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Chapter 12, Problem 12.1P
To determine

The area enclosed within polygon ABDFGA.

Expert Solution & Answer
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Answer to Problem 12.1P

The area enclosed within polygon ABDFGA is AABFDGA=280842.66ft2

Explanation of Solution

Given:

A closed polygon ABDFGA

Concept Used:

The formula for area of a triangle is

  A=s(sa)(sb)(sc)     ....... (1)

The formula for semi-perimeter of triangle is

  s=a+b+c2     ....... (2)

Here, A is area of triangle and s is semi-perimeter of triangle and a,b,c are the three sides of the triangle.

The area of polygon ABDFGA can be calculated by considering smaller triangles which together form the polygon.

  AABFDGA=ADFG+ABDG+AABG     ....... (3)

Calculation:

Consider the triangle DFG

Substitute 219.1ft for a , 567.6ft for b , 588.7ft for c in formula (2)

  s=219.1+567.6+588.72=1375.42=687.7ft

Substitute 219.1ft for a , 567.6ft for b , 588.7ft for c , 687.7ft for s in formula (1)

  ADFG=687.7( 687.7219.1)( 687.7567.6)( 687.7588.7)=687.7×468.6×120.1×99=3831594230=61899.87ft2

Consider the triangle BDG

Substitute 298.3ft for a , 588.7ft for b , 551.8ft for c in formula (2)

  s=298.3+588.7+551.82=1438.82=719.40ft

Substitute 298.3ft for a , 588.7ft for b , 551.8ft for c , 719.40ft for s in formula (1)

  ABDG=719.40( 719.40298.3)( 719.40588.7)( 719.40551.8)=719.40×421.1×130.7×167.6=6635983183=81461.54ft2

Consider the triangle ABG

Substitute 536.6ft for a , 551.8f for b , 610.5ft for c in formula (2)

  s=536.6+551.8+610.52=1698.92=849.45ft

Substitute 536.6ft for a , 551.8f for b , 610.5ft for c

  849.45ft , for s in formula (1)

  AABG=849.45( 849.45536.6)( 849.45551.8)( 849.45610.5)=849.45×312.85×297.65×238.95=18901091350=137481.24ft2

Substitute 137481.24ft2 for AABG , 81461.54ft2 for ABDG , 61899.87ft2 for ADFG in the equation (3),

  AABFDGA=ADFG+ABDG+AABG=61899.87+81461.54+137481.24=280842.66ft2

Conclusion:

The area enclosed within polygon ABDFGA is AABFDGA=280842.66ft2

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