Surveying Formulas

Surveying formulas for all the concepts covered under different courses, B.Tech, Diploma, GATE, SSC, or Other competitive exams, are provided here by our subject matter expert. To solve the surveying problems easily, students should learn and remember the basic formulas based on certain fundamentals such as slope, distance, angle, and geometry.

We present you with a host of formulas (more than 60) for your reference to solve all important surveying operations and questions. 

List of Surveying Formulas (for All Concepts)

Distance and Angles:

1. Distance between two points:

• Euclidean distance: \(d = \sqrt{{(x_2 – x_1)^2 + (y_2 – y_1)^2}}\)
• Horizontal distance: \(d_h = d \cdot \cos(\theta)\)
• Slope distance: \(d_s = \sqrt{{d_h^2 + \Delta h^2}}\)

2. Angle between two lines:

• Interior angle: \(\theta = \arctan\left(\frac{{y_2 – y_1}}{{x_2 – x_1}}\right)\)
• Bearing angle: \(\theta = \arctan\left(\frac{{E}}{{N}}\right)\)

3. Vertical angles:

• Zenith angle: \(\theta_z = 90^\circ – \alpha\)
• Elevation angle: \(\theta_e = 90^\circ – \theta_z\)

4. Slope between two points:

• Slope = \(\frac{{y_2 – y_1}}{{x_2 – x_1}}\)

5. Vertical angle correction:

• Vertical angle correction = \(\frac{{\Delta h}}{{d}}\)

6. Degree to radians conversion:

• \(\text{{Radians}} = \frac{{\pi}}{{180}} \times \text{{Degree}}\)

Traversing:

1. Closing error:

• Angular misclosure: \(\Delta\theta = \sum{\theta_i} – 360^\circ\)
• Linear misclosure: \(\Delta L = \sum{L_i} – L_{\text{closing}}\)

2. Adjusted coordinates:

• Latitude correction: \(\Delta X_i = -\frac{{\Delta L \cdot \sin(\theta_i)}}{{L_{\text{closing}}}}\)
• Departure correction: \(\Delta Y_i = -\frac{{\Delta L \cdot \cos(\theta_i)}}{{L_{\text{closing}}}}\)
• Adjusted coordinates: \(X_i’ = X_i + \Delta X_i\) and \(Y_i’ = Y_i + \Delta Y_i\)

3. Adjusted angles in a closed traverse:

• Adjusted angle = \(\frac{{\text{{Angular misclosure}}}}{{\text{{Number of angles}}}}\)

4. Adjusted bearings in a closed traverse:

• Adjusted bearing = \(\text{{Initial bearing}} + \text{{Adjusted angle}}\)

5. Angular misclosure adjustment:

• Adjusted angle = \(\frac{{\text{{Angular misclosure}}}}{{\text{{Number of stations}}}}\)

Also read: Traverse Surveying – Definition, Types, Methods, Checks

Leveling:

1. Height of instrument (HI):

• HI = BM + Backsight (BS) – Foresight (FS)

2. Differential leveling:

• Elevation of a point: \(E = HI + BS – FS\)
• Change in elevation: \(\Delta E = BS – FS\)

3. Profile leveling:

• Gradient: \(G = \frac{{\Delta E}}{{\text{{Horizontal distance}}}}\)
• Rate of vertical curvature: \(R = \frac{{\Delta E}}{{\Delta L}}\)

4. Rise and fall method:

• Rise or fall = \(\text{{BS}} – \text{{FS}}\)

5. Reciprocal leveling:

• Height of Instrument (HI) = \(\text{{BM}} + \text{{FS}} – \text{{BS}}\)

6. Backsight correction:

• Corrected backsight (BS) = Backsight (BS) + Backsight correction (BC)

7. Foresight correction:

• Corrected foresight (FS) = Foresight (FS) + Foresight correction (FC)

Also read: 5 Types of Levelling Instruments used in Surveying

Area Calculation:

1. Trapezoidal rule:

• Area = \(\frac{{(a + b)}}{2} \times h\)

2. Trapezoidal rule with unequal intervals:

• Area = \(\frac{{h}}{2} \times (y_0 + y_1 + 2(y_2 + y_3 + \ldots + y_{n-2}) + y_{n-1})\)

3. Simpson’s rule:

• Area = \(\frac{{h}}{3} \times (y_0 + 4y_1 + 2y_2 + 4y_3 + \ldots + 2y_{n-2} + 4y_{n-1} + y_n)\)

4. Simpson’s rule with unequal intervals:

• Area = \(\frac{{h}}{3} \times (y_0 + y_n + 4(y_1 + y_3 + \ldots + y_{n-1}) + 2(y_2 + y_4 + \ldots + y_{n-2}))\)

5. Trapezoidal rule with end areas:

• Area = \(\frac{{A_1 + A_2}}{2} \times h\)

Coordinate Geometry:

1. Equation of a straight line:

• Slope-intercept form: \(y = mx + c\)
• Point-slope form: \((y – y_1) = m(x – x_1)\)

2. Midpoint of a line segment:

• Midpoint coordinates: \((x_m, y_m) = \left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\)

3. Distance from a point to a line:

• Distance = \(\frac{{\left| A \cdot x_0 + B \cdot y_0 + C \right|}}{{\sqrt{{A^2 + B^2}}}}\)

4. Equation of a circle:

• Standard form: \((x – h)^2 + (y – k)^2 = r^2\)

5. Distance from a point to a plane:

• Distance = \(\frac{{Ax + By + Cz + D}}{{\sqrt{{A^2 + B^2 + C^2}}}}\)

Curves:

1. Circular curve:

• Radius of circular curve: \(R = \frac{{(L_s)^2}}{24S}\)
• Degree of curve: \(D = \frac{{180L_s}}{{\pi R}}\)

2. Spiral curve:

• Length of spiral: \(L = A \cdot R\)
• Tangent distance: \(T = B \cdot R\)

3. Parabolic curve:

• Vertex coordinates: \((h, k) = \left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\)

4. Compound curve:

• Tangent distance between curves: \(T = \frac{{R_2 \cdot L_1}}{{R_1 + R_2}}\)

5. Spiral curve tangent length:

• Tangent length = \(T = \frac{{A}}{B}\)

Triangulation:

1. Baseline length:

• Baseline length: \(B = \sqrt{{(X_2 – X_1)^2 + (Y_2 – Y_1)^2}}\)

2. Resection:

• Resection coordinates: \(X = \frac{{\sum{(W_i \cdot X_i)}}}{{\sum{W_i}}}\) and \(Y = \frac{{\sum{(W_i \cdot Y_i)}}}{{\sum{W_i}}}\)

3. Triangulation angle:

• Angle between two lines: \(\theta = \arctan\left(\frac{{\Delta Y}}{{\Delta X}}\right)\)

4. Resection coordinates with weights:

• \(X = \frac{{\sum{(W_i \cdot X_i)}}}{{\sum{W_i}}}\)
• \(Y = \frac{{\sum{(W_i \cdot Y_i)}}}{{\sum{W_i}}}\)

Circumference and Area:

1. Circle:

• Circumference of Circle: \(C = 2\pi r\) or \(C = \pi d\)
• Area of Circle: \(A = \pi r^2\)

2. Ellipse:

• Circumference of Ellipse (approximate): \(C \approx \pi (a + b)\)
• Area of Ellipse: \(A = \pi ab\)

Basic Surveying Formulas

1. Distance between two points:
\[d = \sqrt{{(E_2 – E_1)^2 + (N_2 – N_1)^2}}\] where \(E\) and \(N\) represent the Easting and Northing coordinates of the points.

2. Slope distance:
\[SD = \sqrt{{d^2 + h^2}}\] where \(d\) is the horizontal distance and \(h\) is the vertical distance.

3. Bearing between two points:
\[\text{Bearing} = \arctan{\left(\frac{{E_2 – E_1}}{{N_2 – N_1}}\right)}\]

4. Horizontal distance from angle and slope distance:
\[d = SD \cdot \cos{\theta}\] where \(\theta\) is the horizontal angle.

5. Vertical distance from angle and slope distance:
\[h = SD \cdot \sin{\theta}\] where \(\theta\) is the vertical angle.

6. Radius of curvature:
\[R = \frac{{(SD)^2}}{{2h}}\]

7. Earth’s curvature correction:
\[C = \frac{{(SD)^2}}{{2R}}\]

8. Elevation difference between two points:
\[dH = N_2 – N_1\] where \(N\) represents the elevations of the points.

9. Slope (gradient):
\[S = \frac{{\text{{Rise}}}}{{\text{{Run}}}}\] where “Rise” represents the vertical change and “Run” represents the horizontal change.

10. Horizontal distance from slope and vertical distance:
\[d = \sqrt{{\text{{Run}}^2 + \text{{Rise}}^2}}\]

3. Vertical angle from slope and horizontal distance:
\[\theta = \arctan{\left(\frac{{\text{{Rise}}}}{{\text{{Run}}}}\right)}\]

11. Station offset:
\[O = \text{{Offset}} \cdot \cos{\theta}\] where “Offset” represents the perpendicular distance from the line of sight and \(\theta\) is the horizontal angle.

12. Horizontal distance from offset and slope distance:
\[d = \sqrt{{\text{{Offset}}^2 + \text{{SD}}^2}}\]

13. Area of a triangle:
\[A = \frac{{1}}{{2}} \cdot \text{{Base}} \cdot \text{{Height}}\] where “Base” is the length of the base of the triangle and “Height” is the perpendicular distance from the base to the opposite vertex.

14. Vertical exaggeration:
\[VE = \frac{{\text{{Vertical scale}}}}{{\text{{Horizontal scale}}}}\] where the scales represent the ratio of the graphical representation of distances in the vertical and horizontal directions.

15. Leveling correction:
\[LC = B – F\] where “B” is the back (higher) sight and “F” is the fore (lower) sight.

16. Vertical curvature correction:
\[VC = \frac{{(SD)^2}}{{2R_v}}\] where \(SD\) is the slope distance and \(R_v\) is the radius of vertical curvature.

17. Combined horizontal and vertical distance:
\[CH = \sqrt{{d_h^2 + d_v^2}}\] where \(d_h\) is the horizontal distance and \(d_v\) is the vertical distance.

18. Horizontal curve deflection angle:
\[\delta = \frac{{180 \times L}}{{\pi \times R_c}}\] where \(L\) is the length of the curve and \(R_c\) is the radius of the horizontal curve.

19. Elevation of a point on a vertical curve:
\[E = E_v + (D \cdot x) + \left(\frac{{D \cdot x^2}}{{2R_v}}\right)\] where \(E\) is the elevation, \(E_v\) is the elevation at the vertex, \(D\) is the algebraic difference in grade, \(x\) is the distance from the vertex, and \(R_v\) is the radius of vertical curvature.

20. Slope correction for horizontal distance:
\[d_c = d \cdot \cos(\theta)\] where \(d\) is the slope distance and \(\theta\) is the angle of slope.

21. Right triangle solution for missing side:
\[a^2 = b^2 + c^2\] where \(a\) is the missing side, and \(b\) and \(c\) are the other two sides of the right triangle.

22. Trigonometric leveling correction:
\[LC = F – B\] where \(LC\) is the leveling correction, \(F\) is the foresight reading, and \(B\) is the backsight reading.

23. Area of a polygon using coordinates:
\[A = \frac{1}{2} \sum_{i=0}^{n-1} (x_iy_{i+1} – x_{i+1}y_i)\] where \(n\) is the number of vertices, and \(x_i\) and \(y_i\) are the coordinates of the \(i\)-th vertex.

Also read: Rise and fall method with Examples

Frequently Asked Questions – FAQs