Crane radius formula
Treat the boom as a straight line from the foot pin to the tip sheave. Its horizontal reach from the foot pin is the adjacent side of a right triangle, so reach = L × cos θ. The operating radius is measured from the centerline of rotation, so add the foot pin's horizontal offset p from that centerline: R = L × cos θ + p. Take p as positive when the foot pin sits ahead of the centerline toward the load and negative when it sits behind.
Rearranging the same equation gives the other two unknowns:
- Boom angle from length and radius: θ = arccos[(R − p) / L]. R − p cannot be negative or longer than L.
- Boom length from angle and radius: L = (R − p) / cos θ. This has no answer at 90°, where the tip sits over the foot pin.
- Tip height above the foot pin: L × sin θ. Add the foot pin's height above ground for tip height above ground.
These equations are our own derivation from the cosine and sine definitions in OpenStax Precalculus 2e, section 5.4. With p = 0 they give the radius from the foot pin, which is what many study problems mean when they omit the offset.
Worked example: 100 ft boom at 60°
With no pin offset, R = 100 × cos 60° = 100 × 0.5 = 50 ft. If the foot pin sits 3 ft ahead of the centerline, R = 50 + 3 = 53 ft. The tip is 100 × sin 60° = 86.603 ft above the foot pin. Lowering the boom to 59° gives 100 × cos 59° + 3 = 54.504 ft, so that one degree adds 1.504 ft of radius.
Working backward: a 50 ft boom at a 30 ft radius with no offset gives cos θ = 30 / 50 = 0.6, so θ = 53.130°. A 70° boom angle at a 40 ft radius with a 2 ft offset needs L = 38 / cos 70° = 111.105 ft of boom. These fictional inputs illustrate the geometry; they are not taken from any crane, chart or exam question.
Why the radius grows as the boom angle falls
Because cos θ rises as θ falls, lowering the boom always moves the hook out. Near vertical, a small angle change moves the hook the most: one degree is worth about L × sin θ × π/180 of radius. On a 100 ft boom that is roughly 1.7 ft per degree near 80°, and about 0.3 ft per degree near 10°. The calculator shows the exact change for your inputs.
Boom deflection and the load chart
Under load the boom bends and the tip moves outward and down, so the loaded radius is larger than the unloaded geometry gives. Measure or read the actual radius with the load suspended, and use the manufacturer's load chart and procedures, which govern.
The calculator assumes a straight, unloaded boom. A loaded boom bends, a telescopic boom also has clearance between sections, and the carrier and outriggers settle slightly. Each of these moves the tip outward, so the loaded radius is larger than the geometry above. That is our explanation from basic mechanics, not a published deflection figure; the amount depends on the crane, configuration and load.
OSHA requires a boom angle or radius indicator on equipment covered by 29 CFR 1926.1416 (paragraph (e)(1)), and requires following the manufacturer's procedures under 29 CFR 1926.1417 (paragraph (a)). Read capacities from the crane's own load chart at the actual loaded radius, and subtract deductions with the net capacity calculator.
Units and limits
Enter lengths in feet or meters; switching units converts the entered values using 1 ft = 0.3048 m from NIST's SI conversion factors. Boom length here means foot pin to tip sheave center, measured along the boom. Some charts add jib, extension or auxiliary head lengths, and some define radius to the hook with the load suspended. Use your chart's definitions.
Practice reading invented charts at a given radius and boom angle in the load chart practice problems, or try the NCCCO written practice test.