Hip-Knee-Ankle Angle Calculator (Mechanical Axis)
Hip-Knee-Ankle Angle Calculator (Mechanical Axis)
Coronal mechanical alignment from two offsets and two lengths, in both published conventions — with the 4.84 degrees of measurement error that limb rotation alone introduces, and the third of men who are constitutionally varus.
Hip-knee-ankle angle, deviation and 180-degree forms
Two arctangents about the knee centreA weight-bearing full-length AP film: the femoral head centre lies 8 mm medial to the knee centre and 400 mm above it; the ankle centre lies 2 mm medial to the knee centre and 350 mm below it
Two axis inclinations, and two conventions for reporting them
tibial inclination = arctan(ankle offset / tibial vertical length)
deviation (varus positive) = femoral inclination + tibial inclination
hip-knee-ankle angle (180 convention) = 180 minus that deviation
- why the two inclinations ADD
- both are measured from the vertical at the knee centre, one towards the hip and one towards the ankle. In a varus limb the knee centre lies lateral to the line joining hip and ankle, so both the femoral head centre and the ankle centre are medial to it, both offsets are positive on this page’s convention, and the two inclinations add to give the total deviation. A perfectly straight limb gives two zeros and a deviation of zero, which is 180 degrees in the other convention
- the horizontal offsets are on TOP
- because each inclination is measured from the VERTICAL, as on this site’s lateral centre-edge angle page and unlike its acetabular index and Sharp angle pages. Transposing an offset and its length here is unusually obvious — the answer jumps from about 1 degree to about 89 — but the proof asserts the direction of each of the four arguments separately anyway, away from the degenerate point
- two conventions, both in print
- the headline is the DEVIATION, which is how Radiopaedia reports it (“approximately 1-1.5 degrees varus”); the secondary figure is the same alignment as an angle near 180 degrees, which is how it is usually drawn and measured on a workstation. A reader holding one of these numbers needs to see the other, because 178.5 and 1.5 varus are the same limb
- and the sign convention is this page’s own
- because the sources do not agree one. Radiopaedia defines no positive or negative direction and uses the words varus and valgus; Gieroba and colleagues are explicit but use the opposite polarity for their component angles, “valgus tibias are recorded as negative yielding an MPTA of greater than 90 degrees”. Here medial offsets are positive and a positive total is varus, stated on the page so a reader can check which way round the answer is
- atan2 is in this engine and is NOT used here
- atan2(y, x) is the right tool for an angle that can exceed 90 degrees or needs a quadrant. Neither applies: both vertical lengths are strictly positive by construction, each axis inclination is a few degrees, and a plain arctangent of a signed ratio over a positive denominator is unambiguous across the whole anatomical range. Using atan2 here would add a quadrant decision with nothing to decide
- the film, not just the limb
- “The HKA is measured on a weight-bearing, full-length AP lower limb radiograph.” A supine film, a short film or a flexed knee is a different measurement: Sun and colleagues found the rotation-induced error correlated with flexion at 0.933, and that “With the increase of knee flexion, the measurement error of lower limb alignment increased significantly”
Worked example
A weight-bearing full-length AP film: the femoral head centre lies 8 mm medial to the knee centre and 400 mm above it; the ankle centre lies 2 mm medial to the knee centre and 350 mm below it
Femoral inclination = arctan(8 / 400) = arctan(0.0200) = 1.146 degrees
Tibial inclination = arctan(2 / 350) = arctan(0.0057) = 0.327 degrees
Deviation = 1.146 + 0.327 = 1.5 degrees varus, inside Radiopaedia's healthy range of 1 degree valgus to 3.2 degrees varus and close to its "approximately 1-1.5 degrees varus" neutral value
In the other convention: 180 minus 1.473 = 178.5 degrees. Same limb, same measurement, and the number a reader is shown depends entirely on which convention the report used
Reverse one offset: put the ankle centre 12 mm LATERAL to the knee centre, so the input becomes minus 12 mm. The tibial inclination is then minus 1.965 degrees and the deviation is minus 0.8 degrees — valgus, and still inside the healthy range. The sign convention is doing real work here and is stated on the page for that reason
Now put the measurement error beside the answer. Sun and colleagues found limb rotation alone moved the measured hip-knee-ankle angle by up to 4.84 plus or minus 0.66 degrees across 45 limbs, and Gieroba's three observers had a coefficient of repeatability of 3.56 degrees on long-leg radiographs. A deviation of 1.5 degrees varus and one of 4.5 degrees varus are not reliably different measurements on a single film
What a healthy limb measures, in both conventions
| Deviation convention | 180-degree convention | Source | |
|---|---|---|---|
| Neutral adult knee | About 1 to 1.5 degrees varus | About 178.5 to 179 degrees | Radiopaedia |
| Healthy range | 1 degree valgus to 3.2 degrees varus | 176.8 to 181 degrees | Radiopaedia |
| Perfectly straight limb | 0 degrees | 180 degrees | The geometry, not a published norm |
| Constitutional varus of 3 degrees or more | 32% of men, 17% of women | At or below 177 degrees | Bellemans et al, as quoted by Gieroba et al 2023; Bellemans’s cohort size is not stated there |
| Children | Varus from birth to 18-24 months; valgus normal until 7 years | Radiopaedia |
How much of the number is measurement
| Source | Cohort | Figure |
|---|---|---|
| Sun et al, BioMed Eng Online 2021 | 45 lower limbs from 30 patients, CT-derived virtual long-leg radiographs from 20 degrees internal to 20 degrees external rotation | “The measurement errors of the HKA and FTA derived from limb rotation were up to 4.84 plus or minus 0.66 degrees”, average deviation range 4.84 plus or minus 0.66 degrees (range 0.56 to 25.03). Correlation with flexion angle 0.933 |
| Sun et al 2021 | The same 45 limbs | “limb rotation caused the deviation ranges of HKA and FTA measurement to be greater than 3 degrees”. Intraclass correlation for the deviation range 0.976 |
| Gieroba et al, Arthroplasty 2023 | 68 patients awaiting knee arthroplasty, 64 with usable CT, three observers reading three times a week apart | For the arithmetic hip-knee angle on long-leg radiographs: intra-observer Pearson r 0.92 (0.87 to 0.95), inter-observer 0.91, with a coefficient of repeatability of 2.88 degrees within an observer and 3.56 degrees between observers |
| Gieroba et al 2023 | The same 68 | On CT the same measurement repeated to 2.57 degrees within and 2.00 degrees between observers, and radiograph against CT agreed at r 0.85 with a repeatability coefficient of 3.34 degrees. “CT measurements of MPTA, LDFA and aHKA are more reproducible” |
Two axes, two conventions, and an error the size of the range
The hip-knee-ankle angle is the coronal angle between the mechanical axes of the femur and the tibia, measured on a weight-bearing full-length AP radiograph. Radiopaedia draws the first line from the femoral head to the femoral intercondylar notch and the second from the tibial interspinous point to the tibial mid-plafond. Each axis’s deviation from the vertical is an arctangent of a horizontal offset over a vertical length, and because both are measured at the knee centre, the two inclinations add: a straight limb gives zero, and a varus limb — knee centre lateral to the hip-to-ankle line — gives two positive offsets that sum.
Two conventions are in print and this page gives both. The headline is the deviation, which is how normal values are usually quoted: Radiopaedia puts the neutral adult knee at approximately 1 to 1.5 degrees varus and the healthy range at 1 degree valgus to 3.2 degrees varus. The secondary figure is the same alignment as an angle near 180 degrees, which is what a workstation protractor reports. 178.5 degrees and 1.5 degrees varus are the same limb, and a reader holding one needs to see the other. The sign convention here is this page’s own, stated explicitly, because Radiopaedia defines none and Gieroba and colleagues use the opposite polarity for their component angles.
Neutral is not zero, which is the first thing to take from the normal values. A healthy adult mechanical axis sits slightly varus, and constitutional varus is common rather than rare: Gieroba’s paper records, citing Bellemans, that 32% of men and 17% of women carry a constitutional varus of 3 degrees or more. A limb measuring 3.5 degrees varus is outside the healthy range quoted above and inside a configuration that about a third of men have.
An angle is only as good as the two lines it is drawn between. Every threshold on this page assumes the measurement convention stated above, and a different landmark gives a different number from the same hip, heel, knee or spine — not a more or less accurate one. Agreement between observers is part of what a measured angle means, and for several of these angles the published spread between experienced readers is wider than the interval between the thresholds they are read against. Each page here gives the reliability figures with the number of observers and the number of radiographs they came from. This page renders no clinical or surgical decision; it computes the published quantity and states what the literature reports for it, with every threshold and performance figure given with the cohort it came from. A measurement is not a diagnosis, and a figure from a published series is a property of that series rather than a fact about one patient.
Frequently asked questions
What is a normal hip-knee-ankle angle?
Radiopaedia puts the neutral adult knee at approximately 1 to 1.5 degrees varus, with healthy knees ranging between 1 degree valgus and 3.2 degrees varus. In the 180-degree convention that is about 178.5 to 179 degrees for neutral and 176.8 to 181 degrees for the healthy range. Neutral is slightly varus rather than zero.
Is the angle 180 degrees or about 1 degree?
Both, and they are the same measurement. The angle between the two mechanical axes is near 180 degrees; the deviation from straight is near 1 degree. This page prints the deviation as the headline and the 180-degree form beneath it, because a report may use either and the two are easy to confuse.
Which way round is varus?
On this page a positive deviation is varus, which corresponds to the femoral head centre and the ankle centre both lying medial to the knee centre — the knee pointing outwards. The convention is stated because the sources do not agree one: Radiopaedia defines no sign at all and Gieroba and colleagues record valgus as negative for their component angles, which is the opposite polarity.
How much does limb rotation change the measurement?
Up to 4.84 plus or minus 0.66 degrees, in a study that generated virtual long-leg radiographs from CT scans of 45 limbs across 20 degrees of internal to 20 degrees of external rotation. The error correlated with knee flexion at 0.933. That is about the width of the whole healthy range, which is why positioning is the dominant term in this measurement.
Is a varus limb abnormal?
A measurement outside a published healthy range is not by itself a deformity. Gieroba’s paper records, citing Bellemans, that 32% of men and 17% of women have a constitutional varus alignment of 3 degrees or more; Bellemans’s cohort size is not stated there and is not quoted here. Alignment is also age-dependent in children, where Radiopaedia records physiological varus from birth to 18 to 24 months and normal valgus until 7 years.
Related calculators
References
- Radiopaedia. Hip-knee-ankle angle. “The hip-knee-ankle angle (HKA) measures the angulation between the mechanical axes of the femur and tibia”; “The HKA is measured on a weight-bearing, full-length AP lower limb radiograph.” The landmarks verbatim: “The first line is drawn from the femoral head to the femoral intercondylar notch,” “the latter from the tibial interspinous point to the tibial mid-plafond.” The normal figures: the neutral adult knee is “approximately 1-1.5° varus” and healthy knees range “between 1° valgus and 3.2° varus”. “Valgus alignment is normal during development until 7 years”, with physiological genu varum from birth to 18–24 months. The page does not state whether the reported angle is the medial or the lateral angle and defines no positive or negative sign convention, which is why this page states its own convention explicitly and prints both the deviation and the 180-degree form.
- Sun X, Yang B, Xiao S, Yan Y, Liu Z, Yuan L, Wang X, Sun B, Yao J, Fan Y. Effect of limb rotation on radiographic alignment measurement in mal-aligned knees. BioMed Eng Online. 2021;20:119. Forty-five lower limbs from 30 patients were CT-scanned and virtual long-leg radiographs generated in 10-degree steps from 20 degrees internal to 20 degrees external rotation. “The measurement errors of the HKA and FTA derived from limb rotation were up to 4.84 ± 0.66°”, with an average deviation range of “4.84 ± 0.66° (range: 0.56-25.03)”; “limb rotation caused the deviation ranges of HKA and FTA measurement to be greater than 3°.” The error tracked knee flexion closely: “Remarkable correlations were found between the flexion angle and deviation ranges of both HKA (0.933, p < 0.01)”, and “With the increase of knee flexion, the measurement error of lower limb alignment increased significantly.” The intraclass correlation for the HKA deviation range was 0.976.
- Gieroba TJ, Marasco S, Babazadeh S, Di Bella C, van Bavel D. Arithmetic hip knee angle measurement on long leg radiograph versus computed tomography: inter-observer and intra-observer reliability. Arthroplasty. 2023;5:35. Sixty-eight patients awaiting knee arthroplasty, 64 with usable CT, read by three observers in three sittings a week apart. The axes verbatim: the femoral one is “the line from the center of the femoral head” to the knee, the tibial one “a line at the midpoint of the tibia at the level of the knee joint to the center of the tibial plafond”, and “The angle subtended by these two lines is the mHKA”. It reports Pearson’s r and a coefficient of repeatability rather than an ICC: for the arithmetic HKA, intra-observer r 0.92 (0.87–0.95) on the radiograph against 0.94 on CT, inter-observer 0.91 against 0.95, with a repeatability coefficient of 2.88° intra- and 3.56° inter-observer on the radiograph against 2.57° and 2.00° on CT; radiograph and CT agreed at r 0.85 with a coefficient of 3.34°. It also records, citing Bellemans et al, that “32% of men and 17% of women have a constitutional varus alignment of 3˚ or more” — Bellemans’s cohort size is not stated there and is not quoted here. On sign conventions it is explicit where Radiopaedia is silent: “valgus tibias are recorded as negative yielding an MPTA of > 90°”.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/