Introduction
Pitches
Pitched
roofs
Any carpenter knows that the basis for roof framing is math. It is almost impossible to frame a roof without knowing any mathematics, especially trigonometry. It is needed, for example, to calculate the steepness of a roof  the roof pitch.
In Nordic climate, a pitched roof is preferred so that rain water would not be accumulated on the roof, which would happen in case of blocked water drainage with a flat roof. One of the main reasons for blocked drainages is the cold weather which causes blockages when the water turns into ice; other reasons include leaves, debris etc.
For this reason, throughout time, pitched roofs have been preferred
in Nordic countries.
Pictured below is an example of a flat roof and a pitched roof:
As we can see, the simplest and most common pitched roof is shaped as a single triangle. The situation becomes more
complicated when two or more pitched roofs are joined together. This creates a number of different slopes in one spot which need to be attuned together (constructed).

1.
Ridge 2.
Hip 3.
Gable 4.
Eave 5.
Rake edge 6.
Eave 
Source: OÜ Plekk –
Katusemeister
Here we can see the different angles and shapes than can occur on
a roof.
Pitched roofs can be in many different shapes, most common ones are gable and hipped roofs.
There are also roofs that are not so simply shaped. Below are pictured some different shaped pitched roofs:
Source: Shelton Roofing Santa Cruz
Source: Insenerigraafika. Olga Ovtðarenko (eõppematerjal)
Pitches
Carpenters do not refer to the angle of roof as 30° but prefer to use the pitch of the roof. The roof's pitch is its vertical rise divided by its horizontal span (or "run") called slope in geometry or the tangent function in trigonometry.
On the figure below, the various permitted pitches for different roof materials are depicted. For example, in the case of rolled metal roofs it can be a minimal of 5° (as shown on the graph). This means we cannot use rolled metal for flat roofs because it would not be a waterproof construction.
1: lath and shingle 
2: ceramic roof tile 
3: skylight 
4: bitumen roll with triangular lath 
5: asphalt shingles 
6: cement roof tile 
7: bitumen wave plate and tin roof sheets 
8: fibre cement board 
9. rolled metal roof 
In the above graph we
can also see the different values given as a ratio  for a slope of 5° the ratio is 1:12. We know that on traffic signs the slope of a road is also
given but it is shown in percentages.
Let’s have a look how these different values are related to each other:
As we all know, a full circle equals 360 degrees. In the graph below, we can see that in case of a roof pitch, trigonometrically seen, one side of a roof is in the I quarter and the opposite side in the II quarter.
The partition of quarters can be seen in the image below:
If we are talking
about expressing the pitch as a ratio, it will be the ratio of the length of
the segments to each other.
For example: if the
ratio of the roof pitch is 1:1, we are talking about an isosceles triangle and the roof pitch can be given as 45°. To calculate the pitch we use the tangent function, meaning 1:1=1, where
tan 45°=1
To express the roof pitch as a ratio, the height or so
called rise will be given as a value
of 1. The distance from the rise of the roof to the edge is marked with x (also called the run). The value of x shows how many heights till the edge the roof.
For
example: if x=2, then 1:2=0,5. In conclusion tan 0,5 = 26,57°≈ 27°
If
the angle of the roof is more than 45° or in other words the ratio is bigger
than tan > 1, the run will be a constant with the value of 1.
For
example: Given a pitch of 1,5 : 1. This means
x = 1,5 and shows how many runs fit into the rise.
Different ways to calculate the pitch of a roof
Problem: Given a roof with the rise of 4m and a run equalling 12m.
Calculate the pitch of the roof and give it as a ratio!
Here are 2 ways to solve the problem:
Method 1: If the pitch is given as a ratio 1:x, we can do the following to find x:
The pitch of the roof is 1:3
Method 2: We can also use the following solution
A 4 meter rise equals 12 m of horizontal length. Knowing that, we can do the following calculations for a rise of 1m :
The pitch of the roof is 1:3
Pitches can also be shown as percentages and per mille. Percentages are mostly used for roads and hills, per
mills are used for floor slopes and emplacement of water and sewages pipes,
where the pitches are a lot smaller.
A pitch given in percentages equals 1/100
1 m = 100 cm
hence 1% is 1cm per meter
1 m = 1000 mm
therefor 1‰ is 1mm per meter
Given the values for run and rise, we can calculate the
percentage or per mills of a pitch in the following way:
If we need to calculate one of those values, we can convert the formula.
For example: to find the rise, we can convert the formula in the following way:
If
our run has a length of 1m (100cm), then with a pitch of 1% our rise will be 1
cm. A run with a length of 2m will have a rise of 2 cm given the same pitch.
When
calculating in per mills, a run of 1m with a pitch of 1 will
have a rise of 1 mm. A run with a length of 2m will have a rise of 2 mm given
the same pitch.
To find the values for the run, we must again convert the formula:
Sometimes we might need to build a roof with the same pitch but different measurements
In that case, we use the principal of similar triangles to make the calculation.
The calculation can be done using cross multiplication:
When we compare the pitches, they will be the same even with a different rise of the roof.
In reality it would look like the picture below:
Let’s bring some more examples on how pitches are
related to each other. For example, if we know the % of a pitch but want to
express it as an angle using a tangent, we can do it in the following way:
As show on the graph, given the rise and run, we will
know the pitch percentage (up to 45°) and can calculate the tangent using the
same ratio  it will give us the angle.
From that we can derive the angle α = 16,66924423° ≈ 16° 42’
Further explained, to
get this solution we equal the run to be 100 and choose a rise that when
divided by 100 would give us the %value. Since percentage is basically the
relation between run and rise, the percentage value is at the same time the
tangent of the angle.
The above figure shows the given different proportions
the pitch can be the same. If however we
take the run to be 1, we can show
the pitch as follows:
We know already that tan 16°
42’ ≈ 0,3
Stairs and sewers
Another part of construction that uses
pitch (or slope) calculation are stairs and sewers (especially gravity flow
pipes, not so much pressure pipes).
Stairs must have certain measurements so
that people can comfortably use them. To calculate the height of the stairs a
simple formula is used. Who wants to become a construction worker should
definitely remember this formula:
2h+b=630 mm
This formula helps to find the optimal
ratio between the rise and tread of the step.
h= rise of the step
b= tread of the step
The sum of double rise
and the tread gives a constant. From that the pitch can be derived  usually
1:1,5 to 1:2.
For the width of the
steps the formula h + b = 450mm is recommended.
Additionally to the
pitch, the stairs will have a total rise and total run which are also important
measurements.
Image: Wikipedia
When
building multiturn stairs you must also
make sure that it has an uneven number of steps otherwise you must change legs
when walking up the stairs. All steps must also have the same rise  human
brain adapts to a certain rhythm of walking so if one step is of different height than it might cause you to trip on the
steps.
The
above table shows that given the ratio 1:1 the triangle will have the exact
shape as pictured beside  an isosceles
triangle with 45° equalling an angle of 100% and a tangent of 1.
Given
the ratio 1:100, the grade will be as pictured on the side with an angle of 1%
and tangent 0,01. With 1:1000 the same logic applies.
For example
If
we know the value of the tangent then we can write the pitch as a ratio in the
following way:
Given
tan 0,5 = 26° 33’ 54,18’’
Mark
the pitch as a ratio 1 : x where x equals 0,5 (also the tangent)
Hence
the pitch can be given as 1:2
The
value of x in the ratio represents how
many parts of the whole it composes since 1 represents the whole (l=1 on the
graph) , so 1 : 0,5 = 2. You can also
show the pitch as a measurement of a whole.
As
already mentioned, if the rise (h) is greater than the run ( tan > 1 ) ,
then the value of x is above the value of the part pitch is denoted: x:1
For
example, if the value of the tangent is 2 value of the slope is 2 : 1
In
sewage pipes we also use pitches but there it is given in per mille.
Pictured
below is a cross section of a sewage line, which is used for pipe grades
in construction drawings. As you can see the distance between the two wells
(from the centre) is 34500 mm and the difference between the heights of the
wells is 300 mm.
Let’s
look at another example. How to calculate the pitch of a pipe as a ratio and in
percentages. In order to calculate the pitch we must sketch a triangle where h
marks the difference in height and l the distance between two wells.
Given
h=300mm and I=34500. Using crossmultiplication we can first calculate the
pitch as a ratio.
Substituting
x with the value we get the pitch 1:115
We
can also use crossmultiplication when we know the pitch (1 : 115) and want to calculate the
difference between the heights h.
The
same goes for l.
The
pitch given in percentages is: 1:115 x 100 ≈ 0,87%
On
the other hand, if we know the percentage we can calculate the ratio as
follows:
this
can be given as a ratio 1:115
In
order to calculate the pitch in percentages we must use the tangent
option
1: tan = h/l
option
2: ratio 1 : x
tan
= 300 : 34500 ≈ 0,0087 or as a ratio which is basically the same 1 : 115
≈ 0,0087
tan
0,0087= 0,41011655° or 0°24’36,42”
In
the picture below a sewage line in depicted in the ground before it has been
covered with soil.
Area calculation
A big part of construction is calculating volumes. It is necessary for
planning and ordering materials and for price quote calculations.
For example, sometimes only the pitch and the height from the ground are
shown on a project drawing. In order to calculate the area of the roof we must
know the length and width of the roof (on the picture above: 12 000 mm and 6034
mm) But if we only know the pitch we must take the drawing and based on the
width of the building calculate the measurements of the roof.
These
measurements are not only needed for calculating the quantity of needed materials
but also for constructing the roof, for example the length of a rafter.
For example
Given
the width of half the building is 4860mm, then with a pitch of 35° the height
of the roof will be:
tan
35° x 4860 mm = 3403 mm
Knowing
the height we can calculate the length of the rafter.
In
order to calculate the angle between the rafters we can use basic math. All we
need to know is what angels occur when we cross two parallel lines with a third
line.
When
dealing with complicated surfaces we must divide the area into parts.
As
we can see on the graph above there are different ways to make the area
measurements easier. One way is to divide the area into many smaller areas and
calculate their areas separately and then either sum up or subtract the
individual areas.
But
how do act if the roof has more than one angle like shown below:
This is a 2D drawing of a roof which is given in the project drawing, next
to it a 3D drawing is usually also given.
The pitches on the drawing are given on a 45° angle. The hips at the ends
are also at 45°. If we draw a notional line vertically from H,I to K, their
pitch will be 45°. If we know the angle
and the length of the roof, we can calculate the height of the roof that we
will need later in order to calculate the area of the roof.
Since the angle is 45° we know that we are dealing with an isosceles
triangle that has two sides of equal length. This makes our calculations
easier.
For example, the height of the roof is half the width of the roof, or 4000
mm . This can be seen in the figure below , which shows a side view of a roof.
To calculate the length of the hip we will need a calculator.
We can use the Pythagoras theorem since the length of the hip is equal to
the hypotenuse of A, B and H.
The length of A,B and H equals
This
is the height of the hip end, which from the viewpoint of the hip would look
like:
BH
is the hypotenuse and represents the height of the roof. The solution was
obtained by the Pythagorean theorem. The length of the hypotenuse is :
Since
this is not an isosceles triangle, we can draw the conclusions that although
all roof pitches are at 45 degrees, it is not the case for his hip angles .
If
we know these measurements we can start calculating the area of
the roof . We can see that the roof elements are composed of
different geometric shapes. As a reminder, let's see which formulas we will
need.
We
can calculate the area as follows:
The
hip ends of the roof form equal triangles ABH and DEI. Therefore, it is sufficient to compute the area
of one triangle and multiply it by 2 .
2 x
22,63m2 = 45,26m2
The parallelogramshaped
roof parts ( BCHJ and AGHJ ) are also equal, so we only need to calculate the
area of one and multiply it by 2.
To
calculate the area we must also know the value of HJ. This is equal to the base and doing a quick
calculation we know it should equal 6,0 m. Since it has the same pitch and
height as the hip we know the height of the parallelogram to be 5,66m
Thus, the area equals:
And
2 x 33,96m2 = 67,92 m2
DEIJ
is also a parallelogram with a height of 5,66 m and a base of 4,0. Its area
equals:
We
can also see a triangle (KLG) – its side GK has the length 5,66m. Its other
side is 4m. Based on these measurements we can calculate the area. Its
hypotenuse is the side GJ which is 6,93m long.
Therefor
we can calculate the area of the triangle:
Only one part remains – the trapezoid (EIKF). We know that EF is 12,0 m and IK 8,0
m long. We have already calculated 5,66
m to be the height.
Again
we can calculate the area:
Now
we can add all the areas together to get the total area:
45,26
+ 67,92 + 44,64 + 11,32 + 56,60 = 225,74 m2
Overlapping
surfaces
It
is not uncommon that the measurements of a detail vary from what it should be
when constructing all details together. For example floorboards.
In
the picture below You can see a floorboard which has a protruding part and
which makes it wider then it would be when the whole area is covered with floorboards.
When measuring we know that a single floorboard has the width 111 mm, but when
joining it with another board it will only have the width of 100mm.
Therefore
at most times timber salesmen mark the total coverage area of floorboards on
the package.
An example
Given
a floor with measurements 6,27 m x 4,37 m. How much flooring material must we
buy to cover the whole floor given floorboards with an overlap of 100 mm and a
length of 4 m.
We
start by calculating the area of the floor surface:
6,27
m x 4,37 m = 27,40 m2
Next
we calculate the area one floorboard would cover:
4,00
m x 0,1 m = 0,40 m2
Finally
we'll calculate how many floorboards we must buy to cover the whole floor:
27,40
m2 : 0,40 m2 = 68,5 ≈ 69 pieces of floorboard
Another
important part of wooden floors is the baseboard. For the floor size given
above to find the amount of baseboards needed You must calculate the perimeter
of the room. The purpose of the baseboard is to cover the nailheads and often
it is not possible to install the last floorboard close enough to the floor and
a 12 cm gap is left. The baseboard is used to cover the gap as shown on the
picture below.
An example
It
is best to calculate the amount of baseboards needed by running meter because
they are installed on the wall forming a perimeter around the room. Let's
calculate the amount of baseboards needed if the length of the baseboard is
2,5m.
First
let's calculate the perimeter of the room:
(6,27
m + 4,37 m) x 2 = 21,28 rm
rm 
running meters, often used in construction. Equals 1m in SI units
Now
we can calculate the amount of baseboards needed:
21,28
rm: 2,5 rm= 8,51 ≈ 9 pieces of baseboard
Geodesy and Geodetic Measurements
Geodesy
is the science of measuring the size and shape of the Earth and the location of
points on its surface. It includes land and construction surveying.
The
easiest or so called lower class geodesy is used for general construction
(because of small distances). In road and land surveying the accuracy needs to
be higher and the equipment used is a lot more complicated, because with longer
distances the error margin also gets bigger (see example below). There we must
install additional heights (temporary surveying staffs) to measure different
sections and transfer their heights and angles.
In
this paragraph we well look closer into construction surveying, which is simple
and only requires basic math.
As
we know water has the ability to always stay horizontal or as construction
workers say it is horizontally levelled. In the old days water hoses with glass
vials on both ends, that abled to see the water level, were used.
Even
nowadays this method is sometimes used (see picture below) but a lot of more
tools have been developed since that are more exact in determining the level or
plumb.
A
spirit is an instrument designed to indicate whether a surface is horizontal
(level) or vertical (plumb).
In
the picture below is another example of a spirit, but instead of water other liquids are used (usually alcohol such
as ethanol)
The
bubble inside the spirit tube shows if the surface is level or not. If it’s
centred between the lines on the tube, your object is level. If the bubble is
to the right of the lines, your object slopes downward righttoleft. If the
bubble is to the left of the lines, your object slopes downward lefttoright.
Now,
if we need to level longer distances a tool called dumpy level (also builder's
auto level)used. It is an optical instrument used to establish or check points
in the same horizontal plane. The level instrument is set up on a tripod and
set to a levelled condition using foot screws. The operator looks through the
eyepiece of the telescope while an assistant holds a levelling rod vertical at
the point under measurement.
A
better tool You can use is the laser level, which can be operated by only one
person. It consists of a laser beam projector that can be affixed to a tripod,
which is levelled according to the accuracy of the device and which projects a
fixed red or green beam along the horizontal and/or vertical axis.
The
levelling rod, also called level staff, is used with a levelling instrument to
determine the difference in height between points. The metric rod has major
numbered graduations in meters and tenths of meters. When viewed through an
instrument's telescope, the observer can easily visually interpolate a 1 cm
mark to a quarter of its height, yielding a reading with accuracy of 2.5 mm.
Mean sea level
In
common usage, elevations are often cited in height above sea level.
MSL
is a type of vertical datum – a standardised geodetic reference point – that is
used as a chart datum in cartography.
Countries tend to choose the mean sea level at one specific point to be
used as the standard „sea level” for all mapping and surveying in that country.
However, zero elevation as defined by one country is not the same as zero
elevation defined by another. In Estonia and in the Baltic sea region the MSL
of Kronstadt is used. This was denoted by Mihhail von Reineke in 1840 after
extensive measurings in the Baltic sea. Pictured below is the marking of MSL on
the Obvodnõi canal bridge in Kronstadt.
Allikas: http://stengazeta.net/wpcontent/uploads/10005290siniymost_.JPG
Estonia is also planning on going over to the Amsterdam Ordnance Datum, which
has the accuracy of ± 0,2 mm per
kilometre.
In the
field of construction a level is used to measure heights but when mapping
larger areas a tachometer or precision GPS is used, as it enables you to
measure not only height but also the angle.
Below
a simple example of using a level.
Example
In
this example, the geodesist has previously measured the foundation of an
existing building to have a height 0.819 m above sea level. It will be used
when we start marking down the planned foundation of the house with the height
given in the project by the architect.
The
zero point of the construction itself is the top surface of the first floor.
All heights above it will be marked with a ( + ) sign and the ones below (such
as a basement ) denoted by (  ) sign.
On the figure we see that the building has been designed with a height of
1.06 m above sea level (0 height is given by the surface of the first floor).
The height of the ground is marked at the corners of the building. The upper
0.56 m tells us what altitude the ground around the building must remain, the
number below indicates the current height of the ground in nature.
Let’s do a small calculation:
1,06 – 0,56 = 0,50 m
This tells us that the ground must be 50 cm below the surface of the first
floor.
Now we look what the level must show if we start to mark down the
foundation of the building. Still looking at the figure above, we see that the
horizontal beam of the level must have the height of 551 mm, meaning that from
the sea level the horizontal beam must have the height:
819+ 551 = 1370 mm
But if we change the location of the level the result will also change.
Next question that arises is how much should the starting point of the level
be below the horizontal beam so it would correspond to the height of 1.06 m
above sea level.
We know, the horizontal beam is currently 1370 mm above sea level and we
can do the following calculation:
1370 – 1060 = 310mm
From the level we
should be able to read out 0310, as shown in the figure above. Even though
quite good laser tools exist, often construction workers still prefer to
use optical measuring devices for longer distances as it is more exact
since the laser beam will start to fade at larger distances.
Mathematically we
can look at this error as an occurring angle since the beam coming from
the device is straight.
Calculating the error:
For example if we know the angle of the error we can use tangent to compare the values of
x1 and x2
Example
If we want to show the error rate in mm we must first convert the length to get
40 000 mm. If we take the error to be only 0,1°, then
x1 =
tan 0,1° x 40 000 = 69,8 mm is already a very big difference,
x2 =
tan 0,1° x 100 000 = 174,5 mm, which is twice as big
With
a distance of 1 m and the same angle the error rate would only be 1,7 mm
Therefor
the surveyors must constantly correct the measurements in case of long
distances, so a situation as pictured below would not happen.
Source: unknown