---
title: "Elements of simple circular curve"
subject: "Surveying II"
chapter: 6
key: "semester_4/surveying_ii:simple-circular-curve-elements"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/simple-circular-curve-elements
site: BCT Engineering
---

# Elements of simple circular curve

Surveying II · Chapter 6: Curves

- Asked in: 2069 Bhadra, 4 marks
- Video: [Elements of Simple Circular Highway Curve (Notations of Simple Circular Curve)](https://www.youtube.com/watch?v=Tqir3VmwrE4) — CIVIL Solved
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/simple-circular-curve-elements
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-6

## Question

Find the elements of simple circular curve.

## Short answer

Elements: Tangent length T = R tan(Δ/2), curve length L = πRΔ/180, long chord LC = 2R sin(Δ/2), external distance E = R(sec(Δ/2)-1), mid-ordinate M = R(1-cos(Δ/2)). Chainage of T1 (PC) = IP - T, chainage of T2 (PT) = T1 + L.

## Answer

A simple circular curve is a single arc of constant radius that connects two straights (tangents) meeting at the point of intersection (IP).

```text
              V (IP)
             /\
          T /  \ T
           / Δ  \
      T1  /      \  T2
       o'--------'o
         \ M  E /
          \ __ /
          R \  / R
             O
```

With $R$ = radius and $\Delta$ = deflection angle (angle between the forward extension of the first tangent and the second tangent), the elements are:

| Element | Formula |
|---|---|
| Tangent length (T₁V = VT₂) | $T = R\tan\dfrac{\Delta}{2}$ |
| Length of curve | $L = \dfrac{\pi R\Delta}{180}$ |
| Long chord (T₁T₂) | $LC = 2R\sin\dfrac{\Delta}{2}$ |
| External distance (apex distance) | $E = R\left(\sec\dfrac{\Delta}{2}-1\right)$ |
| Versed sine (mid-ordinate) | $M = R\left(1-\cos\dfrac{\Delta}{2}\right)$ |
| Chainage of T₁ | IP chainage − $T$ |
| Chainage of T₂ | Chainage of T₁ + $L$ |
| Angle at centre | $\Delta$ (equal to the deflection angle) |

Other terms: T₁ is the point of curve (PC) and T₂ the point of tangency (PT); the interior (intersection) angle is $180^\circ - \Delta$; the degree of curve is $D = 1718.87/R$ for a 30 m arc.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the tangent length T of a simple curve?
   A. R tan(Delta/2)
   B. 2R sin(Delta/2)
   C. R sin(Delta/2)
   D. R(sec(Delta/2) - 1)
   Answer: A. T = R tan(Delta/2), the distance from IP to each tangent point. (Key point: Tangent length)

2. Which formula is for the long chord?
   A. pi R Delta/180
   B. R(1 - cos(Delta/2))
   C. R tan(Delta/2)
   D. 2R sin(Delta/2)
   Answer: D. LC = 2R sin(Delta/2). (Key point: Long chord)

3. What is the external distance E?
   A. R sin(Delta/2)
   B. R tan(Delta/2)
   C. R(1 - cos(Delta/2))
   D. R(sec(Delta/2) - 1)
   Answer: D. E is IP to mid-point of the curve; the versed sine M = R(1 - cos(Delta/2)) is from the long chord. (Key point: External distance)

4. How is chainage of T2 found?
   A. IP - L
   B. T1 + T
   C. IP + T
   D. T1 + L
   Answer: D. Chainage T1 = IP - T and T2 = T1 + L, using curve length rather than tangent length. (Key point: Chainage of T2)

## Related in Surveying II

- [Degree of curve, elements of simple curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Adegree-of-curve-elements-simple-curve) — Surveying II, chapter 6
- [Simple circular curve by deflection angle method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asimple-curve-deflection-angle-method-sketch) — Surveying II, chapter 6
- [Simple curve by offsets from long chord](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asimple-curve-offsets-long-chord) — Surveying II, chapter 6
- [Setting out simple curve by Rankine's method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asimple-curve-rankine-method-sketch) — Surveying II, chapter 6
- [Simple curve data, roads intersecting at 150°](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asimple-curve-two-roads-150-4deg) — Surveying II, chapter 6
- [Elements of transition curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atransition-curve-elements) — Surveying II, chapter 6

## Related in other subjects

- [Simple circular curve elements and chainages](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-64s-simple-curve) — Transportation Engineering I, chapter 3
- [Car on circular curve, angular rate](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Acar-circular-curve-constant-speed) — Applied Mechanics (Dynamics), chapter 1