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Engineering Graphics and CAD · Lesson 13 of 35

Tolerances and dimensional variation

Specify allowable variation so parts are manufacturable and interchangeable.

Readiness check

Learning objectives

By the end of this lesson you can:

Define tolerance, limits, and nominal (basic) size.
Express tolerances as limits, symmetric or unilateral plus/minus, and general tolerances.
Relate tighter tolerance to higher manufacturing cost.
Compute the tolerance and the upper and lower limits of a dimension.
Choose a sensible tolerance from a feature's function.

Check your starting point

Five to ten minutes.

If a shaft is meant to be 20 but every real shaft is slightly off, what must a drawing add to be makeable?
For "20 plus 0.1, minus 0," what are the largest and smallest acceptable sizes?
Is a tolerance of 0.01 cheaper or more expensive to achieve than 0.2?

Interpretation.

Q1: A tolerance, the allowed variation. Without it, "20" is unmakeable because nothing is exactly 20.
Q2: Largest 20.1, smallest 20.0. If unsure, Skill 13.4 shows the computation.
Q3: More expensive; tighter tolerances cost more. Skill 13.3 explains why.

You need L11-L12 (placement and selection); now you add variation.

0 or 1 weak itemsContinue with this lesson.

2 weak itemsReview Lesson 11, Lesson 12, then return.

3 or more weak itemsWork through the prerequisite examples before continuing.

The core idea

What it is. A tolerance is the total amount a dimension is allowed to vary. Every real dimension differs from its nominal value; the tolerance states how much difference is acceptable.

Why an engineer needs it. Nothing is made exactly to size. A dimension without a tolerance is either unmakeable (if read as exact) or uncontrolled (if ignored). Tolerances make a drawing manufacturable and inspectable, and they are how a designer trades cost against precision.

What problem it solves. It converts an ideal nominal size into a permitted range that manufacturing can hit and inspection can check.

What goes wrong when it is ignored. Untoleranced critical dimensions get made to the shop's guess; over-tight tolerances waste money; over-loose tolerances let parts fail to fit. The tolerance, not the nominal, often decides whether parts work and what they cost.

A simple mechanical example. A shaft nominally 20 that must spin in a bearing might be toleranced 20 minus 0.02, minus 0.04 (always slightly under 20) so it fits. A non-critical outer edge of the same part might carry only a general tolerance of plus or minus 0.5. The two tolerances reflect two very different functions and costs.

Ways to express tolerance:

Limit dimensions: state the two limits directly (for example 20.1 and 20.0).
Plus/minus: a nominal with deviations, symmetric (20 plus or minus 0.1) or unilateral (20 plus 0.1, minus 0).
General tolerance: a note covering all unspecified dimensions (for example to ISO 2768, identified but not detailed here), so only critical dimensions need individual tolerances.

Key relationships:

Tolerance equals upper limit minus lower limit.
Tighter tolerance costs more, because it needs better processes, tooling, and inspection.
Tolerance follows function: tight where it matters, loose where it does not.

Part 3: Dimensioning and technical definition.

Check: explain the decision in your own words before using a CAD command.

The concept. Tolerance zones show allowable variation directly. Their relative positions determine whether mating parts always clear, may transition, or interfere.

The skills, taught in order

Skill 13.1 - Define the terms precisely

Concept. Nominal (basic) size, limits, and tolerance are distinct. Terminology. Nominal/basic size is the intended size; upper and lower limits are the extremes; tolerance is their difference; deviation is the signed difference from nominal. Procedure. For any toleranced dimension, identify the nominal, the two limits, and the tolerance. Reasoning. Clear terms prevent confusing a deviation with a limit or a tolerance. Failure mode. Calling the deviation the tolerance, or the limit the nominal. Check. For 20 plus 0.1, minus 0, state nominal, limits, deviation, and tolerance.

Skill 13.2 - Express tolerances correctly

Concept. The same tolerance can be written as limits, symmetric plus/minus, or unilateral plus/minus. Terminology. Limit form, symmetric, unilateral, general tolerance. Procedure. Choose the form that communicates intent: unilateral when variation is allowed only one way (a hole that must not go undersize), symmetric when either way is fine, limits when the two extremes are clearest. Reasoning. The form signals design intent and eases inspection. Failure mode. Using symmetric tolerance where only one direction is acceptable. Check. Rewrite 20.0 to 20.1 as a unilateral plus/minus.

Skill 13.3 - Relate tolerance to cost

Concept. Tighter tolerances need more capable, slower, more inspected processes, so they cost more. Terminology. Process capability is a process's natural ability to hold a tolerance. Procedure. Assign the loosest tolerance that still meets function; reserve tight tolerances for features that need them. Reasoning. Cost rises sharply as tolerance tightens; needless precision wastes money. Failure mode. Applying a tight tolerance everywhere out of caution. Check. Given two features (a bearing seat and a clearance edge), state which deserves the tighter tolerance.

Skill 13.4 - Compute limits and tolerance

Concept. Limits and tolerance follow by simple arithmetic from a nominal and deviations. Terminology. Upper limit equals nominal plus upper deviation; lower limit equals nominal plus lower deviation; tolerance equals upper limit minus lower limit. Procedure. Add each deviation to the nominal to get the limits; subtract limits to get the tolerance. Reasoning. Reliable arithmetic prevents limit errors that scrap parts. Failure mode. Sign errors on deviations. Check. Compute limits and tolerance for 50 plus 0.05, minus 0.10.

Worked example 1: limits and tolerance of a shaft

Problem. A shaft is specified 20 plus 0.1, minus 0. State the nominal, upper and lower limits, and the tolerance, and verify by arithmetic.

Planning. Add each deviation to the nominal; subtract the limits for the tolerance.

Solution.

Nominal size. 20.
Upper limit. 20 plus 0.1 equals 20.1.
Lower limit. 20 plus 0 equals 20.0.
Tolerance. 20.1 minus 20.0 equals 0.1.
Interpretation. Any shaft between 20.0 and 20.1 is acceptable; this is a unilateral tolerance (variation allowed only above nominal).

Result. Nominal 20, limits 20.1 and 20.0, tolerance 0.1, unilateral upward.

Why the method works. Adding signed deviations to the nominal gives the limits directly; their difference is the tolerance by definition.

How to verify independently. The tolerance equals the sum of the absolute deviations: 0.1 plus 0 equals 0.1, matching the limit subtraction. Both routes agree.

Worked example 2: comparing two tolerance choices

Problem. A feature could be toleranced 20 plus or minus 0.05 or 20 plus 0.2, minus 0. Compare the tolerance bands and decide which suits a bearing seat and which suits a non-critical clearance edge. The complication is matching tolerance to function and cost.

Planning. Compute each band, then reason about function and cost.

Solution.

First option, 20 plus or minus 0.05. Limits 20.05 and 19.95; tolerance 0.10 (a tight, symmetric band).
Second option, 20 plus 0.2, minus 0. Limits 20.2 and 20.0; tolerance 0.20 (a looser, unilateral band).
Bearing seat. A seat that sets a fit needs a tight, well-centred size, so the plus or minus 0.05 band (tolerance 0.10) is appropriate despite higher cost, because function demands it.
Clearance edge. A non-critical edge only needs to be roughly right, so the plus 0.2, minus 0 band (tolerance 0.20) is appropriate and cheaper; spending on tightness here would waste money.
Cost note. The 0.10 band needs a more capable process and more inspection than the 0.20 band; assign it only where function requires.

Comparison. The tight band costs more but is necessary for the bearing seat; the loose band is cheaper and sufficient for the clearance edge. Matching tolerance to function optimizes both fit and cost.

Result. Use plus or minus 0.05 (band 0.10) for the bearing seat and plus 0.2, minus 0 (band 0.20) for the clearance edge; tolerance follows function and cost.

Independent check. Subtract each pair of limits: 20.05 minus 19.95 equals 0.10; 20.2 minus 20.0 equals 0.20. The bands are as stated, confirming the comparison.

Misconceptions and diagnostics

Misconception	Why it seems reasonable	Why it is wrong	Evidence that reveals it	Correction	Diagnostic question
"A dimension is a single exact value."	The nominal looks exact.	Every real dimension varies; the drawing must state a permitted range.	Two acceptable parts measure slightly differently.	Add a tolerance (or rely on the general note).	"What range is acceptable here?"
"Tighter tolerance is always better."	Precision sounds good.	Tight tolerances cost more; needless tightness wastes money.	A loose feature was toleranced tightly, raising cost.	Use the loosest tolerance that meets function.	"Does the function actually need this tightness?"
"The tolerance is the bigger deviation."	The number looks like the tolerance.	Tolerance is the difference between the two limits, not one deviation.	For plus 0.1, minus 0.05, the tolerance is 0.15, not 0.1.	Compute upper limit minus lower limit.	"What is upper limit minus lower limit?"

Practice ladder

Level A - Recognition

Task. For eight toleranced dimensions, state nominal, limits, and tolerance. Deliverable. An eight-row table. Success criteria. At least seven correct, including a unilateral and a non-symmetric case. Answer guidance. Add deviations to the nominal; subtract limits. Common errors. Sign errors on the lower deviation. Difficulty. Low.

Level B - Guided application

Task. Fill a limits table for several features and convert each between plus/minus and limit form, with prompts. Deliverable. A completed conversion table. Success criteria. Correct limits and equivalent forms throughout. Answer guidance. Limit form and plus/minus describe the same band. Common errors. Mismatched conversions. Difficulty. Medium.

Level C - Independent application

Task. Assign tolerances to a part from a short function brief, justifying tight versus loose choices. Deliverable. A toleranced drawing with a brief rationale per critical feature. Success criteria. Tolerances match function; general note covers the rest; arithmetic correct. Answer guidance. Tight only where function demands; loose elsewhere. Common errors. Uniform tolerances ignoring function. Difficulty. Medium.

Level D - Transfer and design

Task. For a given assembly, justify a tolerance set against both cost and function, identifying the one or two features that truly need tight control. Deliverable. A toleranced scheme plus a cost/function justification. Success criteria. Correct identification of critical features; defensible cost trade-off; correct limits. Answer guidance. Follow the function to the few features that set fits. Common errors. Over-tightening non-critical features. Difficulty. High.

Working with AI, and proving it yourself

Use AI as a tutor

Useful AI support:

Ask it to check your limit arithmetic, then verify by hand.
Ask it to explain unilateral versus symmetric with your feature.
Ask it to suggest which features are likely critical, then confirm from function.

Limits:

A text assistant does not know your assembly, so it cannot set your tolerances.
It may state a tolerance value without justifying it from function or cost.

Verify AI output against: hand arithmetic (limits and tolerance), the function-sets-tolerance principle, and the cost-rises-with-tightness trend.

Prove it yourself

A plausible but incorrect AI answer, and how to catch it. You ask, "To be safe, should I put plus or minus 0.01 on every dimension?" and the assistant replies: "Yes, a tight plus or minus 0.01 on everything guarantees quality."

This is a costly error. Detect it with the cost-versus-tolerance principle: tightening every dimension multiplies process and inspection cost with no functional benefit on non-critical features. The evidence is economic and practical: many features do not need 0.01, and holding it everywhere may be beyond the chosen process. Correct conclusion: tolerance follows function; tight only where needed, loose (or general note) elsewhere.

Retrieval and spaced review

Define tolerance in terms of limits.
What is a unilateral tolerance, and when is it used?
Why does tighter tolerance cost more?
Compute the limits and tolerance for 30 plus 0.2, minus 0.1.
What is a general tolerance note for?
How should tolerance relate to function?
Cumulative (L12): How do tolerance choices interact with chain versus baseline dimensioning?
Reconstruction task: From memory, state the limits and tolerance for the shaft 20 plus 0.1, minus 0.

Answers. 1: tolerance equals upper limit minus lower limit. 2: variation allowed in only one direction; used when a feature must not go over (or under) nominal, like a hole that must not be undersize. 3: it needs more capable processes, tooling, and inspection. 4: limits 30.2 and 29.9, tolerance 0.3. 5: it covers all unspecified dimensions so only critical ones need individual tolerances. 6: tight where function requires, loose elsewhere. 7: chained tolerances accumulate, so the scheme and the tolerance size together set the worst-case stack.

Suggested review intervals. 1 day, 3 days, 7 days.

Reference mapping and next step

Read further

Giesecke ch.11
ISO 129-1
general-tolerance note ISO 2768 (identify only).

Standards details must be checked against the current official edition used by your institution or employer.

Finish the lesson

You can now: define nominal, limits, deviation, and tolerance; express tolerances three ways; relate tightness to cost; compute limits and tolerance; and choose tolerance from function.

Self-assessment checklist.

I can compute limits and tolerance from a plus/minus spec.
I can convert between limit and plus/minus forms.
I assign tight tolerances only where function needs them.
I use a general note for the rest.
I never confuse a deviation with the tolerance.

Next lesson: L14 - Limits and fits: holes and shafts (ISO 286). Why it follows: you can now tolerance a single dimension; next you learn to tolerance two mating dimensions together so a shaft and hole go together as intended (clearance, transition, or interference), using the ISO 286 code system.

Required files or submissions: submit your Level C toleranced part with rationale. Optional extension: take a part and re-tolerance it, loosening every non-critical feature to a general note and justifying the two you kept tight.