There are two luer adapters: One is called luer slip, which is just a cone.
The other one is called luer lock, which is a luer slip cone with an internally threaded securing cylinder or cap around it.
Luer lock and luer slip are compatible, so luer lock will fit on luer slip and vice versa.

The luer lock adapter has a 6% cone or a 3.44° opening angle.
So from the mid axis of the cone the opening angle is 1.72° and the slope is 3% (or 0.03 mm/mm).

figure 1: different luer slip and luer lock adapters. pointing right - male side. pointing left - female side
above: tygon tube with male fixed luer lock adapter, filter with left - female luer lock and right - male luer slip
middle: 2 mL syringe with male luer slip, blue cap with left - female luer lock and right - male luer lock
below: 10 mL syringe with male luer lock, stop cock with female luer lock (short 270° thread) and male luer lock with rotating luer lock cap, tygon tube with female luer lock

male piece - luer slip

The luer cone is max. 7.5 mm long, it is typically 4mm at the tip it is 4.45 mm at the end
half angle : 0.225mm / 7.5 mm = 0.03 or a 3% slope    arctan 0.03 =  1.72°.   full angle = 3.44° or 6%

The ID of the luer cone does not seem to be specified (this is why there is no number in the drawing below). The opening is typically about 2 mm and as far as we can see the channel is typically not conical inside. Yet the radius of the rounding at the tip of the cone is specified to be max 0.5 mm

male piece - luer lock

figure 2: male luer lock dimensions

Concentric with the luer cone, there is a cylindrical collar with an internal thread (lower left on picture) The cylinder might also be a loose cap like on the stopcock above right side, so you can fix the luer lock without turning the male piece itself. The cone is usually securely fixed on the female piece with one full turn (5mm of total length of 7.5mm)

The ID of the cylinder body is 8mm and the opening between the threads is 7 mm (so threads are 0.5 mm high). The threads have a 30° shoulder and they are > 0.3mm at the top. The threads have a 2.5mm pitch and they start on both sides (180° offset) so one full turn moves 5 mm.

The luer taper sticks out of the luer lock thread by typically 2 mm

The OD of the locking cylinder is not defined. Our "Braun Injekt" luer lock syringes (see lower left on picture) typically have a flat cylindrical outer shape. The cylinder is 7mm long being slightly conical (9.8mm at the tip and 9.9mm at the syringe body). The loose luer lock cap with an internal thread of the stopcocks above has a structured outside shape with a max OD of 11mm.

female piece - luer slip

the female piece typically has an opening of 4.3mm and it is 9mm deep (so there is enough room for the 7.5mm long cone). With a 6% slope (or 3.44° opening angle) it should be 3.76 mm at its end .
The luer cone typically slides in by 2/3 of its length (5mm).

female piece - luer lock

figure 3: female luer lock dimensions

additionally to the conical 4.3mm opening, there is a 2.5mm pitch right hand thread on the outside of the piece. It starts on both sides, so one full turn moves 5 mm. The width of the thread is > 0.3mm at the top. Needles often have a very short piece of outer thread (only 270° or 3.7mm) that is yet enough to secure it. (like the female side of the stopcock on the picture)

The OD of the thread on the female piece is 7.8 mm, the OD of the body is 6.7mm so the thread is 0.55mm high with 30° shoulders.

REMARK:
This is not an official reference document ! For a proper description of the luer lock standard please refer to ISO 594-2(1998),  ISO 80369 or DIN / EN 1707:1996 or newer standards, which are all based on the original patents of H.A.Wülfing-Lüer. His wife Jeanne Amélie Lüer most likely invented the luer slip connection in the late 19th century (first syringe patent: FR 242646, 1894, US Patent US 583382 from 1897..)  and the rotating cap luer lock standard is based on a publication by A.Ravin from 1952.

A.Ravin: A Luer slip with a rotating lock. J. Lab. Clin. Med. (1952) 39(1): S. 168 PMID 14889106

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