Mathematics 7–10 · Year 9

Gradient of a real ramp: rise over run with a spirit level, checked by angle

Number and algebra

PracticalLow risk

The idea

The gradient of a straight ramp is its vertical rise divided by its horizontal run, the same at every point along it, and it can be written as a ratio 1 : n, as a percentage and as an angle whose tangent is the gradient.

Safety card

Low riskAn adult supervises

Hazards

people using the ramp, including wheelchair users
wet or mossy surfaces
sun exposure

Controls

measure when the ramp is clear and give way to every ramp user
work on dry surfaces only
hats and water

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's excursion and playground supervision rules.

What you need

1.2 m spirit level (or a 1.2 m straight aluminium rule with a 200 mm spirit level taped to it)
5 m steel tape measure with millimetre graduations
30 m fibreglass tape measure
The protractor clinometer from the clinometer practical, or a phone inclinometer app
Chalk and a recording table
A straight ramp in the school grounds, and a flight of steps for comparison

How to do it

Lay the spirit level on the ramp with its upper end touching the surface, lift the lower end until the bubble is centred, and measure the vertical gap between the lower end and the ramp surface in millimetres. The level is then a horizontal run of 1.200 m and the gap is the rise over that run.
Repeat near the bottom, the middle and the top of the ramp; equal gaps show that the gradient is the same all along a straight ramp.
Compute the gradient m = rise ÷ run and write it as 1 : n (n = run ÷ rise), as a percentage (100m) and as an angle θ = tan⁻¹ m; compare the angle with a clinometer reading laid along the ramp.
Measure the length L of the ramp surface from bottom to top with the 30 m tape and compute the total rise L sin θ and the horizontal run L cos θ.
Put the foot of the ramp at the origin with x horizontal and y vertical, write the coordinates of the top, find the midpoint, mark it on the ramp and check its height with the level method; find the distance between the ends by Pythagoras and compare it with L.
Measure the rise and going of one step in the flight beside the ramp and compute the stair gradient for comparison. Compare the ramp with the 1 : 14 maximum gradient that the National Construction Code sets for a fixed or moveable swimming pool access ramp (NCC 2025 Volume One, Specification 16, clause S16C2, read 2026-09-23).

What you should see

As a worked check, an 86 mm gap under a 1.200 m level gives m = 0.0717, which is 1 : 14.0, 7.2 per cent and θ = 4.10 degrees, almost exactly the 1 : 14 benchmark (0.0714, 4.09 degrees). A ramp surface of 8.40 m at that gradient rises 0.600 m over a horizontal run of 8.379 m, so the top is at (8.379, 0.600) and the midpoint at (4.189, 0.300). A 1 mm error in the gap changes the gradient by 0.0008 and the angle by 0.05 degrees, whereas a protractor clinometer read to the nearest degree carries at least ±0.5 degrees, an eighth of the whole 4 degree angle, so rise over run is the more precise method on shallow slopes. Steps are much steeper: a step with a 170 mm rise and a 280 mm going has gradient 0.61, an angle of 31 degrees. The learner knows it worked when the three gap readings agree within 2 mm on a straight ramp and the angle from tan⁻¹ agrees with the clinometer within its reading error.

What changes

This activity lists no variables to change, measure and keep the same.

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Gradient is the length of the sloping surface divided by the height.
The tape length along the ramp surface is the horizontal run.
A 1 : 14 ramp slopes at 14 degrees.
Doubling the angle always doubles the gradient.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K–10 Syllabus (2022), Stage 5 Linear relationships A; page read 2026-09-22MA5-LIN-C-01
Mathematics K–10 Syllabus (2022), Stage 5 Trigonometry A; page read 2026-09-22MA5-TRG-C-01
Australian Curriculum v9AC9M9A03AC9M9M03

Sources

The pages the author read to write this activity.

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