Triangle Solutions Solver equation 

The HP Technical Applications manual for the HP 19BII business calculator presents a Solver equation to compute the sides, angles and areas of various triangles. This is a common problem in geometry and trigonometry.

The solution presented in the Technical Manual, as well as the modified version given here, shows how to use the IF(S( )) function in Solver.

The modified version has one important caveat, namely, to make it work properly you have to label the sides and angles of a triangle in a clockwise direction so that they are consistent with the illustration in Screen 1.

 Furthermore, the first "S" in any of the configurations must always be labeled S1. This drives the position and labeling of the other sides and angles. For example, an ASA configuration would require A to be A3, S to be S1 and A to be A1. Note the clockwise direction of the labeling. It is useful to view or draw the labeling in a circular format so that S1 follows A3, A1 follows S1, S2 follows A1 and so on.

 As a check, after labeling the sides and angles, make sure that the side S1 is adjacent (in a clockwise direction) to A1 and also S3 is opposite A1.

Following these rules will ensure correct orientation of the triangle in question.

 The procedure is as follows:

1. Draw the triangle to be solved.

2. From the 5 configurations available (ASA, SAA, SAS, SSA & SSS) identify which one suits the problem you are dealing with e.g. SAS/side-angle-side. If none matches the problem, you must do an extra calculation for the side variable you need. For example if you had a "AAS" situation you would have to change it to SAA say, by finding the other angle. It is useful to write down the inputs as follows (SAS pattern):

S1 = x; A1 = y; S2 =z.

 3. Enter the three variables e.g. S1, A1, S2 (for a SAS configuration).

 4. Solve for the situation you have identified i.e. press SAS or, highlight the SAS variable and press the spacebar.

Here are a couple of examples. The second example uses A, B, C in place of A1, A2, and A3 and a, b, c as the sides S1, S2 and S3.

 Example 1: A ramp is to be made to ascend a height of 2m, at a slope of 18 degrees with the horizontal. How long will the ramp be? Answer = 6.16m

 Example 2: A typical "pythagoras" right angled triangle with the angle ABC = 90'; AB = 3cm; BC = 4cm. Find the Area; length of AC; angle CAB and angle BCA.

 Answer = Area = 6sq/cm; AC = 5cm; angle CAB = 53.13'; angle BCA = 36.87'

 Here is the Solver equation that you may key into the Solver editor.

0*(S1*S2*S3*A1*A2*A3*ASA*SAA*SAS*SSA*SSS)+IF(S(SSS) OR S(SAS),IF(S(SAS),0*L(S3,SQRT(SQ(S1)+SQ(S2)-2*S1*S2*COS(G(A1)))),0)+0*L(P,(S1+S2+S3)/2)+0*L(A3,2*ACOS(SQRT(G(P)*( G(P)-S2)/(S1*S3))))+0*L(A2,2*ACOS(SQRT(G(P)*(G(P)-S1)/(S2*S3) )))+0*L(A1,ACOS(-COS(0*A1+A2+A3)))-.5*S1*S3*SIN(A3)+IF(S(SSS) ,SSS,SAS),IF(S(SSA),IF(G(FLG)=1 AND S2>S1,0*L(A3,2*ASIN(1)-A3)+0*L(FLG,0),0*L(FLG,1)+0*L(A3,ASIN(S2*SIN(A2)/S1)))+0*L(A1,ACOS(-COS(A2+ A3))),0)+IF(S(SAA),0*L(A3,ACOS(-COS(A1+A2))),0)+0*L(A2,ACOS(-COS(A1+ A3)))+0*L(S2,S1*SIN(A3)/SIN(A2))+0*L(S3,S1*COS(A3)+S2*COS(A2))-.5*S1*S3* SIN(A3)+IF(S(SSA),SSA,IF(S(SAA),SAA,ASA)))

Click here to download the TRIANG.EQN file and you can use Menu, File, Open in Solver to load the equation file

 Editor: In the above tip as well as the Easter.eqn document, we have included the equations. We tested them and then copied and pasted them into the document. If you choose to type them in yourself, be sure that you double check your typing before claiming that the solutions don't work. We will also include electronic versions of these Solver equations on the Internet at www.palmtop.net (the SUPER site) and on this issue of The HP Palmtop Paper ON DISK. In the past we have not printed long equations. However, readers with no other way to get a copy of the equations have requested that we print them.