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Civil & Surveying formula solvers

Percent Grade from Rise and Run

G=100 ΔhLG = \frac{100\,\Delta h}{L}

Civil & SurveyingGeometryPercent grade is one hundred times the vertical rise divided by the horizontal run, measured level, never along the slope.

Grade to Slope Angle

θ=arctan⁡ ⁣(G100)\theta = \arctan\!\left(\frac{G}{100}\right)

Civil & SurveyingTrigonometryConverts a percent grade into the slope angle measured from horizontal, and back again with the tangent function.

Slope Ratio (H:V) to Percent Grade

G=100nG = \frac{100}{n}

Civil & SurveyingSoil MechanicsConverts an embankment slope quoted as n horizontal to one vertical into the equivalent percent grade, and back.

Elevation from Grade and Distance

E2=E1+G L100E_2 = E_1 + \frac{G\,L}{100}

Civil & SurveyingGeometryProjects an elevation along a uniform grade, adding the rise over a measured horizontal distance to the known starting elevation.

Earthwork Volume by Average End Area

V=L (A1+A2)2V = \frac{L\,(A_1 + A_2)}{2}

Civil & SurveyingGeometryVolume between two cross sections, averaging their end areas over the distance between them; the answer is reported in cubic yards.

Earthwork Volume by the Prismoidal Formula

V=L (A1+4Am+A2)6V = \frac{L\,(A_1 + 4A_m + A_2)}{6}

Civil & SurveyingGeometrySimpson's rule applied to earthwork, weighting the middle cross section four times the ends; the answer is reported in cubic yards.

Stockpile Volume (Truncated Pyramid)

V=h3(A1+A2+A1A2)V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)

Civil & SurveyingGeometryVolume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer is reported in cubic yards.

Swell: Loose Volume from Bank Volume

VL=VB(1+S100)V_L = V_B\left(1 + \frac{S}{100}\right)

Civil & SurveyingSoil MechanicsConverts undisturbed bank volume into the loose volume the same soil occupies once excavated, using its percent swell.

Shrinkage: Compacted Volume from Bank Volume

VC=VB(1−Sh100)V_C = V_B\left(1 - \frac{S_h}{100}\right)

Civil & SurveyingSoil MechanicsConverts bank volume into the compacted volume it fills in an engineered embankment, using the soil's percent shrinkage.

Concrete Volume with Waste Allowance

V=L W T(1+w100)V = L\,W\,T\left(1 + \frac{w}{100}\right)

Civil & SurveyingGeometryOrders concrete for a rectangular slab, footing or column by adding a percent waste allowance to the neat volume, in cubic yards.

Asphalt Tonnage from Area and Thickness

M=A t ρM = A\,t\,\rho

Civil & SurveyingConverts a paving area, lift thickness and compacted mix density into the tonnage to order, reported in US short tons.

Radius from Degree of Curve (Arc Definition)

R=5729.578DR = \frac{5729.578}{D}

Civil & SurveyingGeometryConverts degree of curve to radius using the arc definition, where D is the central angle subtending one 100 ft station of arc.

Horizontal Curve Tangent Length

T=Rtan⁡Δ2T = R\tan\frac{\Delta}{2}

Civil & SurveyingTrigonometryDistance from the point of intersection back to the point of curvature, from the curve radius and its total deflection angle.

Horizontal Curve Length from Degree of Curve

L=100 ΔDL = \frac{100\,\Delta}{D}

Civil & SurveyingGeometryLength of a circular curve in 100 ft stations, from the total deflection angle and the degree of curve, on the arc definition.

Horizontal Curve External Distance

E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)

Civil & SurveyingTrigonometryClearance from the point of intersection to the midpoint of the curve, the distance a curve cuts back from the corner.

Horizontal Curve Middle Ordinate

M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)

Civil & SurveyingTrigonometryOffset from the middle of the long chord to the middle of the arc, the number that governs sight distance around obstructions.

Horizontal Curve Long Chord

C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}

Civil & SurveyingTrigonometryStraight-line distance from the point of curvature to the point of tangency, the chord that spans the entire circular curve.

Vertical Curve Length from K Value

L=K AL = K\,A

Civil & SurveyingLength of a parabolic vertical curve as the design K value times the algebraic grade change, K being length per percent of grade.

Elevation on a Parabolic Vertical Curve

E=EBVC+g1x100+A x2200 LE = E_{BVC} + \frac{g_1 x}{100} + \frac{A\,x^{2}}{200\,L}

Civil & SurveyingGeometryElevation at any station on an equal-tangent parabolic vertical curve, measured from the beginning of vertical curve.

High or Low Point on a Vertical Curve

xt=−g1LAx_t = -\frac{g_1 L}{A}

Civil & SurveyingDistance from the BVC to the crest or sag of a parabolic vertical curve, where the profile grade passes through zero.

Crest Vertical Curve Length for Sight Distance

L=A S2200(h1+h2)2L = \frac{A\,S^{2}}{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}

Civil & SurveyingGeometryCrest curve length needed to see an object over the hill, for the case where sight distance is shorter than the curve.

Stopping Sight Distance

d=v tr+v22ad = v\,t_r + \frac{v^{2}}{2a}

Civil & SurveyingDistance to stop from a design speed, adding the reaction distance travelled before braking to the braking distance itself.

Superelevation Rate for a Horizontal Curve

e100+f=v2g R\frac{e}{100} + f = \frac{v^{2}}{g\,R}

Civil & SurveyingBalances banking and side friction against centripetal demand on a horizontal curve, using standard gravity of 9.80665 m/s².

Back Azimuth

αb=α±180∘\alpha_b = \alpha \pm 180^{\circ}

Civil & SurveyingTrigonometryReverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside the zero to three sixty range.

Latitude of a Traverse Leg

Lat=Lcos⁡α\text{Lat} = L\cos\alpha

Civil & SurveyingTrigonometryNorth-south component of a traverse course, from the measured slope-corrected distance and the azimuth of the line.

Departure of a Traverse Leg

Dep=Lsin⁡α\text{Dep} = L\sin\alpha

Civil & SurveyingTrigonometryEast-west component of a traverse course, taken from the measured horizontal distance and the azimuth of the line.

Traverse Closure Error

Ec=(ΣLat)2+(ΣDep)2E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}

Civil & SurveyingGeometryLinear misclosure of a closed traverse, combining the residual sums of the latitudes and departures as a right triangle.

Traverse Precision Ratio

N=PEcN = \frac{P}{E_c}

Civil & SurveyingExpresses traverse accuracy as one part in N, dividing the total perimeter walked by the linear closure error.

Area of a Three-Sided Parcel by Coordinates

A=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]

Civil & SurveyingGeometryShoelace area of a parcel from the coordinates of its three corners, listed counter-clockwise so the result comes out positive.

Differential Levelling Elevation

E2=E1+BS−FSE_2 = E_1 + \text{BS} - \text{FS}

Civil & SurveyingCarries an elevation forward one instrument setup, adding the backsight to the known point and subtracting the foresight.

Stadia Distance from Rod Intercept

D=K s+CD = K\,s + C

Civil & SurveyingHorizontal distance from a stadia rod intercept, with the instrument's stadia interval factor and additive constant.