Friday, May 10, 2019

Area of Several Right Triangles




This post uses trigonometric functions in Excel to analyze the area of right triangles with different angles.

Question:  I have a string of 25 inches, which I use to make right triangle ABC.  AB and BC are the sides and AC the hypotenuse.  Angle ABC is 90 degrees.  Angle BCA ranges from 5 to 85 in increments of 5.   Angle BAC is 90-angle BCA.

Calculate the area of the right triangles defined by angle BCA.

Analysis:  Denote the hypotenuse as X.

Line Segment AB is the Sine(BCA)*X.

Line  Segment BC is the Cos(BCA)*X.

The perimeter is 25 = X +X*Sine(BCA) + X*COS(BCA)

The COSINE and SINE are known from the angle.

Solve for X.

X=25/(1 + Cos + Sine)

The calculation of the hypotenuse is outlined below.


Perimeter
25.000



Degrees
radians
cos
sin
Hypotenuse Perimeter/(1+cos+sin)
5.000
0.087
0.996
0.087
12.000
10.000
0.175
0.985
0.174
11.582
15.000
0.262
0.966
0.259
11.237
20.000
0.349
0.940
0.342
10.957
25.000
0.436
0.906
0.423
10.735
30.000
0.524
0.866
0.500
10.566
35.000
0.611
0.819
0.574
10.448
40.000
0.698
0.766
0.643
10.378
45.000
0.785
0.707
0.707
10.355
50.000
0.873
0.643
0.766
10.378
55.000
0.960
0.574
0.819
10.448
60.000
1.047
0.500
0.866
10.566
65.000
1.134
0.423
0.906
10.735
70.000
1.222
0.342
0.940
10.957
75.000
1.309
0.259
0.966
11.237
80.000
1.396
0.174
0.985
11.582
85.000
1.484
0.087
0.996
12.000








After finding X find length of segments AB and BC from formulas.

The area is AB*BA*.5.


Degrees
Hypotenuse Perimeter/(1+cos+sin)
AB
hypotenuse * cos
BC
Hypotenuse * Sin
area
5.000
12.000
11.954
1.046
6.25
10.000
11.582
11.406
2.011
11.47
15.000
11.237
10.854
2.908
15.78
20.000
10.957
10.296
3.747
19.29
25.000
10.735
9.729
4.537
22.07
30.000
10.566
9.151
5.283
24.17
35.000
10.448
8.559
5.993
25.65
40.000
10.378
7.950
6.671
26.52
45.000
10.355
7.322
7.322
26.81
50.000
10.378
6.671
7.950
26.52
55.000
10.448
5.993
8.559
25.65
60.000
10.566
5.283
9.151
24.17
65.000
10.735
4.537
9.729
22.07
70.000
10.957
3.747
10.296
19.29
75.000
11.237
2.908
10.854
15.78
80.000
11.582
2.011
11.406
11.47
85.000
12.000
1.046
11.954
6.25



Observation:  Note that the triangle with angle  BCA equal to a is identical triangle with BCA equal to 90-a because BCA and BAC must sum to 90 and the two triangles are flipped.

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