A diagonal scale is a scale used to measure three units or a unit and two of its subdivisions accurately.
It is based on the principle of similar triangles and diagonal lines.
Diagonal scales are commonly used in engineering drawings for precise measurements.

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Diagonal Scale in Engineering Drawing

Introduction

A Diagonal Scale is an advanced scale used in engineering drawing to measure three units or a unit and two successive subdivisions accurately.

It is more precise than a Plain Scale and Plane Scale because it can measure very small distances using the principle of similar triangles.

Examples:

• Kilometers, hectometers, and decameters
• Meters, decimeters, and centimeters
• Feet, inches, and fractions of inches

Diagonal scales are widely used in:

• Engineering Graphics
• Surveying
• Mapping
• Civil Engineering
• Mechanical Engineering

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What is a Diagonal Scale?

A diagonal scale is a scale that enables measurement of:

A unit, its subdivision, and a further subdivision.

For example:

• 1 meter
• 0.1 meter (decimeter)
• 0.01 meter (centimeter)

Thus, it provides much higher accuracy than a plane scale.

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Definition

A diagonal scale is defined as:

“A scale used to read three successive units by employing the principle of similar triangles.”

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Purpose of a Diagonal Scale

The main purposes are:

• Measure small distances accurately
• Represent large dimensions on small drawings
• Obtain precise readings
• Read units, subdivisions, and sub-subdivisions

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Principle of Diagonal Scale

The diagonal scale is based on the principle of similar triangles.

Principle

If parallel lines cut two intersecting lines, the corresponding segments are proportional.

This principle allows a very small division to be obtained without physically dividing a line into extremely tiny parts.

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Representative Fraction (R.F.)

The construction of a diagonal scale begins with the Representative Fraction.

Formula

Where:

• Drawing Length = Length on paper
• Actual Length = Real-world length

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Components of a Diagonal Scale

1. Main Scale

Represents the primary unit.

Examples:

• Kilometers
• Meters
• Feet

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2. Subdivisions

Represent smaller units.

Examples:

• Hectometers
• Decimeters
• Inches

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3. Diagonal Lines

Used to obtain even smaller subdivisions.

These diagonals provide the third level of measurement.

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4. Vertical Divisions

Divide the scale height into equal parts.

These divisions work with diagonals to provide accurate readings.

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Construction of a Diagonal Scale

Step 1: Determine the R.F.

Example:

R.F. = 1/1000

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Step 2: Determine Maximum Length

Example:

Maximum distance = 50 m

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Step 3: Calculate Scale Length

Using:

Length of Scale = R.F. × Actual Length

Convert:

50 m = 50000 mm

Then:

Length = 1/1000 × 50000

= 50 mm

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Step 4: Draw Main Scale

Draw a rectangle of length:

50 mm

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Step 5: Divide Main Scale

Suppose:

5 divisions

Each division represents:

10 m

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Step 6: Subdivide First Main Division

Divide into:

10 equal parts

Each part represents:

1 m

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Step 7: Draw Vertical Lines

Construct a rectangle above the first division.

Divide its height into:

10 equal parts

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Step 8: Draw Diagonals

Join:

• Bottom left corner
• Top right corner

Draw parallel diagonals.

These diagonals produce smaller subdivisions.

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Reading a Diagonal Scale

Suppose we need to read:

23.7 m

The reading consists of:

Main Scale

20 m

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Subdivision

3 m

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Diagonal Reading

0.7 m

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Total:

23.7 m

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Why Diagonal Scales Are More Accurate

A plane scale can measure:

• Units
• Subunits

A diagonal scale can measure:

• Units
• Subunits
• Smaller fractions of subunits

This increases accuracy significantly.

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Applications of Diagonal Scale

Civil Engineering

Used for:

• Building plans
• Site layouts
• Highway designs

Surveying

Used for:

• Land measurement maps
• Topographic maps

Mechanical Engineering

Used for:

• Machine layouts
• Precision engineering drawings

Architecture

Used for:

• Floor plans
• Elevations
• Sections

Cartography

Used for:

• Geographic maps
• Route planning

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Advantages of Diagonal Scale

High Accuracy

Measures very small fractions.

Easy Reading

Provides direct measurements.

Better Precision

More accurate than plain and plane scales.

Widely Used

Suitable for engineering and surveying.

No Complex Instruments Needed

Only drawing instruments are required.

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Limitations of Diagonal Scale

More Complex Construction

Requires careful drafting.

Time Consuming

More steps than plain scales.

Larger Space Required

Occupies more drawing area.

Reading Errors Possible

If diagonals are drawn inaccurately.

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Plain Scale vs Plane Scale vs Diagonal Scale

Feature	Plain Scale	Plane Scale	Diagonal Scale
Units Measured	One unit and fraction	Unit and subunit	Unit and two subdivisions
Accuracy	Low	Medium	High
Construction	Simple	Moderate	Complex
Precision	Limited	Better	Excellent
Engineering Use	Basic	Intermediate	Advanced

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FAQ Section

1. What is a Diagonal Scale?

A diagonal scale is a scale used to measure three successive units accurately using the principle of similar triangles.

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2. Why is it called a Diagonal Scale?

Because diagonal lines are used to obtain smaller subdivisions and improve measurement accuracy.

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3. What principle is used?

The principle of similar triangles.

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4. Where is it used?

• Engineering drawing
• Surveying
• Mapping
• Civil engineering
• Mechanical engineering

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Importance in Engineering Drawing

Diagonal scales are important because they:

• Increase dimensional accuracy
• Allow precise measurements
• Reduce scaling errors
• Improve engineering drawing quality
• Help represent large dimensions accurately

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Conclusion

A Diagonal Scale is a highly accurate engineering drawing scale used to measure a unit and two successive subdivisions. It is constructed using the Representative Fraction (R.F.) and operates on the principle of similar triangles. Compared with plain and plane scales, diagonal scales offer superior precision and are widely used in engineering graphics, surveying, mapping, civil engineering, and mechanical design. Because of their ability to read very small measurements accurately, diagonal scales are considered one of the most important scales in engineering drawing.

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