## 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.