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“Fabulous Builders”
Iceland – Estonia
Course manual
1
Table of Contents
Introduction ........................................................................................................................................... 2
1.Formulas ............................................................................................................................................. 3
Triangles ............................................................................................................................................. 3
Polygons.............................................................................................................................................. 3
Volume measures............................................................................................................................... 5
Practical formulas used in carpentry .............................................................................................. 8
2.The use of an electronic calculator.................................................................................................... 9
3.
General computing projects operated with an electronic calculator ...................................... 10
Introduction
This cooperation project aims at combining teaching methods in practical mathematics and
carpentry vocational training. The project runs for 1, 5 years or from the September 1 2013
through the June 1, 2015. Ágústa Unnur Gunnarsdóttir will manage the project on the behalf
of the teachers of Breiðholt College.
Several teachers at the carpentry department will participate in the project, under the head of
the department’s director Stefán Rafnar Jónhansson. Stefán Benediktsson and Sigurður
R.Guðjónsson will gather and edit the course content.
Not only does the project provide Breiðholt College the opportunity to acquire knowledge on
Estonian carpentry teaching methods, ingenuity and skills but also provides the college an
opportunity to share its knowledge in the field. After substantial work and considerations the
college staff came to the conclusion that the project would be confined solely to the carpentry
department. The projects slogan will be “Factual and Fabulous builders”.
2
1. Formulas
Frequently used formulas; area, volume and surface areas.
Triangles
All triangles
Area
½ base (b) * vertical height (h)
Perimeter
Side (a) + side (b) + side(c)
A = a+b+c
Pythagorean Theorem / Right triangle
An equation related to the lengths of the sides A, B and C of a right
triangle. =
C2 =A2 + B2
Polygons
Rectangle
Area
Length x height
A=L*H
Perimeter
3
=>
Length * 2 + with * 2
=>
P =2L * 2W
Proportions of the side lengths of a rectangle which are in the golden ratio
Shorter side = Longer side / 2 * (√5- 1)
P = L/2 * (√5- 1)
=>
Square
Area
A
The length of the two sides in the power of 2
=s
2
Perimeter
P= 4 * S
4 * length of the side
Trapezoid / Trapezium
Area
The bases are the two parallel sides of the trapezoid
½ the vertical height (base + lower base)
½ h (b1 +b2)
Perimeter
Add up the two bases and the two legs
=
Parallelogram
Area
Vertical height * base =
=>
A= h *b
=>
P= 2a +2b
Perimeter
Add up lengths of the sides or
4
P= a +b + c + B
Octagon
Area
2 * side length *2 * (1 + 2√) =>
A=2s2(1+2√)
Perimeter
P = 8 * side
P = 8s
=>
Circles
Area
2 * pi (π) * radius (r)
=>
A =2 πr
=>
P = 2r * π
Perimeter (circumference)
Diameter * pi (π)
N sided polygons
Any polygon total = degrees 180° (N - 2)
Regular polygon only each angle = 180° (N-2) N
Volume measures
Cubes
Length x width x height
5
=>
V=L*W*H
Prism
Lateral surface area (LSA) = area of all faces
=>
B = base
Volume
Base * height =
=>
V=B*h
Total surface area
LSA +2* base
=>
TSA = LSA + 2b
Cylinder
LSA = (lateral surface area)
LSA = 2 π r h
Volume
V = π r2 h
Total surface area (TSA)
TSA= LSA + 2 πr2
Sphere
Surface area of a sphere
4 x pi (π) x radius in the power of 3
6
=>
S= 4 * π x-* r2
Volume of a sphere
4 x π / 3 x radius in the power of 3.
V= 4 * π / 3 * r3
=>
Cone
Volume of a cone
½ π (PI) * radius in the power of 2 * height
=>
V= ½ π r2h
Lateral surface area (LSA) of a cone
π (pi) * r (radius) * s (slant height) =
=>
LSA=π * r* s
Mass
Density x volume
=>
M =density x volume
Trigonometry
Sin function (sin)
The ratio of the side opposite the angle to the hypotenuse= opposite (a)/ hypotenuse (c)
 Sin = a/c
Cosine function (Cos)
Ratio of the adjacent leg (b) to the hypotenuse (c) = adjacent (b)/ hypotenuse (c)
 Cos = b/ c
Tangent function (tan)
The ratio of the opposite leg (a) to the adjacent leg (b) = opposite (a) /adjacent (b)
 Tan = Sin (a) / Cos (b)
Speed formulas
7
Single Belt Transmission
Pulley diameter (1) x number of revolutions (1) = Pulley diameter (2) x number of revolutions

D1 * N1 = D2 * N2
Cutting speed
Diameter x pi (π) x number of revolutions / 60 min =
=>
V=d*π*
=>
I = c * g / 100
n/60min
Interest rates
Interests rates = capital stock * growth rate / 100
When the interests rates are added to the capital stock annually for x number of years =
 Cs= c (1 +v) n
Temperature Conversions
C = 5/9 (F-32) (Fahrenheit to Celsius)
F = 9 /5 c + 32 (Celsius to Fahrenheit)
Practical formulas used in carpentry
Gross building area
External diameter (length) x width =
gA = D * L
=>
Net building area
Gross building area – length x wall thinness =
8
=>
nA = gA – L * wth
Parking spaces
-
Residential housing; One parking space per 75 m2
Industrial buildings; One parking space per 25 m2
Playground
Area in m2 = total nr of apartments (32- nr of apartments) / 2 => Playground = N (32 –
N) / 2
Defining pipe s and flat / area / slope
Slope % = height differential / length * 1000
2. The use of an electronic calculator
An electronic calculator is a portable device used to perform operation of arithmetic. The
most common types of calculators are; a basic pocket calculator, graphic calculator and a
scientific calculator. The Scientific Calculator supports the following operations:
9
+
Addition/ summation
-
Subtraction
X
Multiplication
÷/
Division
,
Decimal point
C
Clear display
AC
Clear all internal values
√
Square root
X2
Square / exponent notation
Xy
X in the power of Y
X1/y
Y square root of X
+M
Memory Addition
-M
Memory Subtraction
RM
Memory Recall
()
Parentheses (calculate expression inside first)
Sin
Sin function
Sin-1
The inverse trigonometric function of sine
Cos
Cos function
Cos
The inverse trigonometric function of cosine
Tan
Tan function
Tan-1
The inverse trigonometric function of cosine
The operation symbol must be typed after the number have been typed.
3. General computing projects operated with an electronic
calculator
Summation
Example: add 70 to 140
Operation: Type 70 and press the + symbol. Type 140 and then press the = symbol. The result
should be 210
Solve these problems.
0.1
10
125 + 75
0.2
150 + 89 + 31
0.3
175 + 25 + 67+ 63
0.4
17,50 + 2,5 + 7,6 + 3,6
0.5
1½
Subtraction
Example: Subtract 70 from 140
Operation: Type 140 and press the – symbol. Type 70 and then press the = symbol. The result
should be 2010.
Solve these problems.
0.6
75 – 125
0.7
150 – 89 – 31
0.8
175 – 25 – 67 – 63
0.9
17.7 – 2.5 – 7.6 – 3.6
0.10
2½-2¾
Division
Example: Divide 140 with 7
Operation: Type 140 and press the ÷ symbol. Then type 70 and press =. The result should be
20.
Solve these problems.
0.11
0.12
0.13
0.14
0.15
125 / 5
150 / 30
175/25
17.5 / 2.5
3¾/1¼
Multiplication
Example: multiply 70 with 140
Operation: type 70, then press x and 140. Finally press the = symbol. The result should be
9800.
Solve these problems.
11
0.16
0.17
0.18
0.19
0.20
125 x 75
150 x 89 x 31
175 x 25 x 67 x 63
17.7 x 2.5 x 7.6 x 3.6
2½x3½
Exponential function
The exponential function is practical to use, when working with high numbers or very low
numbers. The function is written as Xa , which we interpret as x*y (the function is its own
derivative).
Every positive number has a square root, which is written in exponent notation, as X 2
Example: 32 = 9 = 32 = 3 * 3= 9
Example = √9 = 3
Complete the following equations
(1)
Example: Find the 2 in the power of 4
Operation: Type in the number 4, then press the INV symbol and in finally x2. The result
(2)
Example 2: Find 5 in the power of 4
Operation: Type in the number 4, then press the symbol xy and finally press =. The result
should be 1024
(3)
Example: Find -3 in the power of 4
Operation: Type in the number 4, then the xy symbol. Then type in the number -3, and finally
press the = symbol. The result should be 0,015625
Square root
Example: Find the square root of the number 441.
Operation: Type the number 441 and then press the √ symbol. The solution should be 21.
12
Example: Find the fourth root of 16
Operation; Type in the number 16, the press the following symbols INV and x 1/y. Then press
the = symbol. The result should be 2
Complete the following equations
0.27
Find the square root of 27
0.28
Find the square root of 729
0.29
Find the square root of 531.441
0.30
Find the square root of 282.429.536.000
0.31
Find the fourth root of 35
0.32
Find the fifth root 40
0.33
Find the seventh root of 125
0.34
Find the twelfth root of 564
Mixed operations
Example: 10 + 12 * 14 + 15 / 3 – 13
Complete the following equations
0.35
125 + +75 (125 * 75)
0.36
135 + 85 (120 * 70) – 15
0.37
120 + 78 (105 * 70)2
0.38
126 + 80 (103 * 77)3
0.39
115 + 78 (120 * 25) / 25
0.40
105 + 8 (130* 25) / 45
0.41
18 + ((56+44)4 / 80) / 100
0.42
28 ((56 + 64)5 / 180) / 1000
0.43
7 (26 + 35 (56+65) 3 /75) - 1.267
0.44
5(9 +5 (56+ 6) 4 / 175 -0,143
Interest rates
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Interests are calculated as the percentage rate of a hundred (%). i.e. 10/ 1000 = 10%. Thus,
10% interest rate of 100 euro would be 10 euro.
The interest rate can be determined by the following formula
Capital stock (principal amount) (p) * interests rates (r) / 100 = Ni =ca * I /100
Compound interests
When the interest’s rates are added annually to the capital stock (principal amount) for x (n)
number of years, you apply the following formula.
Cs (p) = c (1 +v) n
1) Example: Find 25% interests of 77
Operation: Type in the number 77, then press the * symbol. Type in the number 25 and
then press the ÷ symbol and then type in the number 100. The result should be 19, 25.
2) Example: What is 20 of 80 in percentages %.
Operation: Type in the number 20, following with the ÷ symbol. Then type in the number
85 and then press the * symbol, and finally 100. The result should be 23, 53 %
3) Example: The gross value of a product is 120 ISK, the price includes 25 % VAT (value
added tax), what is the net value (initial price) without the tax?
Operation: Type in the number 120, then press the ÷ sign. Then type in the number 1, 25
and lastly press the symbol =. The result should be 14049, 28.
4) Example: 10000 ISK are deposited to a bank account by locking it in for a period of three
years, with 12% annual interests rate, what will the total amount be after that period?
Operation: Type in the number 10000, then press the x symbol. Then 1, 12, xy, 3
And finally press the = symbol.
5) 1000 ISK are deposited to a bank account by locking it in for a period of three years, with
12 % biannual interested rates, what will the deposited amount be after three years?
Method: Add given values to the calculator, begin by typing 1000 and press x (1 +
0.12/2), xy (3 * 2) and finally press =. The result should be 14185, 19.
Complete the following equations
0.45
Find 35% of 86
0.46
Find 45 of 286
0.47
A net value (initial price) of a product is 125 ISK excluding 25% VAT, what is the net
value of a product including the VAT?
14
0.48
A net value of a product is 255 ISK excluding 25% VAT, what is the net value of a
product including the VAT?
0.49
A gross value of a product is 550 ISK including 23, 3 % VAT, what is the net value
(initial price) of the product excluding the VAT?
0.50
A gross of a product is 750 ISK including 23, 3 % VAT, what is the initial price of the
product excluding the VAT?
0.51
12.500 ISK were locked in a bank account with 18 % interests rates for a period of 3
months. What did the interest rates amount to in the end in ISK?
0.52
32.500 ISK are locked into a bank account with 17, 5% interests rates for 13 months.
What did the interest rates amount to in the end in ISK?
0.53
42.500 ISK are locked into a bank account with 19, 5% interests rates for 2 years.
What is the gross amount in ISK after that period?
0.55
32.500 ISK are locked into a bank account with 17, 5% interests rates for 2 years.
What is the gross amount in ISK after that period?
0.56
42.500 ISK are locked into a bank account with 19, 5% interests rates for 3 years.
What is the gross amount in ISK after that period?
0.57
32.500 ISK are locked into a bank account with 17, 5% interests rates for 6 years.
What is the gross amount in ISK after that period?
0.58
42.500 ISK are locked into a bank account with 19, 5% interests rates for 5 years.
What is the gross amount in ISK after that period?
0.59
50.000 ISK are locked into a bank account with 17, 5% bimanual interests rates for a
period of 10 years. What will the gross value amount to after that period?
0.60
40.000 ISK are locked into a bank account with 19, 5% trimestral interests rates for a
period of 7 years. What will the gross value amount to after that period?
4. Practical mathematics; Area and volume measure
calculations
Utilization ratio
Area measures of yards are calculated in square meters. Area measures of buildings
are calculated in square meters and cubic meters, often referred to as the Gross
15
External area. However, Net internal Area refers to the face of the internal finish or
perimeter, excluding and taking each floor into account.
1.1 A residential building (8, 75 * 15) is based on a 21 * 25 building site. The
buildings wall height is 2, 8 m and the roof slope difference is 1, 3 m.
a)
b)
c)
d)
e)
Calculate the area measure of the building site.
Calculate the area measure and the volume of the residential building.
Calculate the utilization ratio of the building site.
Calculate the buildings net internal area.
Calculate the number of allowed parking spaces and the size of the
playground.
1.2 A residential building (9, 75 * 19) is based on a 26 * 29 plan. The buildings wall
height is 2, 8 m and the roof slope difference is 1, 3.
a)
b)
c)
d)
e)
Calculate the area measure of the building site.
Calculate the area measure and the volume of the residential building
Calculate utilization ratio of the building site.
Calculate the buildings net internal area.
Calculate the number of allowed parking spaces and the size of the
playground.
1.3 A residential building (12 * 20) is based on a 35 * 85 plan. The building has four
floor including a basement, three of the floors have a basement.
f)
g)
h)
f)
16
Calculate the area measure of the building site.
Calculate the area measure and the volume of the residential building
Calculate utilization ratio of the building site.
Calculate the number of allowed parking spaces and the size of the
playground.
5. Roof pitch.
Roof pitches are calculated by the following methods;
(1) by measuring the degree of the slope. (Expressed in degrees)
(2) By calculating the proportions 1:2 of (1 being the vertical rise divided by 2 its horizontal
span)
(3) By measuring the centimetres pr meter.
Example:
A residential building width is 7, 5 meters and the height difference is 0,935.
17
A)
B)
C)
D)
a)
b)
c)
d)
What is the roof slope if measured in degrees?
What is the roofs rafter length, if the roofs overhang is 45 cm?
What is the roofs area measure if the roofs rafter length is 12 m?
How many meters of 25 * 150 mm of roof panelling is needed for the building?
Roofs slope – TangA – 0,935 (7,5 / 2) = 0,24933 = 13,995° ≈ 14°
Roof rafter length = ( 3,75 + 0,45/ cos 14° = 4,33 m
Roof area measurement =2 * 4,33 * (12 +0,90) = 111,71 m2
Roof panelling measurement = 111,7 / 0,15 =744,76
2.1 A residential building width is 7, 0 meters and the height difference is 0, 80.
A)
B)
C)
D)
What is the roof slope measured in degrees
Which is the roof rafter length if, if the roofs overhang is 50 cm
What is the roofs area measure if the roofs rafter length is 12,5 m?
How many meters of 25 * 150 mm of roof panelling is needed if.
2.2 A residential building width 7, 0 meters and the height difference is 0, 80.
A)
B)
C)
D)
18
What is the roof slope measured in degrees
Which is the roof rafter length if, if the roofs overhang is 45 cm
What is the roofs area measure if the roofs rafter length is 12 m?
How many meters of 25 * 150 mm of roof panelling is needed.