1 calculations on completion of this unit the learner should be able to demonstrate an understanding...

11

CalculationsCalculations

On completion of this unit the Learner should be able to demonstrate an understanding of:

The formula used to calculate:

• Areas

• Volumes

• Ratios

• Volumes into weight

• perimeters

• Mean, Mode, Median and range

22

Calculations

Areas of Rectangles, irregular shapes, circles.

Perimeter calculations.

Centreline method used to calculate volumes of concrete.

Volume for calculating quantities of concrete.

Converting volume into tonnage.

Mean = average (find the total and divide by the number youy have

Mode= most commonly occurring number

Median= put the numbers in order and find the middle number

Range= difference between highest and lowest number

33

Surface areasSurface areas

Calculating areas of brickwork and blockwork we are in fact calculating the

surface area. Surface areas in construction are calculated to determine

quantities of materials required to cover that area. Examples of this are

bricks, blocks, floor tiles and block paving. To calculate for these

quantities the following procedures should be followed:

A) Calculate the total area to be covered ( m2)

B) Calculate the area of one unit (a brick,block,tile etc).

C) Divide the total area by the area one unit.

D) Add percentage to compensate for waste.

44

Calculate the following areasCalculate the following areas

Formula required to do this is:- Area = L x H = M2

Calculate the area of a wall 7.5m long x 4.5m high with a window opening of

1.200m x 1.050m

Answer Formula L x H 7.5m x 4.5m = 33.75m2 – area of window

1.200m x 1.050m = 1.260m2

33.75m2 – 1.260m2 = 32.49m2

55

Example 1Example 1

Calculate the total of 0.100m x 0.200m paving block required to cover the drive area shown below, allow 5% for cutting/waste 8 m

4 m

Area = 8 x 4 = 32m2

Area of one paving block = 0.100m x 0.200m = 0.02m2

Total blocks required = 32m2 divided by 0.02m2

= 1600 blocks add 5% for cutting/waste

1600 x 105% = 1680 blocks

100

66

Answer the following questionsAnswer the following questions

Calculate the total 0.3m x 0.3m tiles required to cover the floor area shown below and allow 6% for waste.

6 m

4 m

Area = 6 x 4 = 24 m2

Area of one tile = 0.3m x 0.3m = 0.09m2

Total tiles required = 24m2 divided by 0.09m2

= 266.6 tiles add 6% for cutting/waste

266.6 x 106% = 282.596 tiles (round up to 283 tiles)

100

77

Calculate the total 0.225m x 0.450m air-rated blocks are required to cover the wall area shown below and allow 8% for waste.

9 m

Area = 18 x 9 = 162 m2

Area of one block = 0.225m x 0.450m = 0.10m2

Total blocks required = 162m2 divided by 0.10m2 =1620 blocks

= 1620 blocks + 8% for cutting/waste

1620 x 108% = 1749.6 blocks

100

18 m

88

Calculate the area of insulation required for a cavity wall 4.725M long x 2.475M high in which there is a window 1.810M wide x 1.2M high, and allow 5% for wastage.

(A calculator can be used, but show all your workings.)

Area of wall 4.725m x 2.475m = 11.694m2

Area of window opening 1.810m x 1.2m = 2.172m2

Area of wall – area of window = 11.694m2 – 2.172m2 = 9.522m2

5% allowed for wastage = 9.522m2 x 105% = 9.9981m2

100

Total area of insulation required = 9.9981m2

99

Calculating the area of irregular shapesCalculating the area of irregular shapes

8m

4m

5m

Length = average of two sides

8 + 5 divided by 2 = 6.5m

Length = 6.5m

width = 4m

6.5 x 4 = 26m2

Area = Length x Width

1010

Answer the following questions

7m

5m

6.5m


7 + 6.5 divided by 2 = 6.75m

Length = 6.75m

width = 5m

6.75 x 5 = 33.75m2


4 + 12 divided by 2 = 8m

Length = 8m

width = 8m

8 x 8 = 64m2

12m

8m

4m

1111

2m

4m

8m

3m

10m

7m



Length = 6m

width = 7m

6 x 7 = 42m2



Length = 6m

width = 3m

6 x 3 = 18m2

1212

Calculating the volume of concreteCalculating the volume of concrete..When taking off quantities of brickwork, excavation and concrete for a building, it is essential to calculate the perimeter of the structure.

The formula for this differs depending if the internal or external measurements are being used. Here is an alternative method to calculate the volume of concrete required. This is known as the centreline method

Using the external measurements

The mean perimeter = (2 x L) + ( 2 x W ) Less 4 x Thickness of foundations

Using the internal measurements

The mean perimeter = (2 x L) + ( 2 x W ) plus 4 x Thickness of foundations

1313

Fig. 1 shows a foundation trench for a building.Fig. 1 shows a foundation trench for a building.

0.75m

10.00m

8.0

m

The trench in fig 1 is 1.2m deep. Calculate the volume of material to be excavated.

B

A

C D

Formula when using

External dimensions is:

(2 x L) + ( 2 x W )

Less 4 x foundation width.

Volume = L X W x D = m3Length 2 x 10 = 20

Width 2 x 8 = 16

36m (less 4 x 0.75m)

36m – 3m = 33m

33 x 0.75 x 1.2 = 29.7 m3 of concrete required for this foundation

1414

Fig. 2 shows a foundation trench for a building.Fig. 2 shows a foundation trench for a building.The trench in fig 2 is 1.5m deep. Calculate the volume of material to be excavated.

A0.60m

16.5m3.3

m

A

B

DC

Centre-line calculations

Length 2 x 16.5 = 33

Width 2 x 3.3 = 6.6

39.6m (less 4 x 0.60m)

39.6m – 2.4m = 37.2m

37.2 x 0.60 x 1.5 = 33.48m3 of concrete required for this foundation

Formula when using


(2 x L) + ( 2 x W )


Volume = L X W x D = m3

1515

Centre-line calculationsCentre-line calculations

Length 2 x 10 = 20

Width 2 x 4 = 8

28m (plus 4 x 0.65m)

28m + 2.6m = 30.6m

30.6 x 0.65 x 0.50 = 9.945m3 of concrete required for this foundation.

10m

4m 0.65

Depth of foundation 0.50m

Formula when using

Internal dimensions is:

(2 x L) + ( 2 x W )

plus 4 x foundation width.


1616

Centre-line calculations

10m7

m

6m 4m

3m

4m

The trench is 0.65m wide and 0.50min depth. Calculate the volume of concrete required

Length 2 x 10 = 20

Width 2 x 7 = 14

34m (less 4 x 0.65m) = 31.4m

34m – 2.6m = 31.4m


1717


12m

Calculate the quantity of concrete required for this foundation.

N.B Width 0.60m depth 0.45m

18m

3m

Length 2 x 18= 36

Width 2 x 3 = 6

42m (plus 4 x 0.60m)

42m + 2.4m = 44.4m


1818


12m

Calculate (using the centre line method how many bricks would be required to build the one brick thick inspection chamber. (assume 120 bricks per square metre) depth of brickwork is 0.90m

0.91m

0.7

m

Length 2 x 0.91m = 1.82m

Width 2 x 0.7m = 1.4m

3.22m (plus 4 x 0.225m)

3.22m + 0.9m = 4.12m

L D B/M2

4.12 x 0.90 x 120 = 444.96 bricks required to build the inspection chamber

1919

Material which are ordered by weight such as hardcore (per tonne). Material which are ordered by weight such as hardcore (per tonne). To calculate the quantity required we use the following formula.To calculate the quantity required we use the following formula.

1. calculate the volume as we do for concrete 1. calculate the volume as we do for concrete 2. Use the density method (volume x density)2. Use the density method (volume x density)

Example An area is to be filled with hardcore. The area is 8.5m long Example An area is to be filled with hardcore. The area is 8.5m long by 4.2m wide. The average depth is 0.15mby 4.2m wide. The average depth is 0.15m

Calculate the amount of hardcore requiredCalculate the amount of hardcore requiredVolume = 8.5 x 4.2 x 0.15 = Volume = 8.5 x 4.2 x 0.15 = 5.36m5.36m33 (2 decimal places) (2 decimal places)

Total amount = Total amount = Volume x density = tonnageVolume x density = tonnage 5.36 x 2.000kg/m 5.36 x 2.000kg/m33

= = 10.72 Tonnes10.72 Tonnes (we would order 11 tonne) (we would order 11 tonne)

2020

Answer the following questionsAnswer the following questions

An area is to be filled with hardcore. The area is 6.70 long by 5.50m wide,

with an average depth of concrete of 0.20m

1. Calculate the quantity of hardcore required. (Density = 2.000kg/m3)

2. Total cost of the hardcore at £18 per tonne

Remember Volume x density = tonnage

Volume = 6.70 x 5.50 x 0.20 = 7.37m3 ( 2 decimal places only)

7.37 x 2.000kg/m3 = 14.74 tonne

we would order 15 tonne.

2. Cost 15 x £18 = £270.00

2121

An area is to be filled with hardcore. The area is 8.5m long by 4.2m wide. The average depth is 0.15m

Calculate the amount of hardcore required (Density = 2.000kg/m3)

Remember Volume x density = tonnage

Volume = 8.5 x 4.2 x 0.15 = 5.36m3 ( 2 decimal places only)

= 5.36 x 2.000kg/m3

= 10.72 Tonnes

we would order = 11 Tonne.

2222

An area is to be filled with concrete. The area is 10.5m long by 6.1m wide. The average depth is 0.55m

Calculate the amount of concrete required (Density = 2.400kg/m3)

Remember Volume x density = tonnageVolume = 10.5 x 6.1 x 0.55 = 35.22m3 ( 2 decimal places only)

= 35.22 x 2.400kg/m3

= 84.52Tonnes

we would order = 85 Tonne.

2323

Ratios of materialsRatios of materials

Material such as concrete as mixed together in ratios. A typical mix for concrete would be a 1:3:6 mix this means that

1 Portion of cement is required

3 portions of fine aggregate (sharp sand)

6 portions of course aggregate

To produce the required specified mix of concrete.

It is relatively easy to work out volume of concrete

Volume = (L x W x D).

From the volume we would then need to break this down into the proportions of material required to complete the concrete structure.

2424

Example 1

Total portions is 1: 3: 6 = 10

Average density of concrete is 2400 kg/m3

(2.4 tonne of aggregate to produce a cubic meter of concrete)

Divide 2400 by 10 = 240kg therefore

1 portion of cement = 1 x 240 = 240kg

3 portions of fine aggregate = 3 x 240 = 720kg

6 portions of course aggregate = 6 x 240 = 1440kg

Are required to produce 1m3

2525

Example 2

N.B Ratio of facings to commons (that is 3 facings to every 1 common)

Flemish bond is specified by the customer to construct a wall 23.6 long by 1.36 in height. Calculate the amount of facings and commons that would be required to complete this.

Area of wall 23.6 x 1.36 = 32.095m2

Total no of bricks required = 32.095 x 120 = 3851.52 bricks

(120 bricks per m2 in any one brick wall)

Add together 3 and 1 together = 4

Therefore divide 3851.52 by 4 = 962.88 (is equal to one portion)

Commons 1 x 962.88 = 962.88 bricks

Facings 3 x 962.88 = 2888.64 bricks

check = 3851.52 total

2626

Area of wall 15m x 2m = 30m2

Total no of bricks required = 30 x 120 = 3600 bricks

(120 bricks per m2 in any one brick wall)

Add together 3 and 1 together = 4

Therefore divide 3600 by 4 = 900 (is equal to one portion)

Commons 1 x 900 = 900 bricks

Facings 3 x 900 = 2700 bricks

check = 3600 total

Flemish bond is specified by the customer to construct a wall 15m long by 2m in height. Calculate the amount of facings and commons that would be required to complete this.

2727

Calculating diagonalsCalculating diagonals

Dimension A = 10m

Dim

en

sio

n B

= 8

m

Dimension C =

?

Remember the formula for working out diagonal dimensions is:

A2 + B2

A = 10 x 10 = 100

B = 8 x 8 = 64

100 + 64 = 164 square root of 164

√164 = 12.8m length of dimension C = 12.8 m

2828

Calculating diagonalsCalculating diagonals

Dimension A = 12m

Dim

en

sio

n B

= 7

m

Dimension C =

?

Remember the formula for working out diagonal dimensions is:

A2 + B2

A = 12 x 12 = 144

B = 7 x 7 = 49

144 + 49 = 193 square root of 193

√193 = 13.89m length of dimension C = 13.89m

2929

Half of total roof span

2.500m

2.1

00m

Ris

e o

f ro

ofThe diagram shows a section

through a roof.

(Note rafter AB = AC2 + BC2)

CB

A

Calculation

AC2 = 2.100 x 2.100 = 4.41m2

BC2 = 2.500 x 2. 500 = 6.25m2

AC2 + BC2 = 4.41 + 6.25 = 10.66

√ 10.66

3.26m is the length of rafter AB

3030

3m

The diagram below shows a foundation of a building in the process of being set out. Calculate the length of X mathematically to ensure that the conservatory is square to the rear elevation of the building.

7mX

(Note length of X = √ AC2 + BC2)

B

A

C

Calculation

AC2 = 7m x 7m = 49m2

BC2= 3m x 3m = 9m2

AC2 + BC2 = 49 + 9 = 58

√ 58 = 7.61m is the length of the diagonal X

conservatory

900

3131

Calculating areas of trianglesCalculating areas of triangles

The formula required to do this is :-Area = ½ Base dimension x Height = m2

8m

10m

Base 8 ÷ 2 = 4

4 x 10 = 40m2

3232

Calculate the areas of triangles below

7m 12m 37.7m

8m

12.3

m

19.5

m

A)

Base = 7 ÷ 2 = 3.5m

3.5 x 8 = 28m2

B)

Base = 12 ÷ 2 = 6m

6 x 12.3 = 73.8m2

C)

Base = 37.7 ÷ 2 = 18.85m

18.85 x 19.5 = 367.5m2

A) B) C)

3333

Calculating the area of a circleCalculating the area of a circle

The distance around a circle is called its circumference.

The distance across a circle is called its diameter. diameter

For any size of circle, if you divide the circumference by its diameter, the value of 3.14 will be the answer. This is called (Pi).

Radiu

s

The distance of half the diameter is called the radius.

Example 1. The radius of a circle is 3 in. What is its area ?

Area of circle = (Pi) x R x R

= 3.14 (Pi) x 3 x 3 = 28.26 inch 2

R = 3in

3434

Calculate of the areas of the circles below.Calculate of the areas of the circles below.

R = 4.0m

R = 10cmArea of circle = (Pi) x R x R

(Pi) = 3.14 x 10 x 10 = 314cm 2


(Pi) = 3.14 x 4 x 4 = 50.24m 2

3535

Example 2. The diameter of a circle is 8 cm. What is its area ?

Solution D = 8

R = D divided by 2 = 4cm

Area = (Pi) 3.14 x 4 x 4 = 50.24 cm2

D = 8 cm

Basic rule is to use the radius to calculate the area of a circle, therefore if a question only has the diameter size then dividing by 2 supplies us with the radius size.

3636

Calculate of the areas of the circles below.

D = 15m

D = 300mm

D = 15m

R = D divided by 2 = 7.5m

Area = (Pi) 3.14 x 7.5 x 7.5 = 176.625m2

D = 300mm

R = D divided by 2 = 150mm

Area = (Pi) 3.14 x 150 x 150 = 70650mm2

3737

The radius of a circle is 9cm. What is its area ?

Area of circle = D divided by 2 = radius

(Pi) x R x R

3.14 x 6 x 6 = 113.04 in2

The radius of a circular pool is 4ft. What is its area ?

D = 12 in

R= 9cm

R= 4ft


3.14 x 9 x 9 = 254.34 cm2


3.14 x 4 x 4 = 50.24 ft2

The diameter of a circle is 12 in. What is its area ?

3838

Example 3. The area of a circle is 78.5m2. What is its radius ?

Solution Area divided by (Pi) = R x R

78.5m2 ÷ 3.14 = 25

25 = R x R

Therefore the square root of 25 will give us size of the radius.

We now need to find the square root of 25. Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself. I.e the square root of 10 is 10 x 10 which = 100

We can do this with the aid of a calculator by simply pressing the symbol after entering the square number.So the answer for the sum above is : 25 = 5

3939

12m

6m

R =3m

The figure opposite shows a plan view of a timber floor.

The floor area is

64.28m2

86.13m2

100.28m2

158.14m2

Area of rectangle = L x H = m2

12 x 6 = 72m2 + area of circle


3.14 x 3 x 3 = 28.26 m2

Answer = 72 + 14.13 = 86.13 m2

4040

Example 1

Calculate the total cost of the materials below from the prices given.

Allow 17.5% VAT

1500 Facing bricks are required to construct a wall @ £230/ 1000

1. calculate the cost of 1500 bricks

2. Calculate cost of bricks + vat

1. Cost of bricks = 1500 x £230 = £345.00

1000

2. VAT= 345 x 117.5% = £405.375 (cost of bricks including vat)

100

4141

Example 2

Calculate the total cost of materials using the prices given, allowing

17.5% VAT

400 concrete blocks are required to construct a wall @ £6.50/ 10

1. calculate the cost of 400 blocks

2. Calculate cost of blocks + vat

100mm Concrete blocks required @ £6.50/ 10

Cost of blocks = 400 x £6.50 = £260.00

10

VAT = 260 x 117.5% = £305.50 (cost of blocks including vat)

100

4242


17.5% VAT

A) 2100 Facing Bricks required @ £247/ 1000

Cost of bricks = 2100 x £247 = £518.70

1000

VAT 518.70 x 117.5 = £609.47 (total cost of bricks including vat)

100

4343

Cost of blocks = 650 x £4.95 = £321.75

10

VAT 321.75 x 117.5 = £378.05 (total cost of blocks including vat)

100


17.5% VAT

B) 650 100mm Concrete Blocks required @ £4.95/ 10

4444


17.5% VAT

C) 3.5 Tonne Building Sand required @ £18/ Tonne

Cost of sand = 3.5 x £18 = £63.00

VAT 63.00 x 117.5 = £74.02 (total cost of sand including vat)

100

4545


17.5% VAT

D) 250 Wall Ties required @ £17.00/ 100

Cost of wall ties = 250 x £17.00 = £42.50

100

VAT 42.50 x 117.5 = £49.93 (total cost of wall ties including vat)

100

4646

Assessment questionsQuestion 1

Calculate the total number of bricks required to build the ½ B wall shown, allowing 5% waste.

1) 12.00m x 3.00m = 36m2

Door = 2.00m x 1.00m = 2m2

Window = 2.00m x 1.50m = 3m2

36m2 – 5m2 = 31m2

2) 31 x 60 = 1860 bricks 1860 x 105 = 1953 bricks including waste

100

Main area – door and window

4747

Question 2

The uppermost part of a gable wall forms the shape of a triangle.

Calculate the total number of bricks required to build the ½ B gable wall shown, add 3 ½% for waste.

Area of a triangle = half base x height

9 divided by 2 = 4.5m x 2.0m = 9 m2

9 x 60 = 540 bricks + 3.5 % waste

540 x 103.5 = 558.9 bricks (round up to 559)

100

4848

Question 3

An area is to be filled with hardcore. The area is 6.70m long by 5.50m wide. The average depth of hardcore is 0.20m.

1) Calculate the amount of hardcore required. (Density = 2.000kg/m3)

2) Total cost of the hardcore at £30 per tonne, including VAT at 17 ½ %Area to be filled with hardcore =

6.70 x 5.50 x 0.20 = 7.37m3

Volume x density = tonnage 7.37m3 x 2.000 = 14.74 tonnes

(round up to 15 tonnes) 15 x £30 = £450.00

450 x 17.5 = 78.75

100

total cost = 450 + 78.75 = £528.75

4949

Question 4

An area is to be filled with hardcore. The area is 7.00 long by 5.50m wide, with an average depth of 0.20m

1. Calculate the quantity of hardcore required. (Density = 2000kg/m3)

2. 2. Total cost of the hardcore at £27 per tonne

Area to be filled with hardcore = 7.00m x 5.50m x 0.20m = 7.7m3

Volume x density = tonnage 7.7m3 x 2.000 = 15.4 tonnes

(round up to 16 tonne)

16 x £27= £432

5050

Question 5

Calculate the area of the circle which has a radius of 9m.

Area of circle = (Pi) x R x R = m2

3.14 x 9 x 9 = 254.34m2

5151

Question 6

1750 Facing bricks are required to construct a wall @ £320/ 1000

1. calculate the cost of 1750 bricks

2. Calculate cost of bricks + 17.5 % vat

1750 x 320 = £560

1000

560 x 117.5 = £658

100

5252

Formula when using


(2 x L) + ( 2 x W )



6 x 2 = 12m

4 x 2 = 8m

Less 4 x width of foundation 4 x 0.4m = 1.6m

12m + 8m = 20m – 1.6m = 18.4m

Volume 18.4 x 0.4 x 0.5 = 3.68m3

Question 7

Calculate the concrete required for the foundation shown below, given that the trench width is 0.4m and the concrete is to be 0.5m deep.

N.B all measurements are external

5353

Question 8

Calculate the area of the circle which has a diameter of 29m.

Diameter divided by 2 = radius

Area of circle = (pi) x R x R

3.14 x 14.5 x 14.5 = 660.18 m3

Question 9

Add the following dimensions together 1240mm + 1.2m + 2m +1537mm

1.240

1.200

2.000

1.537

5.977m or 5977mm

5454

Question 10

An area which has a length of 16m and width of 9m requires tiling.

1) Calculate the number of 0.100m x 0.200m tiles required to cover the

area.

2) Add 13% for waste

Area to be tiled = 16m x 9m = 144m2

Area of one tile = 0.100m x 0.200 = 0.02m2

144m2 divided by area of one tile = 144 divided by 0.02 = 7200

Add 13% for waste 7200 x 113 = 8136 tiles required

100

1 calculations on completion of this unit the learner should be able to demonstrate an understanding...

Documents

long x

wide x

cuttingwaste1600 x

522m2 x

floor area

length x widthanswer

drive area

total area of insulation