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Area of a Triangle
Area by Reducing to Triangles

Engineering Aid 3 - Beginning Structural engineering guide book
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The length of AB is called the base (b), and the length of DC, the altitude (h); therefore, your for- mula  for  determining  the  area  of  an  oblique triangle  is  again  A  =  1/2bh. You must remember that in a right triangle h  corresponds  to  the  length  of  one  of  the  sides, while in an oblique triangle it does not. Therefore, for a right triangle with the length of the sides given, you can determine the area by the formula A  =  1/2bh.  For  an  oblique  triangle  with  the length  of  the  sides  given,  you  cannot  use  this formula unless you can determine the value of h, Later in this chapter you will learn trigonometric methods  of  determining  areas  of  various  forms of triangles on the basis of the length of the sides alone. Area of a Rhombus or Rhomboid Figure  1-9  shows  a  rhomboid,  ABCD.  If  you drop  a  perpendicular,  CF,  from  L C to AD, and project  another  from  LA to BC, you will create two  right  triangles,   AAEB  and  ACFD,  and  the rectangle  AECF.  It can be shown geometrically that the right triangles are similar and equal. You  can  see  that  the  area  of  the  rectangle AECF  equals the product of AF x FC. The area of the triangle CFD equals 1 /2(FD)(FC). Because the triangle AEB is equal and similar to CFD, the area  of  that  triangle  also  equals  1/2(FD)(FC). Therefore, the total area of both triangles equals (FD)(FC). The total area of the rhomboid equals the area of the rectangle AECF + the total area of  both  triangles. The  total  area  of  the  rhomboid  equals (AF)(FC)  +  (FD)(FC),  or  (AF  +  FD)(FC).  But AF  +  FD  equals  AD,  the  base.  FC  equals  the altitude.  Therefore,  the  formula  for  the  area  of a  rhomboid  is  A  =  bh.  Here  again  you  must Figure  1-10.-Trapezoid. remember that h in a rectangle corresponds to the length of one of the sides, but h in a rhombus or rhomboid  does  not. Area of a Trapezoid Figure 1-10 shows a trapezoid, ABCD. If you drop  perpendiculars  BE  and  CF  from  points  B and C, respectively, you create the right triangles AEB and DFC and the rectangle EBCF between them. The area of the trapezoid obviously equals the  sum  of  the  areas  of  these  figures. The  area  of   AAEB    equals   1/2(AE)(FC), the   area   of  ADFC    equals   1/2(FD)(FC), and  the  area  of  EBCF  equals  (EF)(FC).  There- fore,  the  area  of  the  trapezoid  ABCD  equals l/2(AE)(FC)   +   (EF)(FC)   +   1/2(FD)(FC),   or However,  2EF  =  EF  +  BC.  Therefore,  the  area of  the  trapezoid  equals But  AE  +  FD  +  EF  =  AD.  Therefore,  the  area of  the  trapezoid  equals AD  and  BC  are  the  bases  of  the  trapezoid  and are usually designated as bl and b2,  respectively. FC is the altitude and is generally designated as h.  Therefore,  the  formula  for  the  area  of  a trapezoid  is Figure   1-9.-Rhomboid. 1-11







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