E4004 Surveying Computations A

Area Problems

To Cut Off an Area by a Line Passing Through a Fixed Point• The bearing and distance BP is known

B

P

X

BrgDist

Brg

• The bearing BX is known• The required area BPX is known

A

• Calculate the bearing and distance PX and the distance BX

To Cut Off an Area by a Line Passing Through a Fixed Point• The angle at B is determined from

the bearing difference

B

P

X

BrgDist

Brg

• The general formula for the area of a triangle is

A

CabABC sin2

1

PBXBXBPPBX sin*2

1

PBXBP

PBXBX

sin

*2

C

B

A

a

b

To Cut Off an Area by a Line Passing Through a Fixed Point

• The bearing & distance of the line PX can be calculated by closing PBX

B

P

X

BrgDist

Brg

• Also check that the area PBX calculates to the correct area by using the CLOSE programA

PBXBP

PBXBX

sin

*2

To Cut Off an Area by a Line Passing Through a Particular Point• A farmer wants to fence off a particular

area from a large paddock.There is an existing trough which must be accessible to stock on both sides of the new fence.

To Cut Off an Area by a Line Passing Through a Particular Point

• The bearings of BC and BD are known.

B

C

D

Brg

Brg

• The bearing and distance BP can be measured.

BrgDist

P

• The required area is A

A

To Cut Off an Area by a Line Passing Through a Particular Point

• Note that there will be two solutions

B

C

D

Brg

Brg

BrgDist

P

• Such that

C’

D’

'" PDDAreaPCCArea

A

To Cut Off an Area by a Line Passing Through a Particular Point

B

C

D

Brg

Brg

BrgDist

P

x

y

A

• Let

CBP PBD

yBD xBC

To Cut Off an Area by a Line Passing Through a Particular Point

B

C

D

Brg

Brg

BrgDist

P

x

y

sin**2

1BPxBCP

sin**2

1BPyBPD

sin**2

1sin**

2

1BPyBPxA

yx

BCD

yxBCD

yxBCD

sin*

*2

sin***2

sin**2

1

A

To Cut Off an Area by a Line Passing Through a Particular Point

B

C

D

Brg

Brg

BrgDist

P

x

y

sin**2

1sin**

2

1BPyBPxA y

x

A

sin*

*2

A

sin**sin*

*2*

2

1sin**

2

1BP

x

ABPxA

sin** xA

sin***sin*sin**2

1*sin* 2 BPAxBPxA

Multiply both sides of the

equation by, x sin()

Re-write in terms of x

sin**sin**2

1xBPx sin**BPA

To Cut Off an Area by a Line Passing Through a Particular Point

B

C

D

Brg

Brg

BrgDist

P

x

y

A

sin***sin*sin**2

1*sin* 2 BPAxBPxA

sin*sinsinsin2

10 2 BPAxAxBP

)sin(sin2

1 BPa

)sin( AbsinBPAc

This equation is in quadratic form and can be solved for x

Make the LHS equal zero

To Cut Off an Area by a Line Passing Through a Particular Point

• Write a program to solve for x in a quadratic given values for a, b and c

• OR write a solver program which will solve for x, a, b or c

02 cbxaxbc

acbbx

2

42

To Cut Off an Area by a Line Passing Through a Particular Point When the Figure is not a Triangle

• It is required to cut off a given area CQRSTD by a line passing through P

C

Q

R

S

T

D

P

• The bearings and distances QR, RS and ST are known whilst the position of P has been located from Q

Brg & Dist

Brg & Dist

• Only the bearings are known for CQ and TD

Brg

Brg

A

To Cut Off an Area by a Line Passing Through a Particular Point When the Figure is not a Triangle

• Extend CQ and DT to intersect at B

C

Q

R

S

T

D

P

• The figure CBDF is the same as that formed in the earlier example provided the required area is made equal to the sum of Area QRSTB and A

Brg & Dist

Brg & Dist

Brg

Brg

B

A

• The dimensions of lines TB and BQ can be calculated by closing QRSTB and the line BP by closing BQP