# Civil Engineering Chapter 1 Homework Averages Record The Distance Averages Are Used

Document Type

Homework Help

Book Title

Landscape Construction 3rd Edition

Authors

David Sauter

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Exercise 1

Construction Math

OBJECTIVE

The objective of this exercise is to properly perform

construction-related math equations.

TEXTBOOK REFERENCE

Information related to this activity can be found in the

Landscape Construction textbook in Chapter 4, Construc-

tion Math.

INTRODUCTION

Whether counting the number of shrubs needed for a

planting bed or determining how much concrete to order,

is essential when working on a project, as contractors

typically build upon primary calculations to determine

subsequent numerical values.

The basic math functions covered by this lab manual

include performing item counts, calculating averages,

making linear measurements, and calculating perim-

eters, areas, volumes, and weights. It is assumed that

the reader has already mastered basic math calculations

such as addition, subtraction, multiplication, and divi-

sion of whole numbers, fractions, and decimals. For the

reader’s reference, a written description of geometric

shapes is provided in Table 1–1, and formulas for use in

perimeter, area, and volume calculations are featured in

Figures 1–2, 1–5, and 1–6.

Table 1–1 Common geometric shapes.

• Square/rectangle/parallelogram: Shapes with

two sets of parallel sides. Squares and rectangles

have right-angled corners and parallelograms have

no right-angled corners.

• Trapezoid: Shape with four sides, but only one set of

parallel sides.

• Circle: Round shape with edges equidistant from center.

• Circle sectors: Partial segments of a circle. Defined

by length of the outside arc or the enclosed angle of a

circle measured in degrees.

• Ellipse: Rounded egg shape with edges at variable

distances from center.

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count, use a plan or installed design to count how many

of a particular object are used in a project. Performing

item counts for bids, estimates, and material ordering

may require separating items into groups of materials

that are the same.

Averages

Averages are used when the contractor needs to calculate

one typical, or average, number from several different

where n equals the number of numbers.

When performing landscape calculations:

• Decimals should not be rounded any smaller

than hundredths (two places).

• Fraction additions require determining the

lowest common denominator.

Linear Measurements

A foundational measurement used in landscape con-

tracting is the linear measurement, or length, of a particu-

lar distance or the dimensions of an object. Items such as

Direct Measurement. Direct measurement involves the

use of a tape measure, measuring wheel, or other measur-

ing device to determine the distance between two points.

To direct measure:

• On flat surfaces, stretch or run the measuring

device between the two points to be measured and

record the distance.

• On sloped surfaces, extend a tape measure

from the highest point along the distance being

method will give an accurate measurement of the

horizontal distance between the two points along

the slope, not the diagonal distance.) (Figure 1–1.)

Pacing. The pacing method of measurement uses an

individual’s typical step to determine distances.

To measure by pacing:

• Determine the average length of your pace. To

do this, set up a course that is a known distance

(e.g., 100 feet), and pace the course three times,

counting the number of steps required to walk

the length of the course. Calculate your average

number of steps. Divide the distance by the

2 Exercise 1 Construction Math

0' on tape

0' on tape

Level tape

Level tape

Plumb bob

Plumb bob

Read measurements

Add measurements together

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pacing practice until you are routinely within

5 percent of an actual, measured distance.

Estimation. Estimating requires determining a distance

based on comparison with a known dimension. When

accuracy is not critical, estimating linear dimensions is a

viable option.

Estimation can be performed by:

• Comparison. This requires visually comparing the

distance to be measured to an object with known

measurements. Items such as door openings

(usually 7 feet high), and sidewalk squares can be

Perimeter Measurements

Perimeter calculations determine the total linear mea-

surement of the outside edges of an area or object. Perim-

eter measurement totals are typically expressed in linear

footage, or LF. To perform perimeter calculations, use the

following steps:

• Identify the subject being measured as one of the

standard geometric shapes identified in Table 1–1

or as an irregular shape.

• If the subject is a standard geometric shape,

perform the necessary linear measurements

required to calculate the shape’s perimeter. Use

the formula listed in Figure 1–2 to discern what

Area Measurements

Area calculations determine the surface area of objects

and spaces found in the landscape. Area measurement

totals are typically expressed in square footage, or SF.

Before performing area calculations, the contractor must

determine if the object or space being measured matches

a typical geometric form or is irregularly shaped.

Area Measurements for Standard Geometric Shapes.

When an area calculation needs to be performed for an

• Perform the necessary linear measurements

required to calculate the subject’s area, as listed in

Figure 1–2.

• Use the formula listed in Figure 1–2 to calculate

the area. See Figure 1–3 for examples.

Area Measurements for Irregular Shapes. Most

objects and spaces in the landscape can be broken down

into a collection of shapes that approximate measurable

geometric forms. Once the dimensions of these shapes

calculate area by using the “sum of shapes” method, fol-

low these steps:

• Analyze the object or space for which an area

measurement is required. Observe what geometric

forms are recognizable within the boundaries of

the overall shape. These shapes should cover the

majority of the space being measured without

much overlap. (Figure 1–4, Step A.)

• Using the formulas in Figure 1–2, take the

necessary linear measurements and perform

the calculations required for each identified

shape.

• Total the answers to obtain the area for the entire

object or space. Be certain all calculations are

footage by 100. To obtain rolls of sod (each sod

roll being 1 SY), divide square footage by 9.

Conversions such as number of unit pavers require

the square footage derived from the previously

mentioned calculations and a conversion

number provided by the manufacturer of

the material (such as 4.5 bricks required for

every square foot of paved area). Multiply the

Exercise 1 Construction Math 3

97171_01_ch01_p001-010.indd 3 14/06/10 8:38 PM

conversion number by the squarefootage to

obtain a material quantity estimate.

Volume Measurements

Volume calculations provide the three-dimensional bulk,

or cubic, measurement of materials used in landscaping.

Volume measurement totals are typically expressed in

cubic footage, or CF. To calculate volumes for landscap-

ing, follow these steps:

• Place the numbers in the following formula:

Area (in SF) × Depth (in inches) = CF

12

Note that the depth is not converted to feet, but is left

in inches. Examples are shown in Figure 1–5. To calcu-

late the volume of a cylinder use the following formula

(Figure 1–6). Both area and height must be expressed as

4 Exercise 1 Construction Math

Circle

R = radius

D = diameter

π = pi, value of 3.14

per: πD

area: πR2

Ellipse

S = short radius

L = long radius

per: 2π

area: πSL

Circle Sector

θ = angle degrees

L = arc length

per: (2R) + L

area: πR2 ×

(

––

)

per = Perimeter

D

R

S

L

L

R

(S2 + L2)

2

360

θ

θ

–––––

97171_01_ch01_p001-010.indd 4 14/06/10 8:38 PM

Exercise 1 Construction Math 5

Calculating the volume of a sphere is a necessary first

step in determining the weight of plant rootballs. Once

the volume has been calculated, a weight conversion can

be used to convert to pounds. The formula for calculating

the volume of a sphere is as follows (Figure 1–6):

4.18 × R × R × R = CI (cubic inches)

• If the measurement carried out is in cubic inches

and must be converted to cubic feet, divide the

cubic inches total by 1,728.

Weight Calculations

Many landscape materials are purchased by weight rather

than by volume. For those materials use the following

Square/Rectangle

per: (2 × 9') + (2 × 9')

= 36 LF

area: 9' × 9' = 81 SF

Parallelogram

per: (2 × 20') + (2 × 16')

= 72 LF

area: 20' × 14' = 280 SF

Trapezoid

per: 12' + 9' + 10' + 10.5' = 41.5 LF

area: ––––– × 10' = 105 SF

9' 16'

9'

10.5'H = 10'10'

12'

14'

(12' + 9')

2

16'

C = 22.6

16'

10' 10'

8'

10'

15'

12'

10'

5'

12'

10'

7'

9' 20'

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6 Exercise 1 Construction Math

A

20'

20' D

C

5'

5'

10'

12'

E15' F

7.5' R

Project area

A. Identify shapes and measure dimensions of each

–

Figure 1–4 Calculating area for irregular shapes.

Volume formula:

Examples:

A. 6" layer of topsoil over square shape of 81 SF

area (in SF) × depth (in inches) = CF (cubic feet)

12

97171_01_ch01_p001-010.indd 6 14/06/10 8:38 PM

Exercise 1 Construction Math 7

CI

1728

Volume of Sphere:

Formula: 4.18 × R × R × R = CI

Example: 4.18 × 8 × 8 × 8 = 2140 CI

Conversion to CF: –––––– = CF

2140

1728

–––––– = 1.24 CF

8" R

• 1 CY aggregate (class 5 aggregate, 1 inch

roadstone, or equal) = 1.25 tons

• 2,000 pounds = 1 ton

Comprehensive Process of Converting

Measurements

If this section has been followed from beginning to end,

it should be apparent that primary measurements and

calculations, such as linear measurements and area

calculations, can be used to determine subsequent

numerical values during the course of a project.

eliminating the need to begin each calculation from

scratch using linear measurements.

PREREQUISITE EXERCISES

None.

MATERIALS REQUIRED

• Calculator

• Writing materials

• 100 foot tape measure or measuring wheel

97171_01_ch01_p001-010.indd 7 14/06/10 8:38 PM

8 Exercise 1 Construction Math

Figure 1–7 Calculating averages problem set.

139 27 18 9 3

)++++

=

n

24 2 16 7

)++ + =

n

614 11 21 39 22

)+++ + =

n

75 6 11 4

)++ + =

n

122 17 83 77

)....+++ =

n

233 76 91 26 10

).....++++ =

n

615 53 6 33 7 2 1 11 3 2

).....++++

=

n

7413 344 288

)...++ =

n

113 5

1

2

1

2

1

2

)++ =

n

2315

1

4

3

8

5

16

)++ =

n

n

695 11

6

8

7

16

1

4

)++ =

n

711 12 9 1

3

8

2

8

5

8

4

8

)+++

=

n

n

EXERCISE DESCRIPTION—PART C

To complete this exercise, perform direct measurements

and estimations of linear dimensions for objects identi-

fied by the instructor.

EXERCISE DESCRIPTION—PART E

To complete this exercise perform area calculations for

each of the shapes shown in Figure 1–8. Place your

responses in the form in Figure 1–9.

© Delmar/Cengage Learning.

97171_01_ch01_p001-010.indd 8 14/06/10 8:39 PM

Exercise 1 Construction Math 9

10'

12'

13' 10'

10' 10'

15'

10'

10'

12'

8' 8'

7'

B.

following materials. Place your responses in the form in

Figure 1–9.

• 1 inch thick layer of concrete sand, express

answers in cubic yards (CY)

• 6 inch thick layer of aggregate base material,

express answers in cubic yards (CY)

• 8 inch thick layer of dry, loose soil, express

answers in cubic yards (CY)

EXERCISE DESCRIPTION—PART G

97171_01_ch01_p001-010.indd 9 14/06/10 8:39 PM

10 Exercise 1 Construction Math

Shape

A B C D E

Measurement

Perimeter

Area

Volume (CY)

6" aggregate

Volume (CY)

8" soil

97171_01_ch01_p001-010.indd 10 14/06/10 8:39 PM

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