7.2 Perpendicularity & Angularity Tolerances with Basic Angles & Tangent Plane

ASME GDTP Technologist Y14.5-2009

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1 Chapter 1: Scope, Certification Logistics, Fundamental Rules & General Dimensioning

1.1 ASME GDTP Technologist Exam Scope, Prometric Logistics, & Certification Rules 1.2 Fundamental Rules of ASME Y14.5-2009 (Rules a through p) 1.3 Engineering Drawing Standards, Drawing Units, & Basic vs. Toleranced Dimensions 1.4 Dimensioning Features: Diameters, Radii, Chamfers, Countersinks, Counterbores, Spotfaces, Keyseats & Slots 1.5 Application of Dimensions & Location of Features: Lines, Leaders, Reference Dimensions, Rectangular vs. Polar Coordinates

2 Chapter 2: Features, Limits of Size, & Material Condition Principles

2.1 Features, Features of Size, & Limits of Size 2.2 Rule #1 (Taylor Principle / Envelope Rule), Its Exceptions & Overrides 2.3 Material Condition Concepts: MMC, LMC, & RFS Calculations

3 Chapter 3: General Tolerancing & Symbology

3.1 Direct Tolerancing, Limit Dimensioning, & Angular Tolerancing 3.2 The 14 Geometric Characteristic Symbols, Feature Control Frames, & Special Modifiers 3.3 Screw Threads, Gears & Splines, Angular Surfaces, Tapers & Statistical Tolerancing 3.4 Feature Control Frame Placement, Combined Frames, Tolerance Zone Definition & Tabulated Tolerances

4 Chapter 4: Datum Reference Frames & Degrees of Freedom

4.1 Theoretical Datums, Datum Feature Simulators, & Physical Datum Features 4.2 Degrees of Freedom Constraint & The 3-2-1 Datum Precedence System 4.3 Planar, Cylindrical, & Width Datum Feature Establishment

5 Chapter 5: Advanced Datum Applications: Targets, Boundary Modifiers, Customized Frames & Restrained Condition

5.1 Datum Target Points, Lines, Areas, & Movable Datum Simulators 5.2 Material Boundary Modifiers (RMB, MMB, LMB) & Datum Shift Calculations 5.3 Functional Datum Feature Selection, Multiple Datum Reference Frames & Simultaneous vs. Separate Requirements 5.4 Rotational Constraint About a Datum Axis or Point & Customized Datum Reference Frames 5.5 Irregular & Complex Datum Features, Mathematically Defined Surfaces, Restrained Condition & Datum Feature Controls

6 Chapter 6: Form Tolerances

6.1 Flatness & Straightness Controls for Surfaces & Features of Size 6.2 Circularity & Cylindricity Controls, Differences, & Inspection

7 Chapter 7: Orientation Tolerances

7.1 Parallelism Controls for Planar Surfaces & Cylindrical Features of Size 7.2 Perpendicularity & Angularity Tolerances with Basic Angles & Tangent Plane

8 Chapter 8: Location: Position Fundamentals & Calculations

8.1 Position Tolerancing Fundamentals, True Position, & Coordinate Conversions 8.2 Bonus Tolerance & Virtual Condition Calculations for Internal & External Features 8.3 Projected Tolerance Zones (Ⓟ Modifier) & Fixed vs. Floating Fastener Formulas 8.4 Zero Positional and Orientation Tolerancing at MMC and at LMC 8.5 Surface vs. Axis Interpretation of Position & the Boundary Concept for Slots, Tabs & Noncircular Features

9 Chapter 9: Location: Patterns, Composite Controls, Coaxial & Symmetrical Relationships, and Functional Gaging

9.1 Hole & Slot Patterns, Multi-Segment Frames, & Composite Position Tolerancing 9.2 Functional Gaging Principles & Inspection Verification of Position 9.3 Coaxial Holes, Counterbored Holes, Non-Parallel Axes, Spherical Features & Bidirectional Position 9.4 Concentricity & Symmetry Tolerances Under ASME Y14.5-2009 9.5 LMC Applied to Position: Wall Thickness, Breakout Protection & Boundary Calculations

10 Chapter 10: Profile Tolerances

10.1 Profile of a Line & Profile of a Surface Fundamentals & Basic Geometry 10.2 Bilateral, Unilateral, & Unequally Disposed Profile Boundaries (Ⓤ Modifier) 10.3 Composite Profile Tolerances vs. Multiple Single-Segment Profile Controls 10.4 Profile Verification, Coplanarity Controls, & Non-Rigid Parts 10.5 Profile Applications: Conicity, Extruded and Irregular Shapes, and Profile-Controlled Datum Features 10.6 Modifier Rules for Profile & Combined Controls (Profile with Position, Orientation and Runout)

11 Chapter 11: Runout Tolerances & Comparative Control Analysis

11.1 Circular Runout vs. Total Runout Interpretation & Dial Indicator Verification 11.2 Comparative Analysis: Runout vs. Form, Orientation, & Concentricity Controls

7.2 Perpendicularity & Angularity Tolerances with Basic Angles & Tangent Plane

Key Takeaways Explain This Topic AI

Perpendicularity (ASME Y14.5-2009 Section 6.4.1) controls surfaces, centerplanes, or axes at a 90° basic angle to one or more datums, an angle that is legally implied by orthogonal drawing geometry without requiring an explicit basic dimension.

Angularity (Section 6.4.3) controls orientation at any specified angle other than 0° or 90° and strictly requires an explicit basic angle dimension (or basic linear dimensions); angularity serves as the universal mathematical superset of all orientation controls.

Multiple datum references in orientation controls constrain independent rotational degrees of freedom (e.g. primary datum locks two tilt axes, secondary datum locks the remaining rotation), yet orientation controls NEVER control location.

The Tangent Plane modifier (Ⓣ) per Section 6.5 directs verification to a simulated true plane contacting the highest peaks of a planar surface, allowing functional mating faces with local valleys, dents, or waviness to pass inspection without failing orientation.

Last updated: September 2026

7.2 Perpendicularity & Angularity Tolerances with Basic Angles & Tangent Plane

Quick Answer: Under ASME Y14.5-2009 , perpendicularity (Section 6.4.1) and angularity (Section 6.4.3) control the orientation of surfaces, centerplanes, or axes relative to one or more datums. Perpendicularity controls features oriented at a 90° basic angle to a datum, which is legally implied by orthogonal drawing views without requiring an explicit basic dimension. In contrast, angularity controls features oriented at any angle other than 0° or 90° and strictly requires an explicit basic angle dimension on the drawing. Both controls support planar and cylindrical tolerance zones, permit material condition modifiers (Ⓜ/Ⓛ) on features of size, and can reference multiple datums to constrain rotational degrees of freedom. When controlling planar mating faces, the Tangent Plane modifier (Ⓣ) (Section 6.5) directs inspection to evaluate a simulated plane contacting the true high points (peaks) of the surface, preventing localized depressions, pits, or waviness from falsely rejecting functional assemblies.

Perpendicularity Controls (ASME Y14.5-2009 Section 6.4.1)

Perpendicularity is the condition of a surface, centerplane, or axis at a 90° basic angle to one or more datum planes or datum axes. Perpendicularity is the most widely utilized orientation control in precision manufacturing.

The Implied 90° Basic Angle Rule

A cornerstone rule of engineering drawing interpretation under ASME Y14.5-2009 Section 2.1.1.2 is the implied 90° angle :

Where centerlines and lines depicting features are shown on a drawing at right angles ( 90 ∘ 90^\circ 9 0 ∘ ), a 90° basic angle is legally implied .

When specifying a perpendicularity control, an explicit boxed [90°] dimension is not required . Drafters may include [90°] for emphasis or clarity, but its absence does not invalidate the drawing.

Planar Surface Perpendicularity

When applied to a planar surface:

Tolerance Zone : Bounded by two parallel planes separated by tolerance t t t , oriented exactly 90° perpendicular to the referenced datum plane or axis.

Controlled Entity : All points of the physical surface must lie between these two parallel planes.

Form Refinement : Perpendicularity automatically refines the surface's flatness .

Modifiers : Material condition modifiers (Ⓜ or Ⓛ) are strictly prohibited on planar surface perpendicularity callouts.

Perpendicularity Applied to Features of Size (Pins and Holes)

When applied to a cylindrical feature of size (such as a dowel pin or reamed hole):

Controlled Entity : The derived median line (DML) (the axis) of the feature.

Tolerance Zone : A cylindrical zone of diameter ⌀ t t t whose axis is oriented exactly 90° perpendicular to the datum plane.

Mandatory ⌀ Symbol : The tolerance value in the feature control frame must be preceded by the diameter symbol (⌀) .

Material Condition Modifiers : Perpendicularity of a feature of size permits the Ⓜ modifier , unlocking bonus tolerance as the feature departs from MMC toward LMC.

Multiple Datum References in Orientation Controls

While form controls prohibit datums and single-datum orientation controls are common, orientation controls can reference two or even three datums in a Datum Reference Frame (DRF) (e.g., [ Perpendicularity | Ø 0.1 | A | B ] ):

Why Multiple Datums Are Used : A primary planar datum A constrains two rotational degrees of freedom (pitch and roll). However, an axis can still tilt or rotate about the remaining axis (yaw). Referencing a secondary datum plane B constrains the third rotational degree of freedom , perfectly locking the orientation of the cylindrical tolerance zone in 3D space.

The Location Invariance Principle : Even when referencing primary, secondary, and tertiary datums, orientation controls NEVER control location . Referencing datum B prevents the tolerance cylinder from tilting relative to B, but it does NOT constrain the distance or coordinate position of the hole from datum B! Position (location) requires a position tolerance ( ⌖ ). ORIENTATION DEGREES OF FREEDOM CONSTRAINT

┌─────────────────────────────────────────────────────────────┐
│ PRIMARY DATUM A (Plane): │
│ • Constrains 2 Rotational DOF (Pitch & Roll) │
│ • Axis must be perpendicular to plane A │
├─────────────────────────────────────────────────────────────┤
│ SECONDARY DATUM B (Plane / Axis): │
│ • Constrains 1 Remaining Rotational DOF (Yaw) │
│ • Axis is prevented from tilting relative to plane B │
├─────────────────────────────────────────────────────────────┤
│ CRITICAL GD&T RULE: │
│ • Orientation controls with A and B lock ROTATION ONLY. │
│ • They NEVER control distance or location from A or B! │
└─────────────────────────────────────────────────────────────┘
Angularity Controls (ASME Y14.5-2009 Section 6.4.3)

Angularity is the condition of a surface, centerplane, or axis at any specified basic angle other than 0° or 90° from one or more datum planes or axes.

The Absolute Basic Angle Callout Requirement

The defining technical requirement of angularity is the mandatory presence of an explicit basic angle :

Unlike perpendicularity (where 90° is legally implied) and parallelism (where 0° is legally implied), there is no default or implied angle for inclined features.

The drawing MUST specify a basic angle (e.g., a boxed dimension such as [30°] , [45°] , or [60°] ), or basic linear coordinate dimensions defining the theoretical angle.

The Toleranced Angle Violation : Specifying an angularity feature control frame alongside a conventional toleranced angle (e.g., 45 ∘ ± 1 ∘ 45^\circ \pm 1^\circ 4 5 ∘ ± 1 ∘ ) is a severe drafting syntax violation . In GD&T, an orientation tolerance replaces the toleranced angle; combining both creates conflicting, unresolvable tolerance zones.

Angularity as the Universal Orientation Control

In mathematical and geometric theory, angularity is the parent superset of all orientation controls:

Parallelism is simply angularity where the basic angle equals 0° .

Perpendicularity is simply angularity where the basic angle equals 90° .

While ISO GPS standards allow the angularity symbol to be used at 0° and 90°, ASME Y14.5-2009 maintains distinct symbols for parallelism, perpendicularity, and angularity to maintain clarity on engineering drawings.

Planar vs. Cylindrical Angularity Tolerance Zones

Planar Surfaces : Two parallel planes separated by tolerance t t t , oriented at the specified basic angle to the datum plane.

Cylindrical Features (Inclined Holes/Pins) : A cylindrical tolerance zone of diameter ⌀ t t t , oriented at the basic angle to the datum plane. Requires the diameter symbol (⌀).

The Tangent Plane Modifier Ⓣ (ASME Y14.5-2009 Section 6.5)

In conventional orientation inspection, every single point on the physical surface must lie between the two parallel planes of the tolerance zone. However, in many engineering applications, this requirement is unnecessarily restrictive and counterproductive.

The Engineering Rationale for Tangent Plane

Consider a machined mating flange or a mounting bracket bolted to a machine base:

When bolted together, the mating contact is established solely across the highest contacting points (peaks) of the surface.

Localized depressions, cast porosity, tool chatter valleys, or slight dishing between fasteners do not cause the mating component to cock or tilt.

Under standard orientation, a single deep valley or localized scratch would cause an otherwise functionally perfect mounting face to fail inspection.

Symbol & Syntactical Placement

The Tangent Plane modifier is represented by a capital letter T enclosed within a circle: Ⓣ .

It is placed directly after the tolerance value in the feature control frame: [ ⟂ | 0.20 Ⓣ | A ] or [ ∠ | 0.15 Ⓣ | A ]

In standard drafting practice, the circle diameter enclosing the letter T matches the height of the tolerance compartment (typically 2h), ensuring consistency across engineering drawings according to ASME Y14.5-2009 Section 3.4 and Fig. 3-1.

Geometric Interpretation & Verification

When the Ⓣ modifier is specified:

A simulated true plane contacting the highest points (peaks) of the actual physical surface is established.

It is this tangent plane —and NOT the individual surface points—that must lie within the specified orientation tolerance zone.

Valleys, gouges, dents, and localized depressions on the actual surface are permitted to extend beyond the tolerance zone boundaries without causing rejection!

Application Limitation : The Ⓣ modifier applies strictly to planar surfaces . It cannot be applied to axes, centerplanes, cylinders, or features of size.

Form Independence : The tangent plane control does not control surface flatness. If surface sealing or flatness is required, an independent flatness control must be specified on the drawing. STANDARD ORIENTATION vs. TANGENT PLANE (Ⓣ)

STANDARD ORIENTATION: TANGENT PLANE MODIFIER (Ⓣ):

[ Perpendicularity | 0.2 | A ] [ Perpendicularity | 0.2 Ⓣ | A ]

┌───────────────────────────┐ ┌───────────────────────────┐
│ ░░░░░░░░░░░░░░░░░░░░░░░░░ │ │===== SIMULATED PLANE =====│◄── Must lie

│ │ │ (Contacts High Peaks) │ in zone!

│ ~~~~ ACTUAL SURFACE ~~~ │ │ │
│ \ (Valley fails) │ │ ~~~~ ACTUAL SURFACE ~~~ │
│ \____/ │ │ \ (Valley IGNORED!) │
└───────────────────────────┘ └───────────────────────────┘
ALL surface points must lie ONLY the contacting tangent

inside the 0.2 mm boundary. plane must lie in the zone.

Master Orientation Comparison Matrix

The following matrix provides a comprehensive comparison of all three ASME Y14.5-2009 orientation controls:

Attribute / Parameter Parallelism (§6.4.2) Perpendicularity (§6.4.1) Angularity (§6.4.3)
Geometric Symbol ∥ ⊥ ∠

Basic Angle to Datum 0 ∘ 0^\circ 0 ∘ (Basic) 90 ∘ 90^\circ 9 0 ∘ (Basic) Any angle other than 0 ∘ 0^\circ 0 ∘ or 90 ∘ 90^\circ 9 0 ∘

Drawing Angle Callout Implied ( 0 ∘ 0^\circ 0 ∘ ) Implied ( 90 ∘ 90^\circ 9 0 ∘ ) Mandatory Basic Dimension (Boxed)

Tolerance Zone Shapes Two parallel planes or Cylinder (⌀) Two parallel planes or Cylinder (⌀) Two parallel planes or Cylinder (⌀)

Datum References Required Minimum 1 (Can reference DRF) Minimum 1 (Can reference DRF) Minimum 1 (Can reference DRF)

Material Condition on FOS (Ⓜ/Ⓛ) Permitted (Axis/Centerplane) Permitted (Axis/Centerplane) Permitted (Axis/Centerplane)

Tangent Plane (Ⓣ) on Surfaces Permitted (Planar surfaces) Permitted (Planar surfaces) Permitted (Planar surfaces)

Overrides Rule #1 on FOS? No (Envelope principle holds) No (Envelope principle holds) No (Envelope principle holds)

Automatically Refines Form? Yes (Limits flatness/straightness) Yes (Limits flatness/straightness) Yes (Limits flatness/straightness)

Controls Feature Location? NEVER NEVER NEVER

Universal Parent Control? Special case of Angularity ( 0 ∘ 0^\circ 0 ∘ ) Special case of Angularity ( 90 ∘ 90^\circ 9 0 ∘ ) Yes (Universal parent control)

Common Exam Traps: Perpendicularity, Angularity, & Tangent Plane

Trap 1: Missing the Explicit Basic Angle for Angularity : Exam questions frequently present an inclined surface with an angularity feature control frame alongside a toleranced angle (such as 30 ∘ ± 0.5 ∘ 30^\circ \pm 0.5^\circ 3 0 ∘ ± 0. 5 ∘ ). This is a major drafting error: Angularity requires an explicit basic dimension (boxed angle).

Trap 2: Believing Multiple Datums Convert Orientation into Location : When a perpendicularity frame references primary and secondary datums (e.g., [ Perpendicularity | Ø 0.1 | A | B ] ), candidates often mistakenly assume the control locates the hole from datum B. Secondary datums in orientation frames only constrain rotational degrees of freedom ; they never control location or center-to-edge distance.

Trap 3: Applying Tangent Plane (Ⓣ) to a Feature of Size or Axis : The tangent plane modifier can only be specified for planar surfaces. Placing Ⓣ on a cylindrical hole or pin axis callout is an invalid syntax error.

Trap 4: Assuming Tangent Plane Verifies Surface Flatness : A surface can have massive waviness or deep hollows and still easily satisfy a tangent plane orientation tolerance, because Ⓣ evaluates only the contacting high points. A separate flatness control is required to control surface form.

Trap 5: Forgetting the ⌀ Symbol on Hole / Pin Perpendicularity : When perpendicularity is applied to a cylinder axis, the tolerance zone is a 3D cylinder. The tolerance value must be preceded by ⌀.

Trap 6: Assuming an Explicit [90°] Dimension is Required for Perpendicularity : Per ASME Y14.5-2009 Section 2.1.1.2, a 90° basic angle is legally implied where lines are drawn perpendicular. A missing [90°] box does not invalidate a perpendicularity control.