---
title: "Effective diameter by three-wire method"
subject: "Metrology"
chapter: 12
key: "semester_3/metrology:three-wire-method-effective-diameter"
source: https://www.bctengineer.com/q/semester_3/metrology/three-wire-method-effective-diameter
site: BCT Engineering
---

# Effective diameter by three-wire method

Metrology · Chapter 12: Measurement of Screw Threads

- Asked in: year not given, 8 marks
- Video: [Three Wire Method: Derivation of Effective Diameter | Thread Measurement in Metrology | Shubham Kola](https://www.youtube.com/watch?v=Z4ZHaGafMh8) — Shubham Kola
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/three-wire-method-effective-diameter
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-12

## Question

Explain the three-wire method of measuring the effective diameter of a screw thread. Derive the best wire size for a $60^\circ$ thread. A M20 $\times$ 2.5 mm thread is measured with best-size wires, and the micrometer reading over the wires is 20.563 mm. Find the effective diameter and compare it with the nominal value of 18.376 mm.

## Answer

#### Method
Three wires of equal diameter $d$ are placed in the thread grooves, two on one side and one on the opposite side. The distance $M$ over the wires is measured with a micrometer (or a comparator with a fixed anvil). The effective diameter is then calculated from $M$, the wire diameter and the thread form.

```text
     wire
    __O__          M measured across
   /     \       the outer surfaces of
  /       \      the wires with a
 /   thread \    micrometer
    O   O
```

For a thread of included angle $\alpha$ and pitch $P$ (wire touching the flanks at the pitch line):

$$
M = E + d\left(1 + \operatorname{cosec}\frac{\alpha}{2}\right) - \frac{P}{2}\cot\frac{\alpha}{2}
$$

For 60 degrees this becomes $M = E + 3d - 0.866P$.

#### Best wire size
The wire should touch the flanks at the pitch (effective) diameter, where the gap width is $P/2$, so that an error in the thread angle has the least effect on the result. The contact point is $P/4$ from the centre line of the groove, and the radius of the wire is $d/2$, so

$$
\frac{P}{4} = \frac{d}{2}\cos\frac{\alpha}{2} \;\Rightarrow\; d_{best} = \frac{P}{2\cos(\alpha/2)}
$$

For 60 degrees, $d_{best} = 0.5774\,P$.

#### Numerical
$d = 0.5774 \times 2.5 = 1.443$ mm

$$
E = M - 3d + 0.866P = 20.563 - 3(1.443) + 0.866(2.5) = 20.563 - 4.330 + 2.165 = 18.398\ \text{mm}
$$

Error $= 18.398 - 18.376 = +0.022$ mm, the thread is slightly oversize in effective diameter.

**Answer: effective diameter = 18.398 mm (+0.022 mm over nominal).**

## Check yourself

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

1. In the three-wire method, how are the wires placed and what is measured?
   A. All three wires on one side; thread depth
   B. Two wires on one side and one on the opposite side; distance M over the wires
   C. One wire in each thread; the pitch
   D. Three wires at 120 degrees; the minor diameter
   Answer: B. M is measured across the outer surfaces of the wires, and the effective diameter is calculated from M, wire diameter and thread form. (Key point: Method)

2. What is the 'best wire' size?
   A. The wire that touches the flanks at the pitch (effective) diameter, so angle errors have least effect
   B. The wire that touches the root of the thread
   C. The wire that touches the crest
   D. The largest wire that fits the groove
   Answer: A. Contact at the pitch line, where the gap width is P/2, makes the result least sensitive to thread-angle error. (Key point: Best wire)

3. For a 60 degree thread, d_best = P / (2 cos 30 deg) equals:
   A. 0.866 P
   B. 1.732 P
   C. 0.5774 P
   D. 0.25 P
   Answer: C. P/(2 x 0.866) = 0.5774 P; for P = 2.5 mm, d = 1.443 mm. (Key point: Best wire value)

4. With M = 20.563 mm, d = 1.443 mm, P = 2.5 mm, what is E from E = M - 3d + 0.866P?
   A. 22.728 mm
   B. 18.376 mm
   C. 16.233 mm
   D. 18.398 mm
   Answer: D. E = 20.563 - 4.330 + 2.165 = 18.398 mm, which is 0.022 mm over the nominal 18.376 mm. (Key point: Effective diameter)

## Related in Metrology

- [Measuring thread pitch, angle and diameters](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Atoolmaker-microscope-thread-measurement) — Metrology, chapter 12
- [Thread errors and effect of pitch error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Ascrew-thread-errors-and-pitch-effect) — Metrology, chapter 12
- [Vernier caliper and micrometer readings](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Avernier-and-micrometer-readings) — Metrology, chapter 4
- [Classify fits from given limits of size](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aclassify-fits-from-limits) — Metrology, chapter 9

## Related in other subjects

- [Screw jack with square thread](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Ascrew-jack-square-thread-numerical) — Machine Design II, chapter 8
- [Screw thread production and change gears](https://www.bctengineer.com/question.md?key=semester_4%2Fmanufacturing_and_production_processes%3Ascrew-thread-manufacturing-change-gears) — Manufacturing and Production Processes, chapter 11
- [Slab continuous over three edges, 5x4 m (225 mm beam)](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-continuous-over-three-edges-5x4-2) — Design of RCC Structure, chapter 9
- [Vernier caliper and micrometer readings](https://www.bctengineer.com/question.md?key=semester_2%2Fworkshop_technology%3Avernier-micrometer-reading) — Workshop Technology, chapter 4