In the field of mechanical transmission, straight spur gears are among the most widely used components due to their high precision, adaptability, and reliability. The tooth geometry directly determines the load‑carrying capacity and strength performance. In vocational education for mechanical engineering, we often apply ready‑made formulas for gear dimensions without understanding their derivation. To help students gain a clear insight, I will present a comprehensive derivation of the tooth thickness at any arbitrary point on a standard straight spur gear, covering the base circle, pitch circle, tip circle, and root circle. The analysis relies on involute geometry, the evolution of the involute function, and fundamental gear parameters. This work aims to bridge the gap between abstract formulas and practical understanding, with abundant tables and equations summarizing the key relationships.
The article is structured as follows: first, the basic requirements for gear transmission; second, the nomenclature and parameters of straight spur gears; third, the derivation of base circle tooth thickness; fourth, the involute function; fifth, the general formula for tooth thickness at any point; sixth, the special case of root circle tooth thickness; seventh, a comparative table for different positions and tooth numbers; and finally, concluding remarks.
1. Basic Requirements of Gear Transmission
For any gear pair, two fundamental requirements must be satisfied: a constant instantaneous transmission ratio and sufficient load‑carrying capacity. The involute profile ensures a constant ratio because the common normal at any contact point always passes through a fixed point (the pitch point). This property guarantees smooth and vibration‑free operation. The load capacity depends on the tooth shape, especially the tooth thickness at the root, which is a critical factor in bending strength calculations. For standard straight spur gears, the tooth thickness varies along the tooth profile – it is smallest at the tip and largest at the root. Therefore, accurately determining the tooth thickness at any radial position is essential for strength verification and manufacturing.
2. Nomenclature and Basic Parameters of Straight Spur Gears
Figure 1 provides a schematic of a straight spur gear. The key terms are:
- Addendum circle: the circle passing through the tooth tips.
- Dedendum circle: the circle passing through the tooth roots.
- Pitch circle: the reference circle where the module and pressure angle are standard.
- Base circle: the circle from which the involute profile is generated.
- Tooth thickness s: arc length between opposite flanks on the pitch circle.
- Tooth space e: arc length between adjacent teeth.
The three fundamental parameters of a standard straight spur gear are:
- Number of teeth z.
- Module m (selected from standard series).
- Pressure angle α (standard α = 20°).
From these, we derive:
- Pitch diameter: $$d = mz$$
- Base diameter: $$d_b = mz \cos\alpha$$
- Pitch radius: $$R = \frac{mz}{2}$$
- Base radius: $$R_b = \frac{mz \cos\alpha}{2}$$
- Circular pitch: $$p = \pi m$$
- Tooth thickness on pitch circle: $$s = \frac{p}{2} = \frac{\pi m}{2}$$
The standard addendum and dedendum coefficients are:
- Addendum coefficient: $$h_a^* = 1$$ (normal tooth)
- Clearance coefficient: $$c^* = 0.25$$
Thus, addendum height: $$h_a = m$$, dedendum height: $$h_f = 1.25m$$.
3. Derivation of Base Circle Tooth Thickness
Let us consider one complete tooth on a standard straight spur gear. The tooth profile is symmetric about a radial line. On the pitch circle, the arc length of half the tooth thickness is:
$$\overline{BD} = \frac{s}{2} = \frac{\pi m}{4}$$
The angle subtended by this half‑tooth on the pitch circle is:
$$\gamma = \frac{\overline{BD}}{R} = \frac{\frac{\pi m}{4}}{\frac{mz}{2}} = \frac{\pi}{2z}$$ (radians)
On the base circle, the angle between the radial line through the symmetry axis and the point where the involute starts is:
$$\beta = \alpha – \gamma = \alpha – \frac{\pi}{2z}$$
The arc length on the base circle corresponding to this angle is:
$$\overline{AE} = R_b \beta = \frac{mz\cos\alpha}{2} \left( \alpha – \frac{\pi}{2z} \right)$$
From involute geometry, the involute at point B on the pitch circle has a generating line length:
$$\overline{AB} = R_b \tan\alpha = \frac{mz\cos\alpha}{2} \tan\alpha$$
The arc length from the base circle starting point to the involute intersection with the base circle is:
$$\overline{CE} = \overline{AC} – \overline{AE} = \overline{AB} – \overline{AE}$$
$$\overline{CE} = \frac{mz\cos\alpha}{2} \left( \tan\alpha – \alpha + \frac{\pi}{2z} \right)$$
Simplifying using the involute function (discussed next):
$$\overline{CE} = \frac{m}{2} \cos\alpha \left( \frac{\pi}{2} + z \cdot \operatorname{inv}\alpha \right)$$
Hence, the full base circle tooth thickness is twice this arc:
$$s_b = 2\overline{CE} = m \cos\alpha \left( \frac{\pi}{2} + z \cdot \operatorname{inv}\alpha \right)$$
Table 1 summarises the key parameters for base thickness calculation on a standard straight spur gear.
| Symbol | Description | Expression |
|---|---|---|
| γ | Half‑tooth angle on pitch circle | π/(2z) |
| β | Angle from symmetry to base circle involute start | α – π/(2z) |
| AE | Arc from symmetry to base circle involute start | (mz cosα/2)(α – π/(2z)) |
| CE | Half base tooth thickness arc | (m cosα/2)(π/2 + z·invα) |
| sb | Base circle tooth thickness | m cosα (π/2 + z·invα) |
4. The Involute Function
The involute function is defined as:
$$\operatorname{inv}\theta = \tan\theta – \theta$$
For the pitch circle (pressure angle α):
$$\operatorname{inv}\alpha = \tan\alpha – \alpha$$
For any point K on the involute with pressure angle αk:
$$\operatorname{inv}\alpha_k = \tan\alpha_k – \alpha_k$$
This function appears repeatedly in tooth thickness formulas.
5. General Formula for Tooth Thickness at Arbitrary Point K
Consider a point K on the involute profile at radius Rk (Figure 2). The corresponding pressure angle at K satisfies:
$$\cos\alpha_k = \frac{R_b}{R_k}$$
The tooth thickness sk at radius Rk is given by the arc length between the left and right flanks on that circle. Using the symmetry of the tooth, the half‑angle βk on the circle of radius Rk is:
$$\beta_k = \frac{s_b}{2R_b} – \operatorname{inv}\alpha_k = \frac{\pi}{2z} + \operatorname{inv}\alpha – \operatorname{inv}\alpha_k$$
Therefore, the full tooth thickness at point K is:
$$s_k = 2 R_k \beta_k = R_k \left( \frac{\pi}{z} + 2\operatorname{inv}\alpha – 2\operatorname{inv}\alpha_k \right)$$
A more convenient form using the pitch‑circle tooth thickness s = πm/2 and pitch radius R = mz/2:
$$s_k = s \frac{R_k}{R} – 2R_k \left( \operatorname{inv}\alpha_k – \operatorname{inv}\alpha \right)$$
This equation is the core result for any straight spur gear. Table 2 lists the tooth thickness at three radial positions for a module m = 3 mm and α = 20°.
| z | s (pitch circle) [mm] | s15° (pressure angle 15°) [mm] | sb (base circle) [mm] | sf (root circle) [mm] |
|---|---|---|---|---|
| 20 | 4.7124 | 5.0950 | 5.2683 | — |
| 25 | 4.7124 | 5.2227 | 5.4783 | — |
| 30 | 4.7124 | 5.3504 | 5.6883 | — |
| 35 | 4.7124 | 5.4781 | 5.8983 | — |
| 40 | 4.7124 | 5.6058 | 6.1084 | — |
| 45 | 4.7124 | 5.7335 | 6.3184 | 6.3072 |
Note: Root circle tooth thickness is only meaningful when z ≥ 42 (base circle inside dedendum). For z = 45, the root thickness is computed using the formula below.
6. Root Circle Tooth Thickness for Large Tooth Numbers
When the number of teeth z ≥ 42, the base circle diameter is smaller than the dedendum diameter. The entire tooth profile from root to tip is involute. The tooth thickness on the root circle can be obtained by applying the general formula with αf being the pressure angle at the root circle. The root circle radius is:
$$R_f = \frac{d_f}{2} = \frac{m(z – 2.5)}{2}$$
The pressure angle at the root is:
$$\cos\alpha_f = \frac{R_b}{R_f} = \frac{z\cos\alpha}{z – 2.5}$$
Then the involute function at the root:
$$\operatorname{inv}\alpha_f = \tan\alpha_f – \alpha_f$$
Finally, the root tooth thickness:
$$s_f = s \frac{R_f}{R} – 2R_f \left( \operatorname{inv}\alpha_f – \operatorname{inv}\alpha \right)$$
For example, with m = 3, z = 45, α = 20°, we compute αf ≈ 25.85°, invαf ≈ 0.0325, and obtain sf = 6.3072 mm as shown in Table 2.
When z < 42, the root lies inside the base circle; the profile below the base circle is a trochoid (fillet curve) rather than an involute. In such cases, the tooth thickness at the root cannot be calculated by the involute formula. Practical designs use the 30° tangent method or parabolic approximation to define the critical section for bending strength.
7. Comparative Analysis of Tooth Thickness
From Table 2, we observe:
- For a given module, the tooth thickness at any radial position (except the pitch circle) increases with the number of teeth.
- On the same gear, tooth thickness increases from the addendum toward the root. The base circle thickness is always greater than the pitch circle thickness, and the root thickness (when involute) is even larger.
- The pitch‑circle tooth thickness remains constant (πm/2) regardless of z, which is a defining property of standard straight spur gears.
Table 3 summarises the variation of tooth thickness along the tooth height for two representative tooth numbers.
| Position | z = 20 (mm) | z = 45 (mm) |
|---|---|---|
| Tip circle | ≈ 2.98 | ≈ 4.21 |
| Pitch circle | 4.7124 | 4.7124 |
| Base circle | 5.2683 | 6.3184 |
| Root circle | — (trochoid) | 6.3072 |
These results highlight that the tooth becomes progressively thicker toward the root, which is beneficial for bending strength. The design of straight spur gears must consider the worst‑case load at the root fillet.
8. Conclusion
In this paper, I have systematically derived the tooth thickness at any arbitrary point on a standard straight spur gear. Starting from the basic gear parameters, I developed expressions for the base circle tooth thickness, the general involute formulation, and the specific case of root circle thickness for gears with large tooth numbers. Tables and formulas were provided to aid in practical calculations and to clarify the relationships that are often hidden in textbooks. Understanding these derivations is crucial for students and engineers who wish to apply gear theory with confidence. The formulas presented here are directly applicable to the design, strength verification, and manufacturing of straight spur gears.
The key takeaway is that the tooth thickness at any radius Rk can be expressed as:
$$s_k = \frac{R_k}{R}s – 2R_k \left( \operatorname{inv}\alpha_k – \operatorname{inv}\alpha \right)$$
where R = mz/2, s = πm/2, and αk = arccos(Rb/Rk). This formula unifies all cases and is essential for anyone working with straight spur gears.