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 90angle BCA.
Calculate the area of the right triangles
defined by angle BCA.
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 90a because BCA and BAC must sum to 90 and the two triangles are
flipped.
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