Learn: Horizontal-curve radius
A vehicle following a horizontal curve needs inward acceleration. Superelevation and available side friction contribute to that demand in a simplified road-curve balance.
What the formula is saying
Square speed v and divide by g(e + f). The denominator combines the supplied superelevation and side-friction contributions; their sum must be positive.
Read the symbols in plain language
- v
- Design speed
Design speed. Rearranging v squared over radius gives the radius for the stated speed and acceleration.
m/sUse m/s as the base unit shown here. Use speed in m/s, not km/h; divide km/h by 3.6. e and f are decimal ratios, so 6% superelevation is 0.06. g is m/s² and radius R is m.
- g
- Gravity
Gravity. Gravity converts the dimensionless combined ratio into acceleration.
m/s²Use m/s² as the base unit shown here. Use speed in m/s, not km/h; divide km/h by 3.6. e and f are decimal ratios, so 6% superelevation is 0.06. g is m/s² and radius R is m.
- e
- Superelevation as decimal
Superelevation as decimal. Use consistent decimal ratios and retain the sign of any adverse superelevation.
ratio / no unitA dimensionless ratio has no physical unit; 0.01 as a ratio is 1% when the percent option is selected.
- f
- Side-friction factor
Side-friction factor. Use consistent decimal ratios and retain the sign of any adverse superelevation.
ratio / no unitA dimensionless ratio has no physical unit; 0.01 as a ratio is 1% when the percent option is selected.
- R
- Result to find
Horizontal-curve radius. Rearranging v squared over radius gives the radius for the stated speed and acceleration.
m
Sort out the units first
Use speed in m/s, not km/h; divide km/h by 3.6. e and f are decimal ratios, so 6% superelevation is 0.06. g is m/s² and radius R is m.
Assumptions before calculating
Assume a steady-speed vehicle, a circular horizontal path and the usual small-angle road-banking approximation. Use a side-friction value and signed superelevation that are appropriate to the stated study conditions.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find R and explain the result in the stated output unit.
- v · Design speed
- 22.22 m/s
- g · Gravity
- 9.81 m/s²
- e · Superelevation as decimal
- 0.06
- f · Side-friction factor
- 0.15
Combine the stated lateral contributions
Use consistent decimal ratios and retain the sign of any adverse superelevation.
(0.06) + (0.15) = 0.21Form the available lateral-acceleration scale
Gravity converts the dimensionless combined ratio into acceleration.
(9.81) × (0.21) = 2.0601 m/s²Solve the circular-path acceleration relation
Rearranging v squared over radius gives the radius for the stated speed and acceleration.
(22.22)^2 ÷ (2.0601) ≈ 239.6623465 m
Does this worked answer make sense?
At fixed e and f, twice the speed requires four times the radius in this model. A larger positive combined lateral contribution reduces the calculated radius, but its permitted value cannot be chosen merely to obtain a smaller answer.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- v · Design speed
- 20 m/s
- g · Gravity
- 9.81 m/s²
- e · Superelevation as decimal
- 0.04
- f · Side-friction factor
- 0.12
Combine the stated lateral contributions
Use consistent decimal ratios and retain the sign of any adverse superelevation.
(0.04) + (0.12) = 0.16Form the available lateral-acceleration scale
Gravity converts the dimensionless combined ratio into acceleration.
(9.81) × (0.16) = 1.5696 m/s²Solve the circular-path acceleration relation
Rearranging v squared over radius gives the radius for the stated speed and acceleration.
(20)^2 ÷ (1.5696) ≈ 254.841998 m
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- v · Design speed
- 25 m/s
- g · Gravity
- 9.81 m/s²
- e · Superelevation as decimal
- 0.05
- f · Side-friction factor
- 0.15
Find: Learn: Horizontal-curve radius
A hint, not the answer
Square speed v and divide by g(e + f). The denominator combines the supplied superelevation and side-friction contributions; their sum must be positive.
Use speed in m/s, not km/h; divide km/h by 3.6. e and f are decimal ratios, so 6% superelevation is 0.06. g is m/s² and radius R is m.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Combine the stated lateral contributions
Use consistent decimal ratios and retain the sign of any adverse superelevation.
(0.05) + (0.15) = 0.2Form the available lateral-acceleration scale
Gravity converts the dimensionless combined ratio into acceleration.
(9.81) × (0.2) = 1.962 m/s²Solve the circular-path acceleration relation
Rearranging v squared over radius gives the radius for the stated speed and acceleration.
(25)^2 ÷ (1.962) ≈ 318.5524975 m
Avoid the common trap
Do not insert 80 km/h as 80 m/s or use 6 instead of 0.06 for a 6% bank. Do not allow e + f to be zero or negative and then accept the resulting algebraic radius.
When this method applies — and when it does not
The output is a model radius for the supplied values, not automatic approval of a minimum design radius. Governing speed policy, comfort, climate, pavement, sight distance and transition design must be checked separately. It does not predict real tire grip in every condition.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Horizontal-curve radius. Speed, friction, curvature and stopping-distance principles. Supplied friction/reaction inputs are study data, not a driving recommendation.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.