For a single-leg angle-beam reading on a flat plate, depth d = S·cos(θ) and surface distance SD = S·sin(θ), where S is the sound path read off the flaw detector screen and θ is the refracted angle of the wedge [S1][S5].
Those two right-triangle relations are the entire geometric core of an angle-beam inspection; every practical complication (skips, multi-leg paths, curved test pieces) is a modification layered on top of them, with half-skip = T·tan(θ) and full-skip = 2·T·tan(θ) for a plate of thickness T [S1][S3].
Flat-Plate Geometry: The Right-Triangle Core
A shear-wave transducer fired at refracted angle θ into a flat plate forms a right triangle between the probe index point, the flaw, and the scanning surface, with sound path S as the hypotenuse [S1][S5]. Depth (vertical leg) is S·cos(θ) and surface distance from the index point to the point above the flaw (horizontal leg) is S·sin(θ), the two values the inspector needs to mark up a weld map [S1][S4]. Standard wedge angles in carbon-steel weld shear-wave work are 45°, 60°, and 70°, with 45° and 60° dominating manual weld scanning because the weld crown prevents placing a probe directly over the root [S1].
Worked example from the sources: a 35 mm sound-path reading with a 45° wedge on a flat 20 mm plate corresponds to a reflector at depth 15.2 mm and 24.8 mm from the index point, the same geometry that the calculator replicates [S1][S2]. Calibration should always be done on an IIW V1 or V2 block, and the verified angle used rather than the nominal label, because a wedge nominally marked 70° can read 68.5° once the acrylic wears [S1]. Material sound velocity for shear waves in low-alloy steel is typically entered as 3250 m/s, and digital instruments use that velocity plus probe zero offset to convert pulse transit time into the sound path the inspector sees on screen [S5].
Skip Distance, Half-Skip, and Multi-Leg V-Paths
Once the beam reaches the back wall, it reflects upward and forms a V-path; the surface distance to that first back-wall strike is the half-skip, equal to T·tan(θ), and the distance where the beam returns to the top surface is the full-skip, equal to 2·T·tan(θ) [S1][S3]. For a 20 mm plate at 45° the half-skip is 20 mm, the full-skip 40 mm, and the first-leg sound path 20/cos(45°) ≈ 28.3 mm, a standard textbook value used to size scan plans [S3].
Leg bookkeeping matters because the math for depth inverts after the back-wall bounce: on the second leg the beam travels upward, so the depth-from-near-surface value drops as sound path continues to rise, and the calculator must reverse the sign of the S·cos(θ) term to avoid misreading a near-surface defect on leg 2 as a flaw deeper than the plate [S1]. Codes such as AWS D1.1, ASME V, and BS EN ISO 17640 specify scanning distances and probe positions based on these skip calculations, which is why the inspector must enter material thickness T as well as the angle θ before trusting any readout [S3]. For a deeper overview of UT angle-beam practice, the ultrasonic flaw detector reference covers probe types, calibration blocks, and DAC setup that the leg formulas assume.
Curved Test Pieces: Why the Flat-Plate Formula Breaks
On a curved surface, the same sound path no longer drops the same depth, because the inner surface lies closer to the beam than the flat-plate model assumes; a 35 mm sound path at 45° in a 100 mm OD pipe puts the reflector at 19.2 mm depth and 31.7 mm surface distance, compared to 15.2 mm and 24.8 mm on a flat plate of equal thickness [S2]. The fix is a curved surface correction (CSC) that uses the projection angle at the ID, Φ = sin⁻¹(R2/R1 · sinΒ), where R1 is the inside radius, R2 the outside radius, and Β the angle of refraction, then solves for reflector depth D = R2 − √(A² + R2² − 2R2·A·cosΒ) with A the leg sound-path length [S2].
Modern digital flaw detectors such as the Evident EPOCH series embed this correction as an option, so the operator only has to enter OD and ID instead of working from nomograms; the old analog workflow relied on a correction factor k read off a graph of refracted angle vs the ratio of wall thickness to outside diameter [S2]. The same approach also sets the maximum allowable refracted angle for a given wall, Βmax = sin⁻¹(R1/R2), which is the angle of impingement limit and is the reason a 70° wedge often will not couple into a small-bore pipe [S2]. Comparable geometry appears in thickness and surface-distance metrology; the surface roughness tester and laser distance meter reference pages cover the same principle of converting a measured time-of-flight into a calibrated distance on non-flat parts.
Sources of Error and What the Inspector Must Verify
The single largest error source is wedge angle drift, so the probe zero and refracted angle must be re-verified on an IIW V1/V2 block at the start of every shift, and a nominally 70° wedge that actually reads 68.5° will systematically mis-size every depth and surface distance on the report [S1]. Couplant thickness, probe index-point position, and the way the inspector peaks the gate on the CRT all shift the effective sound path; the digital read-out only matches the formula if those are controlled [S4][S5].
Second, the inspector must confirm which leg the indication is on, because a 2nd-leg reflection from the root can sit at the same sound-path reading as a 1st-leg indication in the cap, and confusing them puts the mark-up on the wrong side of the weld [S1][S3]. Third, on circumferential scans of pipe, the flat-plate formula overestimates depth and underestimates surface distance, and the error grows as the OD-to-thickness ratio shrinks; a 100 mm OD pipe is already off by 4 mm in depth at 35 mm sound path with a 45° wedge, the example Evident uses to show why CSC is not optional for code work [S2].
Where This Method Fits: Limits and Adjacent Tools
This trigonometry is for locating reflectors, not for sizing them; flaw size in angle-beam UT is handled separately by DAC/TCG, dB-drop methods, or tip-diffraction timing, with code-specific acceptance tied to amplitude rather than to the depth and surface-distance numbers computed here [S7]. It also assumes a homogeneous, isotropic weld zone with a single shear-wave velocity, so austenitic welds with coarse grain and large velocity variation need phased-array or TOFD techniques where a single-angle trigonometric fix-up will not hold [S4].
Adjacent instruments cover the rest of the inspection envelope: a proximity probe handles crack monitoring in rotating machinery, an angle grinder is the surface-prep tool the inspector uses before a re-scan, and a sound level meter supports the on-site safety layer of any in-service UT campaign. None replace the S·cos(θ) / S·sin(θ) pair; together they form the coordinate system the inspector plots every indication against before writing the report.
For weld shops moving to phased-array UT, the same depth and surface-distance output is now generated by the instrument from the focal law, but the underlying trig is identical, and the same IIW block still validates the angle and zero. Track AWS D1.1 Clause 6 inspection tables and the next revision of BS EN ISO 17640 for any change to mandated scan distances, since those numbers are derived directly from the formulas above.
This topic is covered further in In-Line vs Offline CT for Battery Cells: Throughput, Defect Coverage, and When Each Wins.