§1Cones instead of cylinders
A spur or helical pair is two cylinders rolling together; a bevel pair is two cones. Their apexes meet at the point where the shaft axes cross, and the teeth are cut on the cone surfaces.
Everything tapers: the tooth is largest at the outer (heel) end and shrinks toward the apex (the toe), so its module, pitch and depth all vary along the face. By convention the gear is dimensioned at the outer end, where the numbers are largest and the drawing is clearest. Because the cones share an apex, a bevel pair transmits motion between intersecting shafts at any shaft angle — though the right angle is overwhelmingly the common case, and the one worked here.
Contents§2Pitch cone angles
The half-angle of each pitch cone is fixed by the tooth counts and the shaft angle. For the usual 90° shaft angle the two cone angles are complementary and follow directly from the ratio.
The pinion, with fewer teeth, gets the slim cone; the gear, the wide one. A special case names itself: when the two gears are equal (z₁ = z₂), both cone angles are 45° and the pair is a mitre gear — a right-angle drive at 1 : 1. Where the shaft angle is not 90°, the general relation tan γ₁ = sin Σ /(z₂/z₁ + cos Σ) applies, but the right-angle forms above cover the great majority of designs.
Contents§3Cone distance and tooth size
The slant length from apex to the outer end — the cone distance — sets the scale of the gear and the length of its teeth.
The face width is then taken as a fraction of the cone distance — commonly no more than a third — because a tooth cut too far toward the apex becomes impractically small and weak. The outer pitch diameters d₁ and d₂ come straight from the module and tooth counts exactly as for a spur gear, since the outer end is where the gear is dimensioned.
Contents§4The back cone and equivalent teeth
A bevel tooth’s profile is worked out by pretending, at the outer end, that it is a spur tooth — Tredgold’s back-cone approximation, the bevel analogue of the helical equivalent gear.
The back cone is a cone tangent to the pitch cone at the outer end, unrolled into a flat sector; the tooth drawn on it is a spur tooth of z_v teeth, and its involute is used as the bevel profile. This is what lets the entire spur toolkit — profile, strength, contact — carry across to bevels. The equivalent counts are always larger than the real ones (the wide gear’s especially so), which is why bevel pinions resist undercut better than their tooth count alone would suggest.
Contents§5Straight, spiral and hypoid
Three tooth forms trade cost against smoothness and capacity, mirroring the spur-to-helical step.
Straight bevel
Teeth radial to the apex; the simplest, but engages abruptly like a spur — noisier, lower speed.
Spiral bevel
Curved, angled teeth engage gradually; quieter and stronger at speed — the helical of the bevel world. Produces thrust that varies with rotation sense.
Hypoid
Like spiral bevel but the axes are offset, not intersecting; allows a lower drive line and heavy sliding contact — the classic car rear axle.
The progression is the same lesson as spur-to-helical: gradual engagement buys quietness and load capacity at the price of complexity and induced thrust. Hypoids step beyond true bevels — the offset means the pitch surfaces are hyperboloids, not cones — and demand extreme-pressure lubricants because of the sliding they introduce.
Contents§6Worked right-angle pair
A single pair pulls the whole page together.
Pinion z₁ = 20, gear z₂ = 40, module 5 mm (outer end), shaft angle 90°. Cone angles: tan γ₁ = 20/40 = 0.5, so γ₁ = 26.57° and γ₂ = 63.43° (they sum to 90°, as they must). Outer pitch diameters d₁ = 100 mm, d₂ = 200 mm. Cone distance A_o = √(50² + 100²) = 111.8 mm. Equivalent teeth: pinion z_v1 = 20/cos 26.57° = 22.4, gear z_v2 = 40/cos 63.43° = 89.4 — so the gear’s tooth is profiled as an 89-tooth spur tooth, nearly straight-flanked, while the pinion is profiled as a 22-tooth one.
§7Quick reference
The working core of the page on one card rack.
Cone angles (90°)
tan γ₁ = z₁/z₂
γ₁ + γ₂ = 90°
Scale
d = m z · A_o = √(r₁²+r₂²)
face ≤ A_o/3
Profile
z_v = z/cos γ (back cone)
Mitre
z₁ = z₂ → 45°/45°
1 : 1 right angle
Forms
straight · spiral · hypoid