Wheelchair ramp gradient and plywood order
A community centre in Derry needs a ramp from floor level up to a stage that is 0.60 m high. Building guidance for this kind of ramp gives a maximum gradient of 1:12, meaning 1 unit of vertical rise for every 12 units of horizontal run. The floor layout allows a straight horizontal run of 6.50 m from the stage edge to the nearest wall, and no more. The ramp will be 1.20 m wide, and its sloping surface will be cut from plywood costing £28.40 per square metre, sold only in whole square metres. All site measurements were made to the nearest 0.05 m. The caretaker argues that 6.50 m is near enough to the 7.20 m the guidance seems to require and that the ramp should simply be built. The centre manager wants the gradient checked properly before any money is spent.
1. Check whether a 6.50 m horizontal run meets the 1:12 maximum gradient for a 0.60 m rise.
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A gradient of 1:12 means run ÷ rise = 12, so the required run for a 0.60 m rise is:
12 × 0.60 = 7.20 m
Only 6.50 m is available, so the run is short by 0.70 m and the ramp would be steeper than allowed.
Check directly with the ratio: run ÷ rise = 6.50 ÷ 0.60 = 10.833...
This is a gradient of 1:10.8 (to 1 d.p.), not 1:12. A smaller run for the same rise means a steeper ramp, so 1:10.8 is too steep.
The caretaker is comparing the wrong numbers: 6.50 m is not 'near enough' to 7.20 m when it is the run that controls safety.
2. Assuming the ramp is built with the 6.50 m run, calculate the length of the sloping surface and the cost of the plywood needed.
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The ramp surface is the hypotenuse of a right-angled triangle with horizontal 6.50 m and vertical 0.60 m.
Length² = 6.50² + 0.60² = 42.25 + 0.36 = 42.61
Length = √42.61 = 6.5276... m (about 6.53 m)
Surface area = 6.5276 × 1.20 = 7.833... m²
Plywood is sold only in whole square metres, so 7 m² is not enough; order 8 m².
Cost = 8 × £28.40 = £227.20
Two judgement points: use Pythagoras rather than the horizontal run for the surface, and round the area up to the next whole square metre before multiplying by the price.
3. The caretaker says 6.50 m is 'close enough'. Using the measurement tolerance of ±0.05 m, decide whether any plausible measurement could make the ramp compliant.
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The run of 6.50 m, measured to the nearest 0.05 m, lies between 6.475 m and 6.525 m. Take the most favourable case, the longest run:
6.525 ÷ 0.60 = 10.875
The best possible gradient is still 1:10.9, which is steeper than 1:12. Now allow the height to vary too: 0.60 m to the nearest 0.05 m lies between 0.575 m and 0.625 m. The most favourable combination is the longest run with the lowest rise:
6.525 ÷ 0.575 = 11.35
Even this is below 12. So no measurement within tolerance can make the ramp compliant, and rounding error is not the explanation.
The manager should reject the straight ramp and consider a switchback with a landing, a shorter rise (for example a raised platform at the front of the stage), or a platform lift, any of which must still be checked against the 1:12 rule.