How to use the longest length around a corner calculator
- Enter the width of the first passage and the width of the second passage, the two hallways or corridors that meet at a right angle. Use one unit.
- Enter the thickness of the item if it is a board or panel with real width. Leave it at 0 for a thin rod, pipe or pole.
- Choose the unit label you want shown after the results. It is only a label.
- Read the longest length that can be turned around the corner, the angle where it is tightest, and the clearance at a few angles in the table.
Formula
clearance length at angle t = a / cos(t) + b / sin(t) - thickness / (sin(t) x cos(t))
longest item = the smallest clearance over every angle t between 0 and 90 degrees
thin item: longest = (a^(2/3) + b^(2/3))^(3/2)
The item is treated as a stiff straight piece moved flat on the floor around a square inside corner. At each angle the item can only be as long as the clearance at that angle, so the shortest clearance sets the limit.
Passages of 36 and 30 inches
For a thin pole, the longest length is (36^(2/3) + 30^(2/3))^(3/2) = 93.21 inches, and the tightest point is when the pole is turned 43.3 degrees. That is much more than the straight diagonal across the two passages, which is only sqrt(36^2 + 30^2) = 46.86 inches, because the pole has to sweep through every angle between the passages.
If the item is a board 3 inches thick, the clearance formula drops the limit to 87.20 inches. Two passages of 48 inches with a 4 inch board give 127.76 inches, and by symmetry the tightest angle is exactly 45 degrees.
Why the limit is not the diagonal
A common guess is to take the diagonal across the corner. The diagonal is the shortest straight line across, but a long item must rotate from lying along one passage to lying along the other, and while it turns, its ends pinch against the outer walls. The tightest moment gives the answer, and it is always longer than the diagonal but depends on both widths.
What the number leaves out
Real moves add handles, hands, tilting the item up on its edge, door frames and rounded or beveled walls. A margin of several inches is sensible. For furniture on a truck, see the Moving Truck Size Calculator, and for the straight diagonal alone, the Pythagorean Theorem Calculator.
More shape tools
For other shape and geometry questions, try the Bike Spoke Length Calculator, IBU Calculator and Cone Pattern Calculator.
Assumptions and limits
- The two passages meet at a right angle and have straight, parallel walls.
- The item is rigid, straight and moved flat on the floor without tilting.
- The thickness is the same along the whole item, and its ends are square.
- No allowance is made for handles, hands or wall fittings.
Frequently asked questions
What is the longest ladder that fits around a corner?
For a thin ladder, take each passage width to the power of two thirds, add them and raise the sum to the power of three halves. With 36 and 30 inch passages that gives 93.21 inches.
Why does thickness make such a difference?
A thick item cannot hug the inside corner, so it sits farther from it and loses length at every angle. The loss is largest where the item is crosswise to the corner.
Does the answer change if the passages have different widths?
Yes. The tightest angle moves toward the wider passage, away from 45 degrees, but the same method finds the length.
Will a sofa that is shorter than the answer always fit?
No. A sofa has height, depth and legs, and the answer only covers a flat item turned on the floor. Measure the doorways too and tilt it on its end if needed.