---
title: "Gear tooth vernier: chordal thickness"
subject: "Metrology"
chapter: 11
key: "semester_3/metrology:gear-tooth-vernier-chordal-thickness"
source: https://www.bctengineer.com/q/semester_3/metrology/gear-tooth-vernier-chordal-thickness
site: BCT Engineering
---

# Gear tooth vernier: chordal thickness

Metrology · Chapter 11: Gear measurement

- Asked in: year not given, 8 marks
- Video: [Measurement of Chordal Thickness using Gear Tooth Vernier Caliper](https://www.youtube.com/watch?v=1a1e9rMgbyY) — Knowledge Capsules
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/gear-tooth-vernier-chordal-thickness
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-11

## Question

Explain how the tooth thickness of a spur gear is measured with a gear tooth vernier caliper. Calculate the chordal thickness and the chordal addendum (setting depth) for a spur gear with module 5 mm, 30 teeth and $20^\circ$ pressure angle.

## Short answer

The gear tooth vernier sets chordal addendum as depth stop, then measures chordal thickness. For module 5, 30 teeth, 20° pressure angle: chordal thickness = 7.850 mm, chordal addendum = 5.103 mm.

## Answer

#### Gear tooth vernier caliper
It consists of two perpendicular vernier scales. The vertical scale is set to the **chordal addendum** $h_c$ and acts as a depth stop, resting on the top of the tooth. The horizontal scale measures the **chordal thickness** $t_c$, the straight line distance between the two flanks at the pitch circle.

```text
        |<-- t_c -->|
     ___|___________|___   <- horizontal vernier jaws
         \    __   /
          \  /  \ /       tooth
    h_c    ||    ||
    (vertical vernier stop on tooth top)
```
The vertical slide is set to $h_c$ and locked, the instrument is placed over the tooth so that the stop touches the top of the tooth, the jaws are closed on the flanks and the horizontal vernier reading is compared with the calculated chordal thickness. Errors: the measurement depends on the outside diameter of the blank, which must be checked, and the instrument is only accurate to its vernier least count (0.02 mm). Each tooth must be measured separately; several teeth are checked around the gear.

#### Formulae
For a gear with $z$ teeth and module $m$, the circular thickness at the pitch circle is $\pi m/2$ and

$$
t_c = z\,m\sin\!\left(\frac{90^\circ}{z}\right), \qquad h_c = m + \frac{z\,m}{2}\left[1 - \cos\!\left(\frac{90^\circ}{z}\right)\right]
$$

#### Calculation ($m = 5$, $z = 30$)
$$
\frac{90^\circ}{z} = 3^\circ
$$

$$
t_c = 30 \times 5 \times \sin 3^\circ = 150 \times 0.052336 = 7.850\ \text{mm}
$$

$$
h_c = 5 + \frac{150}{2}\left(1 - \cos 3^\circ\right) = 5 + 75(0.001370) = 5.103\ \text{mm}
$$

**Answer: chordal thickness = 7.850 mm, chordal addendum = 5.103 mm.**

## Check yourself

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

1. In a gear tooth vernier caliper, what do the two scales set and measure?
   A. The vertical scale measures chordal thickness; the horizontal sets the module
   B. The vertical scale is set to the chordal addendum as a depth stop; the horizontal scale measures chordal thickness
   C. Both scales measure the outside diameter
   D. The vertical scale measures pitch; the horizontal measures pressure angle
   Answer: B. The vertical vernier rests on the tooth top at hc and the horizontal jaws read the straight chord tc at the pitch circle. (Key point: Instrument use)

2. Which formula gives chordal thickness?
   A. tc = z m cos(90/z)
   B. tc = pi m / 2
   C. tc = z m sin(90/z)
   D. tc = m + (z m / 2)(1 - cos(90/z))
   Answer: C. The chord of the arc of tooth thickness is z m sin(90 deg / z); pi m/2 is the circular (arc) thickness. (Key point: Chordal thickness formula)

3. For m = 5, z = 30, what is the chordal thickness?
   A. 7.854 mm with no difference possible
   B. 5.103 mm
   C. 150.0 mm
   D. 7.850 mm
   Answer: D. tc = 150 sin 3 deg = 150 x 0.052336 = 7.850 mm. (Key point: Numerical tc)

4. For m = 5, z = 30, what is the chordal addendum hc?
   A. 5.103 mm
   B. 5.000 mm
   C. 7.850 mm
   D. 4.897 mm
   Answer: A. hc = 5 + 75(1 - cos 3 deg) = 5 + 75(0.00137) = 5.103 mm. (Key point: Chordal addendum)

## Related in Metrology

- [Vernier caliper and micrometer readings](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Avernier-and-micrometer-readings) — Metrology, chapter 4
- [Pitch, bearing contact and composite gear tests](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Agear-testing-methods-notes) — Metrology, chapter 11
- [Sine bar principle, setting and error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Asine-bar-setting-and-error) — Metrology, chapter 5

## Related in other subjects

- [Spur gear terminology and standard tooth system](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aspur-gear-terminology) — Machine Design I, chapter 9
- [Tooth forces on a spur gear pinion](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aspur-gear-tooth-forces-numerical) — Theory of Mechanism and Machine I, chapter 8
- [Spur gear pair dimensions numerical](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aspur-gear-pair-geometry-numerical) — Theory of Mechanism and Machine I, chapter 4
- [Compound gear train speeds, torques and tooth forces](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Acompound-gear-train-forces) — Machine Design I, chapter 9