Welding math
Fillet Weld Leg and Throat
A fillet is specified by its leg but carries load across its throat, and the constant 0.7071 that converts between them is just the cosine of the forty-five degree angle an equal-leg fillet makes.
The formula
t = leg x cos45 = leg x 0.7071, and leg = t / 0.7071
The theoretical throat of an equal-leg fillet is its leg length multiplied by the cosine of forty-five degrees, and the leg is the throat divided by that same factor.
What each symbol means
| Symbol | Meaning | Units |
|---|---|---|
| leg | Leg length measured along either fusion face from the root | in |
| t | Theoretical throat, the shortest distance from root to a flat face | in |
| w | Face width across the hypotenuse of the fillet triangle | in |
Why it works
Draw an equal-leg fillet in cross-section and you have a right isosceles triangle sitting in the corner of the joint. The two legs run along the fusion faces. The hypotenuse is the weld face. The throat is the perpendicular dropped from the root to that hypotenuse, and because the triangle is isosceles the throat is the leg times the cosine of forty-five degrees, which is 0.70710678.
The distinction matters because the two dimensions serve different masters. Drawings and welding symbols specify leg size, because leg is what a fillet gauge measures against the plate surfaces and what an inspector can check without cutting anything. Design calculations use throat, because the throat plane is where a fillet in shear actually fails. Roughly seventy percent of the leg is what carries the load.
Two caveats attach to the word theoretical. First, the formula assumes a flat face; a concave bead has a smaller actual throat than the theoretical value, which is why concavity on a load-carrying fillet is treated seriously. Second, it assumes equal legs at ninety degrees. Unequal legs or a joint angle other than ninety require the general triangle geometry, not this shortcut.
Worked examples
Throat of a quarter-inch fillet
- Specified leg size
- 0.25 in
- Recognise the equal-leg fillet as a right isosceles triangle and take the cosine of 45 degrees: 0.70710678.
- Multiply the leg by that factor: 0.25 x 0.70710678.
Result: 0.177 in
A quarter-inch fillet carries load across a throat of a little over seventeen hundredths of an inch. That throat, not the quarter inch stamped on the drawing, is the dimension a designer sizes the joint by.
Leg needed to reach a required throat
- Required theoretical throat
- 0.25 in
- Rearrange the relationship to solve for the leg: leg = throat / cos 45.
- Divide: 0.25 / 0.70710678.
Result: 0.354 in
Reaching a quarter-inch throat needs a leg of about three eighths of an inch, comfortably larger than the throat itself. Confusing the two dimensions in this direction produces a fillet undersized by nearly thirty percent.
In practice
- A fillet gauge reads leg, and reads it correctly only when the gauge sits flat against both members. On a convex bead a carelessly placed gauge rocks and flatters the reading.
- The 0.7071 factor applies to a ninety degree joint. On a skewed tee the throat is a different fraction of the leg, and on an acute angle it shrinks fast.
- The theoretical throat ignores any root penetration beyond the joint corner. Some procedures qualify a deeper effective throat, but that has to be demonstrated by macro-etch, never assumed.
- Concavity reduces the actual throat below the theoretical value while the leg stays unchanged, so a fillet can pass a leg check and still be short of throat.
Related topics
Practise the arithmetic under pressure
The welding-math domain in the question bank is where these formulas get tested the way they get tested on paper — with distractors built from the mistakes people actually make.