---
title: "Spur gear pair dimensions numerical"
subject: "Theory of Mechanism and Machine I"
chapter: 4
key: "semester_6/theory_of_mechanism_and_machine_i:spur-gear-pair-geometry-numerical"
source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/spur-gear-pair-geometry-numerical
site: BCT Engineering
---

# Spur gear pair dimensions numerical

Theory of Mechanism and Machine I · Chapter 4: Spur Gears

- Asked in: year not given, 6 marks
- Video: [GEARS BASICS - Nomenclature and Main Relations in Just Over 10 Minutes!](https://www.youtube.com/watch?v=NF8fpuczU5I) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/spur-gear-pair-geometry-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-4

## Question

Two 20 degree full-depth involute spur gears of module 5 mm have 24 and 60 teeth. The pinion rotates at 1200 rpm. Calculate (a) the pitch circle diameters, (b) the centre distance, (c) the speed of the gear, (d) the circular pitch and tooth thickness, (e) addendum, dedendum, clearance and whole depth, (f) outside and root diameters, (g) base circle diameters.

## Short answer

Pitch circle diameters: d1 =120 mm, d2 =300 mm. Centre distance =210 mm. Gear speed =480 rpm. Circular pitch =15.708 mm, tooth thickness =7.854 mm. Addendum =5 mm, dedendum =6.25 mm, clearance =1.25 mm, whole depth =11.25 mm. Outside diameters: pinion 130 mm, gear 310 mm. Root diameters: pinion 107.5 mm, gear 287.5 mm. Base circle diameters: pinion 112.76 mm, gear 281.91 mm.

## Answer

Standard 20 degree full-depth system: addendum $= m$, dedendum $= 1.25m$, clearance $= 0.25m$, tooth thickness $= p_c/2$ (no backlash).

#### (a) Pitch circle diameters
$$
d_1 = mz_1 = 5 \times 24 = 120\ \text{mm}, \qquad d_2 = mz_2 = 5 \times 60 = 300\ \text{mm}
$$

#### (b) Centre distance
$$
C = \frac{d_1 + d_2}{2} = \frac{120 + 300}{2} = 210\ \text{mm}
$$

#### (c) Speed of gear
$$
N_2 = N_1\frac{z_1}{z_2} = 1200 \times \frac{24}{60} = 480\ \text{rpm}
$$

Velocity ratio $= z_2/z_1 = 2.5$.

#### (d) Circular pitch and tooth thickness
$$
p_c = \pi m = \pi \times 5 = 15.708\ \text{mm}, \qquad t = \frac{p_c}{2} = 7.854\ \text{mm}
$$

#### (e) Tooth proportions
| Quantity | Formula | Value (mm) |
|---|---|---|
| Addendum | $m$ | 5 |
| Dedendum | $1.25m$ | 6.25 |
| Clearance | $0.25m$ | 1.25 |
| Whole depth | $2.25m$ | 11.25 |

#### (f) Outside and root diameters
| Gear | Outside $d_a = d + 2m$ | Root $d_f = d - 2.5m$ |
|---|---|---|
| Pinion | 130 mm | 107.5 mm |
| Gear | 310 mm | 287.5 mm |

#### (g) Base circle diameters ($d_b = d\cos\phi$, $\cos 20^\circ = 0.9397$)
$$
d_{b1} = 120\cos 20^\circ = 112.76\ \text{mm}, \quad d_{b2} = 300\cos 20^\circ = 281.91\ \text{mm}
$$

(Base circle radii: 56.38 mm and 140.95 mm.)

**Answer: $d_1 = 120$ mm, $d_2 = 300$ mm, $C = 210$ mm, $N_2 = 480$ rpm, $p_c = 15.708$ mm, $t = 7.854$ mm; addendum 5 mm, dedendum 6.25 mm, clearance 1.25 mm, whole depth 11.25 mm; outside diameters 130 and 310 mm, root diameters 107.5 and 287.5 mm; base circle diameters 112.76 and 281.91 mm.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For module 5 mm with 24 and 60 teeth, the centre distance is:
   A. 420 mm
   B. 105 mm
   C. 210 mm
   D. 150 mm
   Answer: C. d1 = 5 x 24 = 120 mm and d2 = 5 x 60 = 300 mm, so C = (120 + 300)/2 = 210 mm. (Key point: Pitch diameters and centre distance)

2. The pinion (24 teeth) runs at 1200 rpm and drives a 60-tooth gear. The gear speed is:
   A. 480 rpm
   B. 3000 rpm
   C. 1200 rpm
   D. 720 rpm
   Answer: A. N2 = N1 x z1/z2 = 1200 x 24/60 = 480 rpm. (Key point: Speed of gear)

3. For a 20° full-depth system with module 5 mm, the dedendum and clearance are:
   A. 5 mm and 1.25 mm
   B. 6.25 mm and 1.25 mm
   C. 6.25 mm and 5 mm
   D. 5 mm and 6.25 mm
   Answer: B. Dedendum = 1.25m = 6.25 mm and clearance = 0.25m = 1.25 mm (addendum = m = 5 mm). (Key point: Tooth proportions)

4. Which pair of results is correct for this gear set?
   A. p_c = 5 mm, r_b1 = 60 mm
   B. p_c = 15.708 mm, r_b1 = 60 mm
   C. p_c = 7.854 mm, r_b1 = 56.38 mm
   D. p_c = 15.708 mm, r_b1 = 56.38 mm
   Answer: D. p_c = pi x m = 15.708 mm and the pinion base radius = 60 cos20° = 56.38 mm. (Key point: Circular pitch and base circle)

## Related in Theory of Mechanism and Machine I

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