Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Calculate rise divided by horizontal run.
- Convert a decimal slope to a percentage.
- Distinguish percentage grade from an angle in degrees.
Before you start
Division, percentages, and an introduction to tangent. The angle calculation is an optional extension.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A simplified road profile rises 6 metres over a horizontal distance of 120 metres. What are its slope, percentage grade, and angle above horizontal?
Why this math matters
A hill's steepness is a rate of change. The same idea describes the rise of a graph, the pitch of a roof in a drawing, and the change in elevation on a map. The numbers become meaningful when you know exactly which two distances are being compared.
Set up the model
A useful answer starts with clear assumptions:
- The profile is a straight line with constant slope over this interval.
- The 120 m distance is horizontal, not the distance travelled along the sloping surface.
- This is a classroom geometry model, not a road or accessibility design specification.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
What does a 5% road grade mean?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A simplified road profile rises 6 metres over a horizontal distance of 120 metres. What are its slope, percentage grade, and angle above horizontal?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Keep rise and run in the same units
rise = 6 m; horizontal run = 120 m
Both measurements are in metres, so their units cancel in the ratio. If one were in centimetres, you would convert it first.
Find the slope and grade
m = 6 / 120 = 0.05; grade = 0.05 × 100% = 5%
A 5% grade means a rise of 5 units for every 100 horizontal units. It also means 0.05 metre of rise per horizontal metre.
Convert the ratio to an angle
tan θ = 0.05; θ = arctan(0.05) ≈ 2.86°
Use the inverse tangent function in degree mode. Grade is a ratio expressed as a percentage; the angle measures rotation from horizontal.
The result
The slope is 0.05, the grade is 5%, and the angle is approximately 2.86°.
Walking in the opposite direction gives a signed slope of −0.05 when the horizontal coordinate increases along that return direction. The steepness has the same magnitude. A 100% grade means equal rise and horizontal run, which corresponds to 45°, not 90°.
Common mistakes to catch
- Dividing by the sloping road length instead of the horizontal run changes the ratio.
- A 5% grade is not an angle of 5°.
- An average slope over a route does not describe every short section of that route.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
A trail profile rises 9 m over a horizontal run of 150 m. What is the percentage grade?
Show a hint
Divide rise by horizontal run, then multiply by 100%.
Reveal answer and explanation
6%
9/150 = 0.06, so the grade is 0.06 × 100% = 6%. Every 100 horizontal metres corresponds to 6 metres of rise in this constant-slope model.
Practice 2
In a constant-slope sketch, a 4% grade extends for 250 horizontal metres. How much elevation is gained?
Show a hint
Write 4% as 0.04 and rearrange slope = rise/run.
Reveal answer and explanation
10 m
rise = 0.04 × 250 m = 10 m. The result is a length; the slope ratio is dimensionless.
Take the idea with you
Choose two points on any straight-line graph. Name the horizontal and vertical units before finding the slope. A slope of 5 dollars per hour means something very different from a slope of 5 metres per metre.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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