Civil Engineering

Horizontal Curve Calculator

Set out a simple circular curve between two straights. Enter the radius and the deflection angle and this calculator returns the curve length, the tangent length back to the intersection point, the long chord, the mid-ordinate, the external distance and the degree of curvature. It also checks the radius against the minimum needed for your design speed, superelevation and side friction factor.

Simple circular curve · IRC 38 minimum radius · 20 m arc degree of curvature

Working on the profile rather than the plan? The vertical curve calculator covers crest and sag curves, grade difference, K value and the sight-distance curve length.

🗺 Curve Geometry and Design Speed

Inputs convert when you switch. Results are always reported in SI — see the note under the results.

Radius R (m)

Deflection Angle Δ (degrees)

The angle between the two straights, not the interior angle.

Design Speed V (km/h)

Superelevation e (%)

IRC caps this at 7 % on plain terrain, 10 % in hills.

Side Friction Factor f

IRC uses 0.15 for design.

📊 Curve Elements

Press Calculate Curve to compute the curve.

ƒ Governing Formulae

Curve elements from radius and deflection angle

L_c = R Δ (Δ in radians)
T = R tan(Δ/2)
C = 2R sin(Δ/2)
M = R [1 − cos(Δ/2)]
E = R [sec(Δ/2) − 1]
D = 1145.92 / R (20 m arc)

Minimum radius for the design speed

R_min = V² / [127 (e + f)]
V in km/h, e as a decimal, R_min in metres

R — radius of the circular curve

Δ — deflection angle between the two straights

L_c — length of curve, measured along the arc

T — tangent length, from the tangent point to the intersection point

C — long chord, straight line between the two tangent points

M — mid-ordinate, from the middle of the long chord out to the curve

E — external distance, from the curve out to the intersection point

D — degree of curvature, the angle subtended by a 20 m arc

e — superelevation, the crossfall banking the road into the curve

f — coefficient of side friction developed between tyre and road

127 — combines the km/h to m/s conversion with g = 9.81 m/s²

The degree of curvature here uses a 20 metre arc. That is Indian and general metric practice, giving D = 1145.92/R. North American practice uses a 100 foot arc, D = 5729.58/R with R in feet, and railway work sometimes uses the chord definition rather than the arc definition. The three give different numbers for the same curve, so check which convention a drawing is using before comparing.

This is a simple circular curve only. There is no transition or spiral curve — no clothoid length, no shift, no offset to the transition. A real road curve at speed needs a transition at each end so that superelevation and curvature build up gradually, and the tangent points move as a result. There is also no widening on curves, no check on sight distance around the inside of the curve, and no compound or reverse curve geometry. Use this for the circular portion and add the transition separately.

R_min is the absolute minimum, not a design target. It is the radius at which the full assumed side friction is being used up, leaving nothing in reserve. Design practice is to use a comfortably larger radius wherever the terrain allows, and IRC tabulates both an absolute minimum and a desirable minimum for each speed. A curve that only just passes this check is one where every vehicle is relying on the tyre friction assumption being right.

📝 Worked Example 1 — 250 m Curve at 80 km/h

Given

Two straights meeting at a deflection angle of 35°, to be joined by a circular curve of 250 m radius. Design speed 80 km/h, superelevation 7 percent, side friction factor 0.15.

Step 1 — curve and tangent lengths

Δ = 35° = 0.610865 rad, so Δ/2 = 17.5°
L_c = 250 × 0.610865 = 152.72 m
T = 250 × tan 17.5° = 250 × 0.315299 = 78.82 m
The tangent points therefore sit 78.82 m back from the intersection point along each straight.

Step 2 — chord and offsets

C = 2 × 250 × sin 17.5° = 500 × 0.300706 = 150.353 m
M = 250 × (1 − cos 17.5°) = 250 × 0.046283 = 11.571 m
E = 250 × (sec 17.5° − 1) = 250 × 0.048527 = 12.132 m
Note that the curve length exceeds the long chord by only 2.4 m over 150 m — a gentle curve.

Step 3 — degree of curvature

D = 1145.92 / 250 = 4.5837° per 20 m of arc.

Step 4 — the speed check

R_min = 80² / [127 × (0.07 + 0.15)] = 6400 / 27.94 = 229.1 m
The 250 m radius provided exceeds it, so the curve is safe at 80 km/h — though with only 9 percent in hand.

Curve length L_c 152.72 m · Tangent length T 78.82 m

Long chord C 150.353 m

Mid-ordinate M 11.571 m · External distance E 12.132 m

Degree of curvature 4.5837° per 20 m arc

R_min 229.1 m against 250 m provided — PASS

These are the calculator's default inputs. Press Calculate Curve without changing anything and you should get exactly these figures back. The tangent length is the number you need first on site: it fixes where the curve begins relative to the intersection point you have already pegged.

⚠ Worked Example 2 — The Same Curve, 20 km/h Faster

Given

Exactly the same 250 m curve, same 35° deflection, same 7 percent superelevation and 0.15 friction factor. The only change is that the design speed is raised from 80 to 100 km/h.

The geometry does not move at all

L_c = 152.72 m, T = 78.82 m, C = 150.353 m, M = 11.571 m, E = 12.132 m, D = 4.5837°.
Every setting-out dimension is identical, because none of them depends on speed.

And the curve is no longer safe

R_min = 100² / [127 × 0.22] = 10 000 / 27.94 = 357.9 m
Against the 250 m provided, that is a shortfall of 108 m, or 30 percent.

At 80 km/h R_min = 229.1 m — 250 m passes

At 100 km/h R_min = 357.9 m — 250 m FAILS

A 25 percent increase in speed demands a 56 percent larger radius

Why the penalty is so steep

Minimum radius goes with the square of speed, so raising the design speed by a quarter multiplies the required radius by 1.25² = 1.5625. Superelevation cannot rescue it either: even at the 10 percent hill-road maximum, R_min would still be 315 m. The only way to run 100 km/h through this deflection is a flatter curve, which means moving the intersection point or accepting a longer alignment. This is the calculation that decides route corridors, and it is why speed limits and geometry have to be settled together rather than one after the other.

📐 Inputs, Units and Accepted Ranges

Input	SI unit	Imperial unit	Accepted	Notes
Radius R	m	ft	> 0	Radius of the circular arc
Deflection angle Δ	degrees	degrees	> 0 and < 180	At 180° the straights are parallel and never meet
Design speed V	km/h	mph	> 0	Drives the minimum radius only
Superelevation e	%	%	0 – 12, IRC caps at 7 or 10	Entered as a percentage, used as a decimal
Side friction f	—	—	0 – 0.5, IRC uses 0.15	e + f must be greater than zero

📚 Reference Tables

Minimum radius by design speed — f = 0.15, computed as V²/[127(e+f)]

V, km/h	R_min at e = 7 %	R_min at e = 10 %	R_min at e = 4 %
30	32.2 m	28.3 m	37.3 m
40	57.3 m	50.4 m	66.3 m
50	89.5 m	78.7 m	103.6 m
60	128.9 m	113.4 m	149.2 m
65	151.3 m	133.1 m	175.1 m
80	229.1 m	201.6 m	265.2 m
100	357.9 m	314.9 m	414.4 m
120	515.4 m	453.5 m	596.7 m

These are absolute minima. IRC also publishes a desirable minimum, typically 40 to 60 percent larger, and using the absolute value should be reserved for genuinely constrained locations.

Degree of curvature and radius — 20 m arc definition, D = 1145.92/R

R, m	D, degrees	Character	Suitable up to
50	22.92	Very sharp	~35 km/h
100	11.46	Sharp	~50 km/h
150	7.64	Moderate	~65 km/h
250	4.58	Gentle	~83 km/h
400	2.86	Flat	~105 km/h
600	1.91	Very flat	~129 km/h
1000	1.15	Near straight	~166 km/h

Speeds shown are the maximum for which each radius meets the absolute minimum at e = 7 % and f = 0.15.

How the curve elements scale with deflection angle — R = 250 m throughout

Δ, degrees	L_c, m	T, m	C, m	M, m	E, m
10	43.63	21.87	43.578	0.951	0.955
20	87.27	44.08	86.824	3.793	3.851
35	152.72	78.82	150.353	11.571	12.132
60	261.80	144.34	250.000	33.494	38.675
90	392.70	250.00	353.553	73.223	103.553
120	523.60	433.01	433.013	125.000	250.000

Mid-ordinate and external distance are nearly equal at small angles and diverge sharply beyond about 60°. At 120° the external distance is twice the mid-ordinate, and the tangent length has grown to nearly twice the radius.

IRC superelevation limits

Terrain	Maximum e	Reason for the cap
Plain and rolling	7 %	Slow vehicles sliding inward
Hilly, snow free	10 %	Steeper accepted where radii are constrained
Hilly, snow bound	7 %	Reduced friction under ice
Urban, built up	4 %	Wide range of vehicle speeds, frequent stops

❓ Frequently Asked Questions

How do you calculate the elements of a horizontal circular curve?

Every element follows from the radius and the deflection angle between the two straights. Curve length is the radius times the deflection angle in radians. Tangent length, the distance from the intersection point back to the start of the curve, is the radius times the tangent of half the deflection angle. The long chord is twice the radius times the sine of half the angle, the mid-ordinate is the radius times one minus the cosine of half the angle, and the external distance is the radius times the secant of half the angle minus one.

What is the minimum radius of a horizontal curve?

The minimum radius is the design speed squared divided by 127 times the sum of the superelevation and the coefficient of side friction, with speed in kilometres per hour and superelevation as a decimal. The constant 127 comes from converting kilometres per hour to metres per second and dividing by gravitational acceleration. At 80 kilometres per hour with 7 percent superelevation and a friction coefficient of 0.15 the minimum radius is 229 metres.

What is the degree of curvature and how does it relate to radius?

Degree of curvature is the angle subtended at the centre by a standard length of arc, and it is simply another way of expressing sharpness. Indian and metric practice uses a 20 metre arc, so the degree equals 1145.92 divided by the radius in metres. North American practice uses a 100 foot arc, giving 5729.58 divided by the radius in feet. A larger degree means a sharper curve, which is the opposite convention to radius.

What is the difference between mid-ordinate and external distance?

Both measure how far the curve departs from the straight lines, but in opposite directions from the mid-point of the curve. The mid-ordinate is the distance from the middle of the long chord out to the curve, and it tells you how much clearance the curve needs inside the chord. The external distance is measured from the curve outward to the point of intersection of the two tangents, and it tells you how much land is needed beyond the curve. For small deflection angles the two are nearly equal; they diverge as the angle grows.

Why is superelevation limited to about 7 percent?

Superelevation is capped because a slow moving or stationary vehicle on a steeply banked curve tends to slide inward, and heavy vehicles with a high centre of gravity risk toppling toward the inside. IRC limits superelevation to 7 percent on plain and rolling terrain and 10 percent in hilly areas without snow, with 4 percent in urban areas where speeds vary widely. Once that cap is reached, any further increase in the safe radius has to come from a flatter curve rather than more banking.

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Horizontal Curve Calculator — multicalci.com. Simple circular curve only — no transition or spiral, no curve widening, no sight distance check around the curve, and no compound or reverse curve geometry. The minimum radius is an absolute limit rather than a design target. Results are indicative and must be verified against IRC 38 or the relevant highway standard.