

Setting Out Lines and Arcs 
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Introduction 

This page leads a section of pages that look at how paving and drainage works are set out using basic geometry, mathematics and a few simple measuring and positioning tools. 
All paving and landscape projects need to be 'set out'  this is the process of using geometry to establish lines, levels, curves and arcs, as well as positioning other features in relation to a given point, typically a house, or other building.
On larger projects, automatic levels, theodolites, laser levels and even GPS are used to assist in the setting out, but there are some basic settingout skills that require nothing more technical than a string line, an accurate spirit level, a tape measure, a handful of marker pegs, preferably 12mm steel road pins, and occasionally, a calculator, all of which should be readily available to most diy'ers. 

To illustrate some of these basic principles used in setting out, the plan for a typical driveway shown opposite will be used as a case study. This is a fairly common layout scenario for a block paved driveway.
The brief is to create a drive 7 metres wide, which is to align with the left hand edge of the front door, with a curved sweep on the right hand side to accommodate the double width garage and a smaller curve on the left hand side to tie in with a 1.2m wide pathway along the front of the property. 

The drainage of the pavement is catered for by existing surface water gullies collecting from downspouts located on both front corners of the property. The general level of the site is flat, with no more than 50mm of fall from the paving level at the front of the house to the threshold with the public footpath. 
Setting out a perpendicularA perpendicular is a line that is at exactly 90° to a base line. They are essential to accurate setting out, and are constructed using one of the fundamental principals of building work, the 345 triangle. From Pythagoras' theorem, we can establish that any triangle that has side lengths in the ratio of 3:4:5 must be a right angled triangle. 

It doesn't matter if the side lengths are 3m, 4m and 5m, or 60 feet, 80 feet and 100 feet  as long as the 3:4:5 ratio is maintained, the triangle will have a right angle. With longer side lengths, however, accuracy of measurement can be become a problem, so we try to use a triangle with a maximum hypotenuse length of around 15 metres, although this is not always possible. With larger triangles, we will perform more check measurements to ensure accuracy.
Putting this knowledge to practical use, we can see from the construction plan for the driveway that the shape of the drive is constructed from a series of rectangles and arcs. The most important line to be setout is the lefthand edge of the driveway, line AB marked in red on the drawing, which is perpendicular to the front of the house, and is aligned with the lefthand edge of the doorway. To establish this perpendicular, we use a 345 triangle. 

The dimensions on the drawing states that the distance from the left hand edge of the front doorway (A), to the right hand corner of the property (C) is 7m + 3m = 10m. However, most landscape plans are drawn at a scale of 1:100 or greater, and will usually have an accuracy of around ±75mm. For our right angled triangle we need to be as exact as possible with our measurements to ensure we get a true right angle at 90°, and not at 88° or 91°, so a measure is made along the front of the house and a distance of 8m is marked with crayon or chalk at point D, just in front of the garage door.  
8m has been chosen as the length of the base line AC because it is a multiple of 4, the largest side of the 345 triangle other than the hypotenuse. The base line is regarded as true, and so the longest available edge, ie, the '4' always goes along the base line.  
From these two fixed points, A and D, we now mark out two arcs. These can be scratches on the ground, sand lines, spray paint, crayon or anything that can make a mark. From Point A we mark out a 6m arc, ie, the '3' side of the 345 triangle, and from Point D we mark out a 10m arc, ie, the '5', the hypotenuse. Where these two arcs intersect, Point E, marks the apex of a 345 triangle.  
If you have two tape measures, these can be used in place of arc marking, provided that you can anchor the tapes ends securely at points A and D. Pull the two tapes towards point E, and where the 6m mark on the tape from A meets the 10m mark on the tape from B, is exactly point E. Drive in a steel pin to mark the point.  
If we now draw a line from point A to point E, we have a line that is perpendicular to the front of the property, and aligned as required with the left hand edge of the doorway. A stake or steel pin is knocked into the ground a point E. A string line is securely fastened to point A, and pulled tight, past the marker pin at E and extended all the way to the threshold of the drive. As long as the line is just touching the marker pin at E, the perpendicular line created from the 345 triangle is extrapolated and point B can be established and marked with a steel pin.  
The accuracy of the perpendicular can be checked by measuring the hypotenuse BD. We know that AD = 8m, and we can measure that AB = 12m. Pythagoras tells us that the hypotenuse, BD is equal to the square root (SQRT) of (AD) ² + (AB) ²
BD = SQRT (8² + 12²)The distance from point B to point D is measured and checked against the calculation for accuracy. In this example, a measurement of between 14.40m and 14.45m will be satisfactory as a check. If the measure is outside this tolerance, check the perpendicular again, setting out an alternative 345 triangle from the base line, if necessary. 
Setting out arcs and curvesArcs and curves are much simpler to set out. Any arc has 3 important points, namely the start tangent point, the end tangent point and the origin. A tangent point is the point where an arc just touches or intersects a straight line or other arc. In the drawing below, Points C, F, and H are all tangent points. 

Referring back to our construction plan, we can see that two arcs are used on the right hand side of the driveway to make an 'S' curve. The 4.5m arc has its origin at G, and a tangent point with the building at C. The 3m arc has its origin at I, with a tangent point at H, where it meets the line that continues towards the public footpath. Where these two arcs 'kiss', at point F, is also a tangent point, common to both arcs.  
It should be noted that these two arcs are very simple, in that they are orthogonal, ie, their origins and major tangent points coincide with other construction lines that are 'square', ie true perpendiculars or parallels to the base line of the front of the house. In some situations, the origin of an arc might be inaccessible, for example, it could lie within the building itself, or be obscured by trees. In such cases, accurate setting out of arcs can be achieved by the use of chords (straight lines connecting two known points on an arc), or 'inverse arcs'. This is dealt with on the Setting Out Obstructed Arcs page.
To establish the 4.5m arc, a measurement is made along the face of the building, and the origin marked at 4.5m from C, ie at point G. The tape measure can now be anchored at point G, and used to scribe out the arc by swinging it around, from C towards H, marking in spray paint or similar. Establishing the 3m arc is a little more tricky. We can see that its origin, I, is located on a perpendicular from the building, aligned with the corner, C, and is 6 metres from the base line. Perpendiculars can be constructed, as detailed above, to locate the exact position of its origin, I, the tape then anchored at I and the resulting arc scribed in the same manner as the 4.5m arc. If all goes according to plan, the two arcs will 'kiss' at point F  the common tangent point. 

All that remains now, is to drive in marker pins at regular intervals along the arcs. The separation between marker pins will depend upon the radius of the arc. The larger the arc, the greater can be the separation between two marker pins. For these two arcs with radii of less than 5 metres, we would typically use marker pins at 600900mm separation.
On the plan opposite, the marker pins have been positioned at 750mm centre to centre (c/c) along the arcs. Once in position, a taut string line can be fastened to the pins to create a line guide for the laying of the arcs. 

Using the two basic methods outlined above, it is now possible to set out the whole driveway with marker pins and a taut string line. However, this only gives us a line guide, a guide to the shape of the driveway. Next, we have to establish levels, to ensure that the driveway is sloping (falling) in the right direction to drain away any surface water. The height of the taut string line above ground level is adjusted at the marker pins to give us a guide to both line and level.
See next page for details on setting levels 
It is essential for accuracy that the pins are driven into the ground deep enough to be secure. They must be absolutely plumb, ie, vertical, otherwise, when the string line level is adjusted up or down the pin, the position is amended as well as the height, causing an error in the layout.
As can be seen in the diagram opposite, which is a crosssectional view of the line, in green, and the pin, brown, when the pin is vertical, moving the line level up or down the pin makes no difference to position, whereas on the leaning pin, adjusting the line level results in the line position being displaced to the right or the left. 
SettingOut Pages 