameter (OD) Pitch Diameter (D) and the Addendum (a) OD = D + (2 X a) Outside Diameter (OD) Normal Diametral Pitch (P) and the Pitch Diameter (D) OD = D + 2/Pn Helix Angle (A) for Parallel Shaft Drive Number of Teeth (N), Pitch Diameter (D) and the Diametral Pitch (P) Cos A = N / (Pn X D) Addendum (a) Normal Diametral Pitch (Pn) a = 1 / Pn Lead (L) Pitch Diameter (D) and Pitch Helix Angle L = (pi*D) / Tan

Circular Pitches and Equivalent Diametral Pitches Table

 Circular Pitch Diametral Pitch Module Arc Thickness of Tooth on Pitch Line Addendum Working Depth of Tooth Dedendum or Depth of Space Below Pitch Line Whole Depth of Tooth 4 0.7854 32.3402 2.0000 1.2732 2.5464 1.4732 2.7464 3 - 1/2 0.8976 28.2581 1.7500 1.1140 2.2281 1.2890 2.4031 3 1.0472 24.2552 1.5000 0.9549 1.9098 1.1049 2.0598 2 - 3/4 1.1424 22.2339 1.3750 0.8753 1.7506 1.0128 1.8881 2 - 1/2 1.2566 20.2117 1.2500 0.7957 1.5915 0.9207 1.7165 2 - 1/4 1.3963 18.1913 1.1250 0.7162 1.4323 0.8287 1.5448 2 1.5708 16.1701 1.0000 0.6366 1.2732 0.7366 1.3732 1 - 7/8 1.6755 15.1595 0.9375 0.5968 1.1937 0.6906 1.2874 1 - 3/4 1.7952 14.1488 0.8750 0.5570 1.1141 0.6445 1.2016 1 - 5/8 1.9333 13.1382 0.8125 0.5173 1.0345 0.5985 1.1158 1 - 1/2 2.0944 12.1276 0.7500 0.4775 0.9549 0.5525 1.0299 1 - 7/16 2.1855 11.6223 0.7187 0.4576 0.9151 0.5294 0.9870 1 - 3/8 2.2848 11.1169 0.6875 0.4377 0.8754 0.5064 0.9441 1 - 5/16 2.3936 10.6116 0.6562 0.4178 0.8356 0.4834 0.9012 1 - 1/4 2.5133 10.1062 0.6250 0.3979 0.7958 0.4604 0.8583 1 - 3/16 2.6456 9.6010 0.5937 0.3780 0.7560 0.4374 0.8154 1 - 1/8 2.7925 9.0958 0.5625 0.3581 0.7162 0.4143 0.7724 1 - 1/16 2.9568 8.5904 0.5312 0.3382 0.6764 0.3913 0.7295 1 3.1416 8.0851 0.5000 0.3183 0.6366 0.3683 0.6866 15/16 3.3510 7.5798 0.4687 0.2984 0.5968 0.3453 0.6437 7/8 3.5904 7.0744 0.4375 0.2785 0.5570 0.3223 0.6007 13/16 3.8666 6.5692 0.4062 0.2586 0.5173 0.2993 0.5579 3/4 4.1888 6.0639 0.3750 0.2387 0.4775 0.2762 0.5150 11/16 4.5696 5.5586 0.3437 0.2189 0.4377 0.2532 0.4720 2/3 4.7124 5.3903 0.3333 0.2122 0.4244 0.2455 0.4577 5/8 5.0265 5.0532 0.3125 0.1989 0.3979 0.2301 0.4291 9/16 5.5851 4.5479 0.2812 0.1790 0.3581 0.2071 0.3862 1/2 6.2832 4.0426 0.2500 0.1592 0.3183 0.1842 0.3433 7/16 7.1808 3.5373 0.2187 0.1393 0.2785 0.1611 0.3003 2/5 7.8540 3.2340 0.2000 0.1273 0.2546 0.1473 0.2746 3/8 8.3776 3.0319 0.1875 0.1194 0.2387 0.1381 0.2575 1/3 9.4248 2.6947 0.1666 0.1061 0.2122 0.1228 0.2289 5/16 10.0531 2.5266 0.1562 0.0995 0.1989 0.1151 0.2146 2/7 10.9956 2.3100 0.1429 0.0909 0.1819 0.1052 0.1962 1/4 12.5664 2.0213 0.1250 0.0796 0.1591 0.0921 0.1716 2/9 14.1372 1.7967 0.1111 0.0707 0.1415 0.0818 0.1526 1/5 15.7080 1.6170 0.1000 0.0637 0.1273 0.0737 0.1373 3/16 16.7552 1.5160 0.0937 0.0597 0.1194 0.0690 0.1287 1/6 18.8496 .5053 0.0833 0.0531 0.1061 0.0614 0.1144

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