# Tributary area on an L-shaped floor

> The re-entrant corner of an L-shaped floor is where hand tributary sketches drift. Here is why, how clipping to the outline fixes it, and the area-balance check that proves it.

**Category:** Fundamentals  
**Author:** Sam Rivera (Structural engineer · Educator)  
**Published:** 2026-05-19

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An L-shaped floor is where the half-bay tributary sketch quietly drifts. Near the re-entrant, inside corner, a support's nearest region wraps around the corner and stops being a neat rectangle, so the quick shortcut fails. The exact method is to build each support's nearest-point region and clip it to the real outline, then confirm the areas sum to the floor area. StructLoads does this clipping automatically, which is its main advantage over a hand sketch on irregular plans.

## Why the rectangle rule breaks

On a regular grid, the lines halfway to a support's neighbours enclose a rectangle, which is why the half-bay rule is so quick, as covered in [how to calculate tributary area for a column](/blogs/how-to-calculate-tributary-area-for-a-column). An L-shaped floor has a re-entrant corner where the boundary turns inward. A support near that corner draws floor that bends around it, and the halfway lines no longer close a simple rectangle. The geometry is still well defined, but it is a polygon, and reading it off by eye is where errors creep in.

## The exact method: build and clip

The reliable approach is the same nearest-point partition used everywhere, a clipped [Voronoi diagram](https://en.wikipedia.org/wiki/Voronoi_diagram). For each support you build the region of the plane closest to it, then intersect that region with the actual L-shaped floor polygon. The intersection is the tributary area, whatever shape the outline forces it into. This is not an approximation but a standard, exactly solvable operation: polygon intersection is one of the regularized boolean set operations that computational geometry libraries such as [CGAL](https://doc.cgal.org/latest/Boolean_set_operations_2/index.html) implement with exact arithmetic, which is why a tool can guarantee the wrapped region around a re-entrant corner while a sketch can only estimate it. Multiply the clipped area by the floor [load](https://en.wikipedia.org/wiki/Structural_load) and you have the support's load, exactly as on a regular floor.

## Regular grid versus L-shape

The difference is entirely in the geometry step, not the load step.

| Step | Regular grid | L-shaped floor |
| --- | --- | --- |
| Tributary region | Read off as a rectangle | Polygon clipped to the outline |
| Re-entrant corner | None | Region wraps the inside corner |
| Hand accuracy | Exact | Drifts, easy to miscount |
| Best method | Half-bay rule | Build and clip the region |

So the same support that is trivial on a grid needs real geometry on an L-shape, which is the practical reason irregular floors take longer by hand.

## The area-balance check

The check that proves the partition on any floor is area balance: the tributary areas of all supports should sum to the floor area. This applies just as well to an L-shape, where the floor area is the rectangle minus the bite. If the tributary areas add up to the L-shaped area, every square metre is assigned once, nothing lost or double counted. A mismatch is a flag, and it is the first thing to check on an irregular plan, in line with basic [structural principles](https://www.designingbuildings.co.uk/wiki/Structural_principles).

## A practical example

Take a 10 by 10 metre floor with a 4 by 4 metre bite out of one corner, so the area is 100 minus 16, which is 84 square metres. A column near the inside corner of the L draws a region that wraps the corner, larger and oddly shaped compared with its grid neighbours. By hand you might approximate it as a rectangle and lose a few square metres; by clipping, you get the true area, and the sum across all columns lands on 84. The reliability of that sum is why a [free tributary area calculator](/blogs/free-tributary-area-calculator) is worth more on irregular plans than on a clean grid.

## Key takeaways: L-shaped tributary areas

On an L-shaped floor the half-bay sketch drifts near the re-entrant corner, where the tributary region wraps and is no longer a rectangle. Build each support's nearest region and clip it to the real outline, then confirm the areas sum to the floor area. StructLoads does the clipping and the balance check automatically, as a preliminary figure a qualified engineer confirms. The full background is in [how tributary areas actually work](/blogs/tributary-areas-explained).

## Quick answers

### How do you calculate tributary area on an L-shaped floor?
Build each support's nearest-point region, then clip it to the actual L-shaped outline and measure its area. Near the re-entrant corner the region wraps around the inside corner and is a polygon, not a rectangle, so you compute the polygon area rather than a half-bay rectangle. The areas should sum to the floor area.

### Why is an L-shaped floor harder than a rectangle?
Because the re-entrant, inside corner breaks the neat half-bay rectangles. A support near that corner draws floor that bends around the corner, and the lines halfway to its neighbours no longer enclose a simple rectangle. Hand sketches tend to over or under count there, while a clipped geometric method handles the wrapped region exactly.

### What is the area-balance check on an irregular floor?
It is the test that the tributary areas of all supports sum to the actual floor area. On any floor, including an L-shape, every square metre should belong to exactly one support, nothing double counted or lost. If the sum matches the L-shaped floor area, the partition is consistent; a mismatch means a region was missed or overlapped.

### When does the half-bay sketch not work?
It does not work on L-shaped, skewed or holed floors, or with very uneven bays, because the tributary regions stop being rectangles. The half-bay shortcut is exact only on a regular grid. On irregular outlines you compute each region by clipping to the floor, which is exactly where a geometric tool is more reliable than a hand sketch.