external area calculation.docx

8/10/2019 EXTERNAL AREA CALCULATION.docx

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CALCULATION OF EXTERNAL AREA OF A SIMPLE CURVEUSING TURALDES EXTERNAL RULE

HENRY P. TURALDEAssociate Professor

College of Engineering

Camarines Sur Polytechnic CollegesNabua, Camarines Sur, Philippines

Finding the area bounded by the curve and its tangents, herein referred to as the external area,is conventionally done by finding the component areas of regular shapes then making use of itscombinations to get the area of the problem section. An alternative solution using Turaldes ExternalRuleformula shows a simplified approach in the calculation of the external area.

Sample Problem 1: Find the area of the shaded part of a square with an inscribed quadrant of a

circle

A.Conventional M ethod

For this problem, a square has an inscribed quadrant of a circle and the curve is tangent to the twosides of the square. The shaded portion is the external area.

Figure 1

A common approach is by calculating the areas of the square and the quadrant of the circle. For example,

suppose the side of the square isR.

.Calculate the area of the quadrant of a circle.

.Subtract the area of the quadrant of the circle from the area of the square to determine the area outside the circle

within the square.

B.Turaldes External Rule

The area of the external is one-third the product of the length of

curve and its external distance.

(Turaldes xternal Rule Formula)

= 90o


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Where Lc = length of curve

Lc =

, that is the circumference of the circle divided by 4

Lc = = 1.5708R

E = external distance of the curve (distance from vertex to the curve)

() = (

() ),

(Since 90)

0 Therefore, =

()()

Sample Problem 2: Find the area of the external of a simple curve with a central angle of 60

and with radius R


A common approach to this problem is done by calculating the area of the triangles less the area of thesector of a circle.

Figure 2

Calculate the area of the quadrilateral.

() Where T= tangent distance

T = R Tan ()

Since 60, Tan () = 0.5774

[ ()] () 0

Calculate the area of the sector of the circle.

, Since = 60, 06 ,

Subtract the area of the sector of the circle from the area of the quadrilateral.

0

06

00

= 60


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(Turaldes xternal Rule Formula)Lc

= length of curve

Lc =

, Since = 60

Lc = 1.0472R

E = external distance of the curve (distance from vertex to the curve)

( () )

Since = 75, 0 Therefore, =

( )( )

Sample Problem 3: Find the area of the external of a simple curve with a central angle of 120and with radius R

Figure 3


Calculate the area of the quadrilateral.

() T = R Tan ()

Since = 120, Tan () = 1.7321

[ ()] ()

Calculate the area of the sector of the circle.

,

Since = 120, 0,Subtract the area of the sector of the circle from the area of the quadrilateral.

0 069

= 120

o


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(Turaldes External Rule Formula)Lc

= length of the curve

Lc =

,

Since = 120, Lc = 2.0944RE = external distance of the curve (distance from vertex to the curve)

( () ),

( = 120), Therefore, =

( )()

Looking through the formula

A.

Conventional M ethod

Tan (/2)

=

60

[Tan (/2)] (

60 )

= [ Tan (/2) ( )]B.

Turaldes External Rule

(Turaldes External Rule Formula)

Lc =

( () )

[ ] [(

() )]

(

) [

()

]

Figure 4


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Table 1Comparison of Calculated Areas

Central angle,

Area by

Conventional

Method

Area by

Turaldes External

Rule Formula

Difference in

Area

Relative

Precision

45 00 00 0 -50 000 000 0 -55 00 00 0 -60 00 00 0 -65 000 000 0 -70 009 0090 000 -75 0 0 000 -80 0 0 000 -

85 0 06 000 -90 0 R 0 000 1/11095 06 06 000 1/90

100 09 0 000 1/80105 0 09 0006 1/60110 06 06 000 1/60115 066 06 000 1/60120 06 069. 00 1/50125 00 0 00 1/50130 00 0 00 1/40135 6 6 00 1/40

Note: The difference in calculated areas between the two methods increases as the central angle

increases.

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