Bevel Gear Dimensions

Calculate pitch cone angles, equivalent tooth count and full dimensional analysis for straight bevel gear pairs on intersecting shafts

Inputs

Z_1

Driver tooth count

Z_2

Driven tooth count

\theta

Shaft angle θ

Formula Interpretation

Table 1 — Cone Angles & Equivalent Teeth

Driver Pitch Cone Angle $\gamma_{01}$

\tan\gamma_{01} = \dfrac{\sin\theta}{\dfrac{Z_2}{Z_1}+\cos\theta}

Pitch cone angle $\gamma_{01}$ of the driver depends on the tooth ratio $Z_2$ / $Z_1$ and shaft angle $\theta$ . For θ=90°: tanγ₀₁ = Z₁/Z₂ = 1/i.

Driven Pitch Cone Angle $\gamma_{02}$

\tan\gamma_{02} = \dfrac{\sin\theta}{\dfrac{Z_1}{Z_2}+\cos\theta}

Pitch cone angle $\gamma_{02}$ of the driven gear. For θ=90°: tanγ₀₂ = Z₂/Z₁ = i. Together γ₀₁ + γ₀₂ = 90°.

Shaft Angle Constraint

\theta = \gamma_{01} + \gamma_{02}

The shaft angle $\theta$ equals the sum of both pitch cone angles $\gamma_{01}$ + $\gamma_{02}$ . For the common right-angle bevel gear pair θ = 90°.

Back Cone Angle

\alpha_1 = 90°-\gamma_{01},\ \ \alpha_2 = 90°-\gamma_{02}

Back cone angle $\alpha$ = 90° − $\gamma_0$ . The back cone is the reference surface for laying out the equivalent spur gear tooth profile.

Equivalent Tooth Count

Z_{c1} = \dfrac{Z_1}{\cos\gamma_{01}},\quad Z_{c2} = \dfrac{Z_2}{\cos\gamma_{02}}

Equivalent tooth count $Z_c$ = $Z$ / cos $\gamma_0$ . The virtual spur gear with Zc teeth on the back cone is used for bending strength calculations (Lewis equation).

Table 2 — Straight Bevel Gear Dimensions

Pitch Circle Diameter

D_1 = mZ_1, \quad D_2 = mZ_2

Pitch circle diameter $D$ = module $m$ × teeth $Z$ . The large-end module m is the standard module at the outer (large) face of the cone.

Tip Circle Diameter

D_{a1} = D_1 + 2m\cos\gamma_{01}, \quad D_{a2} = D_2 + 2m\cos\gamma_{02}

Tip circle $D_a$ is projected onto the plane perpendicular to the shaft. $D_a$ = $D$ + 2 $m$ cos $\gamma_0$ . The cosine term accounts for the cone half-angle.

Cone Distance

A = \dfrac{D_1}{2\sin\gamma_{01}} = \dfrac{D_2}{2\sin\gamma_{02}}

Cone distance $A$ = half the pitch diameter divided by sinγ₀. It is the slant length from apex to the large-end pitch circle, shared by both meshing gears.

Tooth Angles

\beta = \arctan\!\dfrac{m}{A}, \quad \delta = \arctan\!\dfrac{1.25m}{A}

Addendum angle $\beta$ = arctan( $m$ / $A$ ); dedendum angle $\delta$ = arctan(1.25 $m$ / $A$ ). Both are measured along the pitch cone surface.

Tip / Root Cone Angles

\gamma_{b} = \gamma_0 + \beta, \quad \gamma_{d} = \gamma_0 - \delta

Tip cone angle $\gamma_b$ = γ₀ + β (tooth tips flare outward). Root cone angle $\gamma_d$ = γ₀ − δ (roots taper inward). Used for grinding and inspection.

Knowledge Points

Intersecting-Shaft Transmission

Bevel gears transmit power between two shafts whose axes intersect in space. The most common arrangement is θ = 90° (right-angle bevel), but any shaft angle from a few degrees to nearly 180° is possible. The pitch cones of a mating pair share the same apex point.

Equivalent Spur Gear (Back Cone)

For tooth-strength analysis, a bevel gear tooth is idealised as a spur gear tooth on the ''back cone'', a cone whose apex is also at the gear apex but whose half-angle is the back cone angle α = 90° − γ₀. The equivalent tooth count Zc = Z/cosγ₀ is used in Lewis-equation strength calculations.

Types of Bevel Gear Teeth

By tooth direction: straight (simplest), helical/skew (smoother), and spiral/curved (quietest and highest load capacity). Bevel gear shafts typically have only one bearing on each side of the gear due to space constraints, so deflection and axial thrust must be carefully managed.

Worked Example

Two shafts intersect at a right angle ( $\theta=90°$ ). The bevel gear pair has a speed ratio $i=2$ . Find the pitch cone angles γ₀₁ and γ₀₂.

Step 1 — Pitch cone angle of the driver

\tan\gamma_{01} = \dfrac{\sin 90°}{\dfrac{Z_2}{Z_1}+\cos 90°} = \dfrac{1}{i} = \dfrac{1}{2} = 0.5

\gamma_{01} = \arctan(0.5) \approx 26°34'

Step 2 — Pitch cone angle of the driven gear

\tan\gamma_{02} = \dfrac{\sin 90°}{\dfrac{Z_1}{Z_2}+\cos 90°} = i = 2

\gamma_{02} = \arctan(2) \approx 63°26'

Verification — sum of cone angles equals shaft angle

\gamma_{01} + \gamma_{02} = 26°34' + 63°26' = 90° = \theta \checkmark

Result: γ₀₁ ≈ 26°34′, γ₀₂ ≈ 63°26′. Their sum = 90° = θ ✓

Extended Knowledge

•The axial (thrust) force on a bevel gear is $F_a = F_t \cdot \tan\gamma_0 \cdot \sin\alpha$ , where $\alpha$ is the pressure angle. This force must be taken up by the shaft bearings; tapered roller bearings or angular-contact bearings are typically used.
•Spiral bevel gears (also called ''hypoid'' when the axes are offset) have a curved tooth trace with a helix angle typically between 25° and 45°. They offer quieter operation, higher load capacity, and are used in automotive differentials and high-speed marine gearboxes.
•For the equivalent-tooth bending-strength check, the full Lewis equation uses Zc in place of Z. Because Zc > Z (division by a cosine < 1), a bevel gear is weaker in bending than a spur gear with the same tooth count and module, and the large-end module is used conservatively.

Related Standards & Articles

Authoritative references for the formulas used in this calculator

Standard

Gear Design Symbol Reference (GB/T 2821 / ISO 701)

A complete notation reference for gear design calculations — geometric parameters, load factors, strength properties, and safety factors — per GB/T 2821-2003 and ISO 701:1998, with full LaTeX symbol rendering.

Standard

Gear Transmission Types: Selection Overview (Involute, Bevel, Worm & Planetary)

A comprehensive comparison of all major gear transmission types — involute cylindrical, bevel, worm, planetary, few-tooth-difference, and harmonic drives — covering key characteristics, transmission ratios, power capacities, speed ranges, and typical industrial applications for mechanical design selection.

Bevel Gear Dimensions

Calculate pitch cone angles, equivalent tooth count and full dimensional analysis for straight bevel gear pairs on intersecting shafts

Inputs

Z_1

Driver tooth count

Z_2

Driven tooth count

\theta

Shaft angle θ

Formula Interpretation

Table 1 — Cone Angles & Equivalent Teeth

Driver Pitch Cone Angle $\gamma_{01}$

\tan\gamma_{01} = \dfrac{\sin\theta}{\dfrac{Z_2}{Z_1}+\cos\theta}

Pitch cone angle $\gamma_{01}$ of the driver depends on the tooth ratio $Z_2$ / $Z_1$ and shaft angle $\theta$ . For θ=90°: tanγ₀₁ = Z₁/Z₂ = 1/i.

Driven Pitch Cone Angle $\gamma_{02}$

\tan\gamma_{02} = \dfrac{\sin\theta}{\dfrac{Z_1}{Z_2}+\cos\theta}

Pitch cone angle $\gamma_{02}$ of the driven gear. For θ=90°: tanγ₀₂ = Z₂/Z₁ = i. Together γ₀₁ + γ₀₂ = 90°.

Shaft Angle Constraint

\theta = \gamma_{01} + \gamma_{02}

The shaft angle $\theta$ equals the sum of both pitch cone angles $\gamma_{01}$ + $\gamma_{02}$ . For the common right-angle bevel gear pair θ = 90°.

Back Cone Angle

\alpha_1 = 90°-\gamma_{01},\ \ \alpha_2 = 90°-\gamma_{02}

Back cone angle $\alpha$ = 90° − $\gamma_0$ . The back cone is the reference surface for laying out the equivalent spur gear tooth profile.

Equivalent Tooth Count

Z_{c1} = \dfrac{Z_1}{\cos\gamma_{01}},\quad Z_{c2} = \dfrac{Z_2}{\cos\gamma_{02}}

Equivalent tooth count $Z_c$ = $Z$ / cos $\gamma_0$ . The virtual spur gear with Zc teeth on the back cone is used for bending strength calculations (Lewis equation).

Table 2 — Straight Bevel Gear Dimensions

Pitch Circle Diameter

D_1 = mZ_1, \quad D_2 = mZ_2

Pitch circle diameter $D$ = module $m$ × teeth $Z$ . The large-end module m is the standard module at the outer (large) face of the cone.

Tip Circle Diameter

D_{a1} = D_1 + 2m\cos\gamma_{01}, \quad D_{a2} = D_2 + 2m\cos\gamma_{02}

Tip circle $D_a$ is projected onto the plane perpendicular to the shaft. $D_a$ = $D$ + 2 $m$ cos $\gamma_0$ . The cosine term accounts for the cone half-angle.

Cone Distance

A = \dfrac{D_1}{2\sin\gamma_{01}} = \dfrac{D_2}{2\sin\gamma_{02}}

Cone distance $A$ = half the pitch diameter divided by sinγ₀. It is the slant length from apex to the large-end pitch circle, shared by both meshing gears.

Tooth Angles

\beta = \arctan\!\dfrac{m}{A}, \quad \delta = \arctan\!\dfrac{1.25m}{A}

Addendum angle $\beta$ = arctan( $m$ / $A$ ); dedendum angle $\delta$ = arctan(1.25 $m$ / $A$ ). Both are measured along the pitch cone surface.

Tip / Root Cone Angles

\gamma_{b} = \gamma_0 + \beta, \quad \gamma_{d} = \gamma_0 - \delta

Tip cone angle $\gamma_b$ = γ₀ + β (tooth tips flare outward). Root cone angle $\gamma_d$ = γ₀ − δ (roots taper inward). Used for grinding and inspection.

Knowledge Points

Intersecting-Shaft Transmission

Equivalent Spur Gear (Back Cone)

Types of Bevel Gear Teeth

Worked Example

Two shafts intersect at a right angle ( $\theta=90°$ ). The bevel gear pair has a speed ratio $i=2$ . Find the pitch cone angles γ₀₁ and γ₀₂.

Step 1 — Pitch cone angle of the driver

\tan\gamma_{01} = \dfrac{\sin 90°}{\dfrac{Z_2}{Z_1}+\cos 90°} = \dfrac{1}{i} = \dfrac{1}{2} = 0.5

\gamma_{01} = \arctan(0.5) \approx 26°34'

Step 2 — Pitch cone angle of the driven gear

\tan\gamma_{02} = \dfrac{\sin 90°}{\dfrac{Z_1}{Z_2}+\cos 90°} = i = 2

\gamma_{02} = \arctan(2) \approx 63°26'

Verification — sum of cone angles equals shaft angle

\gamma_{01} + \gamma_{02} = 26°34' + 63°26' = 90° = \theta \checkmark

Result: γ₀₁ ≈ 26°34′, γ₀₂ ≈ 63°26′. Their sum = 90° = θ ✓

Extended Knowledge

•The axial (thrust) force on a bevel gear is $F_a = F_t \cdot \tan\gamma_0 \cdot \sin\alpha$ , where $\alpha$ is the pressure angle. This force must be taken up by the shaft bearings; tapered roller bearings or angular-contact bearings are typically used.
•Spiral bevel gears (also called ''hypoid'' when the axes are offset) have a curved tooth trace with a helix angle typically between 25° and 45°. They offer quieter operation, higher load capacity, and are used in automotive differentials and high-speed marine gearboxes.
•For the equivalent-tooth bending-strength check, the full Lewis equation uses Zc in place of Z. Because Zc > Z (division by a cosine < 1), a bevel gear is weaker in bending than a spur gear with the same tooth count and module, and the large-end module is used conservatively.

Related Standards & Articles

Authoritative references for the formulas used in this calculator

Standard

Gear Design Symbol Reference (GB/T 2821 / ISO 701)

Standard