10.1 Circular Curve Elements

Horizontal circular curves turn a route alignment from one tangent to the next so the finished centerline does not kink where those tangents meet. Domain III.E of the 2022 Civil Engineering Surveying (CES) test plan from the California Board for Professional Engineers, Land Surveyors, and Geologists (BPELSG) asks you to calculate radius, curve length, and tangent length, then extend the same elements into compound and reverse geometry. This section is the simple-curve toolkit those later geometries reuse. California street, highway, and most civil site alignments use a simple circular curve—constant radius—between two tangents.

Circular-curve geometry

Why the PI is off the centerline

The incoming and outgoing tangents, extended, meet at the point of intersection (PI). Construction never occupies the PI. The curve is a circular arc of radius R about center O. The arc starts at the point of curvature (PC), where radius OPC is perpendicular to the back tangent, and ends at the point of tangency (PT), where radius OPT is perpendicular to the forward tangent. The central angle at O between those radii is the intersection angle Δ (delta). That same Δ is the deflection between the two tangents.

Degree of curve: arc definition vs chord definition

Sharpness is often stated as degree of curve D instead of R.

Arc definition (US highway, Caltrans, typical California civil work): D is the central angle in degrees subtended by 100 ft of arc:

D = 5729.58 / R, with R in feet.

The constant is 18,000/π rounded to 5729.58. Inverse: R = 5729.58 / D.

Chord definition (railroad and some older transit practice): D_c is the central angle subtended by a 100-ft chord:

R = 50 / sin(D_c / 2).

California highway and street problems on CES use the arc definition. Use the chord definition only when the stem is clearly railroad or it specifies a 100-ft chord. Do not mix definitions in one solution.

With arc D known, arc length is the 100-ft proportion L = 100 Δ / D (Δ and D in degrees). That identity is the same as L = R Δ_rad = R Δ π / 180.

Formulas for the six elements

Keep Δ in degrees for the trig of Δ/2, and convert to radians only inside L = R Δ_rad. CES lets you bring two handheld calculators; a degree/radian mode error is a full miss.

T is a tangent distance, not an arc. LC is always shorter than L. E is always larger than M on the same simple curve.

Worked complete set: R = 800 ft, Δ = 36°00'

Δ/2 = 18°00'. Calculator in degrees.

• tan 18° = 0.3249197, so T = 800 × 0.3249197 = 259.94 ft
• L = 800 × 36 × π / 180 = 160π = 502.65 ft
• sin 18° = 0.3090170, so LC = 1,600 × 0.3090170 = 494.43 ft
• cos 18° = 0.9510565, so M = 800 (1 − 0.9510565) = 39.15 ft
• E = 800 (1/0.9510565 − 1) = 800 × 0.0514622 = 41.17 ft

Arc degree of curve: D = 5729.58 / 800 = 7.162°. Check with the 100-ft rule: L = 100 × 36 / 7.162 = 502.65 ft (matches).

Railroad chord-definition D_c for the same 800-ft radius: sin(D_c/2) = 50/800 = 0.0625, so D_c/2 = 3.5833° and D_c = 7.167°. On this flat curve the two D values differ by only 0.005°, but they are still different definitions. A California highway alignment with R = 800 ft is D = 7.162°, not 7.167°. Using the railroad D in L = 100 Δ/D is the wrong constant for Caltrans-style highway work.

Checks that catch arithmetic errors: LC (494.43 ft) < L (502.65 ft); E (41.17 ft) > M (39.15 ft); T (259.94 ft) is longer than E, as it must be, because T runs all the way to the PI while E is the short external from the PI to the curve.

Inverse: given T and Δ, find R

T = R tan(Δ/2) rearranges to R = T / tan(Δ/2).

Worked inverse: T = 320.00 ft, Δ = 40°00'. Δ/2 = 20°00', tan 20° = 0.3639702. R = 320.00 / 0.3639702 = 879.19 ft.

Then L = 879.19 × 40 × π / 180 = 613.79 ft, and D = 5729.58 / 879.19 = 6.517°. If a stem gives T and R instead, Δ = 2 arctan(T/R). If it gives L and R, Δ° = 180 L / (π R). If it gives L and arc D, Δ = L D / 100.

Exam traps on simple-curve elements

1. Radian mode. tan(18) in radians is not tan 18°.
2. L = R Δ with Δ left in degrees. 800 × 36 = 28,800 ft is not a curve length.
3. Chord D on a highway radius. Streets and Caltrans highway alignments use D = 5729.58 / R.
4. Swapping M and E. M is from the long chord; E is from the PI.
5. Calling T the stationing distance from PC to PT. Stationing uses L.

A stem that gives the long chord and Δ still inverts LC = 2 R sin(Δ/2) for R, then L = R Δ π/180. This section's complete set starts from R and Δ (and the inverse from T and Δ) so you compute the elements rather than reuse a retired numeric stem.