01
Pythagorean Theorem · Diagonal Clearance
Interactive
Diagonal clearance87.73"
ExampleA 36″ × 80″ door → c = √(36² + 80²) = 87.7″ diagonal, the longest dimension that passes when tilted.
Check a doorway fit →The geometry ItemFits uses to calculate whether an item fits — through doorways and hallways, up stairs and ramps, and into vehicles, elevators, and storage. This is the math that turns “will it fit?” into a real answer, not a tape-measure guess.
The straight-line limits — how far an item stretches across an opening or through an enclosed space, and the smallest opening it actually needs.
ExampleA 36″ × 80″ door → c = √(36² + 80²) = 87.7″ diagonal, the longest dimension that passes when tilted.
Check a doorway fit →The longest straight line inside a room, elevator, container, or vehicle cargo area — the true upper bound on what can be slid in diagonally.
ExampleA 5 × 4 × 7 ft elevator → d = √(5² + 4² + 7²) = 9.5 ft, the longest item you can angle corner-to-corner.
Check an elevator fit →Checks whether the visible face of an object can pass through an opening. Area alone isn't a verdict, but it's a fast first filter.
ExampleA 30″ × 20″ TV face (600 in²) clears a 32″ × 80″ door (2,560 in²) — but you still check each dimension.
Check a doorway fit →Instead of “will it fit?”, estimates the smallest opening the item needs, including a safety margin M.
ExampleA 60″ × 30″ piece → narrowest projection 30″, + 2″ margin = a 32″ minimum opening.
Check a doorway fit →What happens to an item's footprint when you turn or tip it to clear a tight spot — and the angle that opens up the most room.
The effective width of an item as it rotates through an opening.
ExampleA 60″ × 12″ shelf at 30° → W = 60·cos30° + 12·sin30° = 52 + 6 = 58″ effective width.
Vertical clearance during a rotated doorway or hallway maneuver.
ExampleThat 60″ × 12″ shelf at 30° → H = 60·sin30° + 12·cos30° = 30 + 10.4 = 40.4″ tall while turning.
For couches, refrigerators, mattresses, and appliances moved at an angle.
ExampleA 72″ × 30″ mattress tilted 40° → H = 72·cos40° + 30·sin40° = 55.2 + 19.3 = 74.5″ bounding height.
When an item is too tall upright but may fit once tilted — solves for the angle the opening height S forces.
ExampleAn 84″ item under a 78″ opening → θ = cos⁻¹(78 ÷ 84) = 21.8° minimum tilt.
The canonical matrix behind rotating an object through a doorway or around a corner.
ExampleRotating the point (10, 0) by 90° → R·(10, 0) = (0, 10).
Pitch, yaw, and roll combined for 3D fit simulation and collision testing.
ExampleYaw a 40 × 20 × 8″ box 90° → its 40 × 20″ footprint swaps to 20 × 40″ before the next axis. Order matters.
Close to what a doorway solver checks: the axis-aligned box a rotated object actually sweeps out, compared against the opening.
ExampleA 60″ × 12″ panel at 30° → projected box W·H = 58 × 40.4 ≈ 2,340 in².
Check a doorway fit →Finds the rotation angle that leaves the most clearance through an opening — the math behind “rotate it this way” results.
ExampleA 33″-wide item through a 36″ door → C* = 36 − 33 = 3″ clearance at its best orientation.
The largest circular rotation zone that fits inside a rectangular opening or tight space.
ExampleA 36″ × 80″ doorway → r_max = min(36, 80) ÷ 2 = an 18″ rotation circle.
When the object — or the door it passes — has to rotate open, not just slide. The floor it sweeps is part of the fit.
The path a door, appliance door, cabinet, hatch, or vehicle door traces as it opens (θ in radians).
ExampleA 32″ door swinging 90° (π⁄2 rad) → s = 32 · 1.571 = 50.3″ of arc the edge travels.
The floor area a rotating panel needs to open without hitting furniture or walls.
ExampleA 32″ door opening 90° → A = π · 32² · 90⁄360 = 804 in² (5.6 ft²) of floor swept.
Does the item overlap an obstacle? Cheap rejection first, then exact rotated-shape tests when it's close.
If two projected shapes don't overlap on at least one axis, they aren't colliding — the exact test for rotated rectangles, walls, and obstacles.
ExampleIf box A spans [0, 30] on an edge-axis and box B spans [34, 60], the 4″ gap proves they can’t touch.
Axis-aligned bounding boxes give a cheap collision check before the heavier geometry — good for walls, openings, vehicles, and rooms.
ExampleBox A x-range [10, 50], box B x-range [55, 90] → 55 > 50, so no x-overlap and no collision — skip the exact test.
Cornering, turn corridors, stair pitch, ramp length, and point-to-point reach through a route.
Use these for motion planning, corner-to-corner clearance, object-space distance checks, and route geometry.
Example2D: (0,0) → (8,6) ft gives d = 10 ft. 3D: add a 4 ft rise → d = √(8² + 6² + 4²) = 10.8 ft.
Check a hallway turn →The circle a rectangular item sweeps when rotated about its center — if the space can't hold this radius, the pivot is blocked.
ExampleA 60″ × 30″ table → r = √(30² + 15²) = 33.5″, the circle it needs to spin in place.
For hallway turns where the item rotates through the corner instead of moving straight. M is the safety margin.
ExampleAn 84″ × 36″ sofa mid-turn projects up to ~60″; add a 3″ margin → a 63″ corridor at the pivot.
Check a hallway turn →Flags long, skinny, awkward items that pass a doorway flat but fail mid-rotation. Higher ratio, higher turn-risk.
ExampleAn 80″ × 16″ ladder → AR = 80 ÷ 16 = 5.0 — high ratio, high turn-risk even though it slides through flat.
Core measurement for route analysis, object movement, and 3D solver geometry.
ExampleFrom (1, 2, 0) to (5, 5, 4) ft → d = √(4² + 3² + 4²) = 6.4 ft.
Pitch sets how much a long object must tilt — and whether it clears the ceiling, landing, or truck bed on the way up.
ExampleA stair rising 7″ over an 11″ run → α = tan⁻¹(7 ÷ 11) = 32.5° pitch.
Check a stair carry →The ramp length needed to reach a target rise at a chosen safe angle — practical for pickups, deliveries, and DIY moves.
ExampleTo climb a 24″ rise on a 10° ramp → L = 24 ÷ sin10° = 138″ (11.5 ft).
Check a stair carry →How the solver turns a whole movement path into one verdict — the tightest point, the safety margin, and the route cost.
Measures the tightest point during the whole move, not just the start or end. This is the core idea behind doors, hallway turns, stairs, elevators, and vehicle loading.
ExampleIf clearance dips to 0.5″ at one doorframe mid-move, C_min = 0.5″ — that point sets the verdict, not the 4″ at the start.
When movement is a route problem: g(n) is the cost already traveled, h(n) estimates the cost remaining to the goal.
ExampleTraveled g = 12 ft, straight-line estimate h = 5 ft → f = 17 ft for that route node.
The solver rule in one line: the item fits only if clearance stays above the safety margin for every point on the path.
ExampleWith a 1″ margin, a path that never drops below 1″ fits; a single 0.7″ pinch point doesn’t.
Turns raw clearance into a margin relative to the item size — the threshold behind “fits tight” versus “safe fit.”
ExampleA 3″ clearance on a 36″ item → M_r = 3 ÷ 36 × 100 = 8.3% — tight but workable.
Will it fit isn't the whole story — these are the rough weight and pressure checks for whether you can actually move it safely.
A crude moving-load estimate for one person, two people, or equipment. It ignores grip, stairs, and shape, but users get it instantly.
ExampleA 180 lb dresser split 2 ways → 90 lb each; over a ~50 lb comfortable limit, so use 3 people (60 lb) or a dolly.
For safes, aquariums, appliances, and gym equipment — a warning signal for concentrated weight, not a structural verdict.
ExampleA 600 lb safe on four 3″ × 3″ feet (36 in²) → P = 600 ÷ 36 = 16.7 psi on the floor.
How much actually fits when you're filling a space, not just passing through it.
Basic vehicle, box, trailer, storage-unit, or container volume. Pair it with length/width/height and opening checks.
ExampleA 4 × 4 × 8 ft truck bed → V = 128 ft³ — yet a 9 ft pipe still won’t lie flat, so check dimensions too.
Check a vehicle fit →How full a storage unit, cargo space, trailer, or container really is.
Example60 ft³ of boxes in a 128 ft³ bed → U = 60 ÷ 128 × 100 = 47% full.
How efficiently repeated items occupy an available space.
Example20 boxes of 1.5 ft³ in 64 ft³ → E = 30 ÷ 64 = 47%; the rest is unavoidable gaps.
Ties directly to storage units, cargo areas, moving trucks, trailers, closets, and multi-item loading.
ExampleA 128 ft³ truck with 90 ft³ loaded → R = 38 ft³ left.
Room capacity, floor-plan coverage, garage storage, closet layouts, and footprint-based fit checks.
Example120 ft² of furniture in a 200 ft² room → U_A = 60% floor coverage.
Footprint is easier than volume when the problem is floor space — rooms, storage units, trailers, and grouped deliveries.
ExampleA 6×3 sofa + 4×2 table + 3×3 chair → 18 + 8 + 9 = 35 ft² of floor needed.
How many identical items can be stacked vertically inside a space — for storage-unit and room-capacity checks.
ExampleA 96″ ceiling with 18″ bins → n_max = ⌊96 ÷ 18⌋ = 5 bins high (90″, 6″ to spare).
How many same-size items fit in rows and columns on a floor or cargo base — simple rectangular packing.
ExampleA 120″ × 80″ floor with 24″ × 18″ totes → N = ⌊120⁄24⌋ · ⌊80⁄18⌋ = 5 · 4 = 20 totes.
Object count per available area — for shelving, warehouse-like arrangements, or “how many fit in this footprint?”
Example30 items across 50 ft² → D = 30 ÷ 50 = 0.6 items per ft².
Measure the door’s clear width and height, then compare them against the item. For tall or wide pieces, use the diagonal clearance c = √(w² + h²) and the rotated-width projection W(θ) = w·cosθ + h·sinθ to find the best tilt angle. ItemFits runs these automatically and returns a verdict.
The diagonal of an opening is c = √(width² + height²). A 36″ × 80″ door has an 87.7″ diagonal — the longest straight dimension an item can pass when tilted corner-to-corner.
Start with cargo volume V = L · W · H for raw capacity, but volume can pass while a single dimension fails — so also check the 3D space diagonal √(l² + w² + h²) and each opening the item must clear on the way in.
A rough estimate is weight per handler, W_total ÷ n. It ignores grip, stairs, and awkward shape, but it’s enough to flag when an item likely needs two people or equipment instead of one.
Stair pitch α = tan⁻¹(rise ÷ run), combined with the item’s length and the ceiling and landing limits, sets the maximum length you can carry up tilted. Steeper stairs reduce the length that clears.
Yes. The solver evaluates clearance along the entire movement path — the minimum-clearance rule C_min = min C(t) — not just the entrance, so the tightest point decides the verdict. See our methodology for the full approach.
ItemFits runs this geometry for you in seconds — with 3D visualization, tilt strategies, and a clear verdict for doors, stairs, hallways, elevators, and vehicles.
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