KappAbel
Digranesskóli, Iceland
1. Introduction
We wanted to find out if there was any difference in building- and electricity expense of
two houses with the same cubic measure and floor size if there would be any difference
in the shape. We decided to design two sports halls, one cubic shaped and another
hemispherical and see if there was any difference in the building and electricity cost.
Inside the houses there were supposed to be two handball courts, each one 20 m x 40 m
plus a security space which is one meter. Then we realized that it would not be possible
to have the house completely hemispherical because it would be difficult to stand upright
close to the walls. Then we changed the hemispherical house into a cylinder with a
hemispherical roof so the floor would be used completely.
The questions we wanted to get some answers to were:
How many cubic meters of concrete do we use for each house?
How much does the concrete weigh?
How much does the concrete cost?
How much does it cost to light up the courts with maximum lighting for average 10 hours
a day?
How much parquet do we need to parquet the courts?
How many oak trees do we need to parquet the courts?
Is there any difference in surface area?
2. Few basic calculation
2.1. Hemispherical house with circle floor
We knew that if the handball courts needed to fit inside the house the
Diameter of the floor would have to be as long or even longer than the
Diagonal of both courts. Then we used the calculation formula
2.1.1 a2 + b2 = c2 where a and b stand for side length and c the diagonal.
2.1.2 c = c2 where c stands for the diagonal.
2.1.1 422 + 422 = 3528
2.1.2 3528 59,5 m
42
We decided to have the diameter 59,5 m on the inside and 60 meters
on the outside because the walls were supposed to be 25 cm thick.
Then we used the calculations formulas
2.1.3 r = d / 2 where r stands for radius and d diameter.
2.1.4 F = r2 x π where r stands for the radius of the floor and F the floor’s area.
2.1.3 59,5 / 2 = 29,75 m
2.1.4 29,752 x π 2.780,5 m2
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KappAbel
Digranesskóli, Iceland
We realized that it would not be possible to put the hemispherical right down on the floor
because then we would not have minimum headroom by the walls of the house. That’s
why we had 3 m tall walls that the hemispherical came onto. Then we could calculate the
cubic measure of the house with these 3 formulas
2.1.5 R1 = r2 x π x h whereR1 stands for the cylinders cubic measure, r radius and h
height.
2.1.6 R2 = 4 π x r3 / 6 where R2 stands for the hemispherical cubic measure and r radius.
2.1.7 R = R1 + R2 where R stands for the cubic measure of the house, R1 the cylinders
cubic measure and R2 the hemispherical cubic measure.
2.1.5 29,752 x π x 3 8.342 m3
2.1.6 4 x π x 29,753 / 6 55.147 m3
2.1.7 8.342 + 55.147 = 63.489 m3
We also wanted to find out surface area and used the following formulas to find out what
that would be.
2.1.8 L = d x π x h where L stands for lateral surface, d diameter and h height.
2.1.9 Sh= 4 π x r2 / 2 where Sh stands for the surface area of a hemispherical and r
radius.
2.1.10 S = L + Sh where S stands for the surface area, L lateral surface and Sh the surface
area of a hemispherical.
2.1.8 60 x π x 3 565,5 m2
2.1.9 4 x π x 302 / 2 5.655 m2
2.1.10 565,5 + 5.655 = 6220,5 m2
2.2. Cubic house
Like we said in the introduction we were going to have the same floor size and cubic
measure in both houses. We chose to have the cubic house with a quadrate floor and to
find out the side lengths we used the formula
2.2.1 F = l where F stands for the floor area and l for side length.
2.2.1 2.780,5 52,7 m
Then we found out the height with the formula
2.2.2 h = R / F where h stands for the height, R the house cubic measure and F the floor
area.
2.2.2 63.489 / 2.780,5 22,8 m
We found out the surface area with the formula
2.2.3 S = l x 4 x h + F where S stands for the surface measure of the house, l the side
length of the walls, h the height and F the floor area.
2.2.3 53,2 x 4 x 23,3 + 53,22 7788,5 m2
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KappAbel
Digranesskóli, Iceland
3. Concrete
To find out the expense and weight of the concrete we had to know what one cubic meter
of concrete costs and how much it weighs. We contacted a concrete factory in Iceland
called Steypustöin MEST and got the information we needed
1 cubic meter of concrete weighs 2,5 ton
1 cubic meter of concrete costs 15.780 ISK
We also had to decide how thick the walls should be so we could
calculate the amount of concrete that would be used in the
houses. It was decided that the walls should be 25 cm thick.
3.1 Hemispherical house
We found out the cubic measure of the concrete with following formulas:
3.1.1 Rc1 = r2 x π x h where Rc1 stands for the cylinder cubic measure, r radius and h
height. It’s all on the outside.
3.1.2 Rc2 = r2 x π x h where Rc1 stands for the cylinder cubic measure, r radius and h
height. It’s all on the inside.
3.1.3 Rc = Rc1 – Rc2 where Rc stands for the cubic measure of the concrete, Rc1 the
cubic measure of the house with walls and Rc2 the cubic measure of the house without
walls.
3.1.4 Rh1 = 4 π x r3 / 6 whereRh1 stands for the cubic measure of the hemispherical and r
radius. It’s all on the outside.
3.1.5 Rh2 = 4 π x r3 / 6 whereRh2 stands for the cubic measure of the hemispherical and r
radius. It’s all on the inside.
3.1.6 Rh = Rh1 – Rh2 where Rh stands for the cubic measure of the concrete, Rh1 the
cubic measure of a hemispherical with walls and Rh2 the cubic measure of a
hemispherical without walls.
3.1.7 F = r2 x π x h where F stands for the floor area, r radius and h thick floor.
3.1.8 C = Rc + Rh + F where C stands for the concrete amount, Rc the cubic measure of
the concrete in the cylinder, Rh the cubic measure of the concrete in the hemispherical
and F the floor area.
3.1.1 302 x π x 3 8.482 m3
3.1.2 29,752 x π x 3 8.341,5 m3
3.1.3 8.482 – 8.341,5 = 140,5 m3
3.1.4 4 x π x 303 / 6 56.549 m3
3.1.5 4 x π x 29,753 / 6 55.147 m3
3.1.6 56.549 – 55.147 1.402 m3
3.1.7 29,752 π x 0,25 = 695 m3
3.1.8 140,5 + 1.402 + 695 = 2.237,5 m3
To find out the weight of the concrete we used the formula
3.1.9 CW = m3 x W where CW stands for the concrete weight, m3 a cubic meter and
W weight of concrete per one cubic meter.
3.1.9 2.237,5 x 2,5 5.594 ton
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KappAbel
Digranesskóli, Iceland
To find out the cost of the concrete we used the formula
3.1.10 CE = m3 x E where CE stands for concrete expense, m3 cubic meter and E the
expenses of one cubic meter of concrete.
3.1.10 2.237,5 x 15.780 = 35.307.750 ISK.
3.2 Cubic house
To calculate cubic measure of the cubic house we used the following formulas
3.2.1 R1 = l x b x h where R1 stands for the cubic measure of two walls on the outside, l
side length on the outside, b thick wall and h height on the inside.
3.2.2 R2 = l x b x h where R1 stands for the cubic measure of two walls, l side length on
the outside, b thick wall and h height. It’s all on the inside.
3.2.3 R3 = l x b x h x 2 where R3 stands for the cubic measure of the floor and roof, l and
b side lengths and h thick. This is all on the outside.
3.2.4 R = R1 + R2 + R3 where R stands for the cubic measure of the concrete, R1 the
cubic measure of two walls, R2 the cubic measure of the two other walls R3 the cubic
measure of the floor and the roof.
3.2.1 53,2 x 0,25 x 22,8 x 2 606,5 m3
3.2.2 52,7 x 0,25 x 22,8 x 2 601 m3
3.2.3 53,22 x 0,25 x 2 1.415 m3
3.2.4 606,5 + 601 + 1.415 = 2.622,5 m3
To find out weight and expense of the concrete we used the same formulas as in the
hemispherical house.
3.1.9 2.622,5 x 2,5 6.556 tonn
3.1.10 2.622,5 x 2,5 6.556 tonn
4. Electricity use and expense
We were informed that lamps are usually put in 15 m height in sports halls. We also got
to know that competition lighting would have to be 800 Lux, but Lux is a measure for
brightness. So that can be possible we need 0.8. kW lamps that give the asked brightness
on 16m2. Then we could find out the number of lamps with the formula
4.0.1 L = F / 16 where L stands for lamps and F floor area.
4.0.1 1764 / 16 110 lamps
To find out how many kW would be used in the hall per one hour we used
the formula.
4.0.2 kW/klst = L x kW where kW/klst stands for kilowatts per hour, L
lamps and kW kilowatts.
4.0.2 110 x 0,8 = 88 kW
We modelled on that the lights would be turned on in average 10 hours a day. To
calculate the daily use we used following formulas
4.0.3 D = H x kW where D stands for daily use, H stands for hour and kW kilowatt.
4.0.3 10 x 88 = 880 kW
According to the catalogue of Orkuveita Reykjavíkur which is an electricity company in
Iceland the kW-hour costs 8,28 ISK. Then we calculated the daily cost with the formula
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KappAbel
Digranesskóli, Iceland
4.0.4 ISK = kW x 8,28 where ISK stands for daily expense and kW stands for kilowatts.
4.0.4 880 x 8,28 = 7286 ISK.
We found out the yearly expense with the formula
4.0.5 Y = ISK x d where Y stands for yearly expense, ISK daily expense and d days.
4.0.5 7286 x 365 = 2.659.390 ISK
5. Oak trees and parquet
We decided to parquet the 42 m x 42 m which would be used as
handball courts with oak trees. We got information from Arnór
Snorrason, a flora specialist in Mógilsá, which is a forest in Iceland,
that in a Scottish oak forest 80 years old trees average height is 23
meters and the circumference in 1,3 meter height is average 1,07 m.
To find out the cubic measure of the trunk we used the following
formulas.
5.0.1 r = U / π / 2 where r stands for radius and U for circumference.
5.0.2 R = r2 x π x h x f where R stands for cubic measure of the
trunk, r radius, h height of the trunk f form factor which is 0,42.
5.0.1 1,07 / π / 2 = 0,17 m
5.0.2 0,172 x π x 23 x 0,42 0,87 m3
Arnór told us that we could only use 40% of the trunk and to find out how much useable
material for the floor is in one tree, we used this formula.
5.0.3 G = R x % where G stands for useable material and R for cubic meter of the tree.
5.0.3 0,87 x 0,4 = 0,35 m2
We bargained that the parquet was 0,02 m thickness. To find out the cubic measure of the
floor we used the following formula
5.0.4 R = l x b x h where R stands for cubic meters, l for length, b for width and h
thickness.
5.0.4 42 x 42 x 0,02 = 35,28 m3
Then we could find out how many trees we had to cut down with the formula
5.0.5 T = R /G where T stands for trees, R stands for cubic meters of the floor boards and
G for useable material of the tree.
5.0.5 35,28 / 0,352 100 tré
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KappAbel
Digranesskóli, Iceland
6. Conclusion
Floor
Highest height
Cubic measure
Surface area
Cubic measure
of concrete
Cost of concrete
Weight of
concrete
Number of oak
trees
Yearly lighting
cost.
Hemispherical
house
2.780,5 m2
32,75 m
63.489 m2
6.220,5 m2
2.237,5 m3
Cubic house
Collation
2.780,5 m2
22,8 m
63.489 m2
7788,5 m2
2.622,5 m3
No difference
9,95 m
No difference
1568 m2
385 m3
6.706.500 ISK.
5.594 ton
42.014.250
ISK.
6.556 ton
100 trees
100 trees
No difference
2.659.390 ISK. 2.659.390 ISK.
No difference
35.307.750 ISK.
962 ton
7. Final words
In this Project we have worked with:
- Different shapes
- Cubic measure
- Surface area
- Pythagoras
- Construction expense
- Concrete
- Electricity
- Oak trees
- 3-D
- Reporting
- Design
- English translating
In this project it surprised us how much the concrete costs and how much it costs to
lighten up a house like this. The thing that was the most surprising is how many 80 years
old oak trees are needed to parquet the court in only one house.
So it will be our generation’s project to find another carpet with the same utility and
doesn’t destroy the forests of the earth.
So then it might be worth considering how many oak trees are cut down every year to
parquet houses in the whole world?
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KappAbel
Digranesskóli, Iceland
8. Bibliography
8. 1 Authority by word of mouth
1. órur Gumundsson, maths teacher in Digranesskóli, 2007
2. Stefán Pálsson, educational deputy in Rafheimar, 2007
3. Arnór Snorrason, wood specialist in Mógilsá, 2007
4. Kristinn Alexandersson, civil engineer in VSO, 2007
5. Steypustöin Mest, Concrete factory, 2007
6. HSÍ, Handball Federation in Iceland , 2007
7. Bjarki Brynjarsson, employee in Askar Capital, 2007
8.2 Written Authority
8. http://www.hsi.is, 2007
9. http://www.orkuveita.is, 2007
10. http://waynesword.palomar.edu/trjuly99.htm, 2007
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