Design Equations and Formula Circular Pitches and Equivalent Diametral Pitches Table

Gear Design and Selection Application

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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
a = Addendum
a_G = Addendum of Gear
a_P = Addendum of Pinion
b = Dedendum
c = Clearance
C = Center Distance
D = Pitch Diameter
D_G = Pitch Diameter of Gear
D_P = Pitch Diameter of Pinion
D_B = Base Circle Diameter
D_O = Outside Diameter
DR = Root Diameter
F = Face Width
h_k = Working Depth of Tooth
h_t = Whole Depth of Tooth
m_G = Gear Ratio
N = Number of Teeth
N_G = Number of Teeth in Gear
N_P = 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

Resources:

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