This
geometry problem asks you to find the length of each side of an Isosceles triangle
ABC where AB=BC given the perimeter of the triangle and the angle ABC. I ask you to set up the solution in an Excel
spreadsheet where triangle perimeter and length of angle ABC are inputs, which
can be modified by user of spreadsheet.
Question
One: You form
an Isosceles triangle ABC with 12 inches of string. You know AB=BC and you denote the length of
both sides X. Set up a spreadsheet where
you can find the length of all sides of the triangle for different values of angle
ABC.
Answer: We know that AB+BC+AC =12. Denote AB and BC as X. Denote the angle ABC as a. We know that half of BC is X*sin(a/2) so BC
is 2*X*sin(a/2).
this means 2X +2sin(a/2)X =12
rearrange to get x
X =
12/(2*(1+sin(a/2))
The table below is from an Excel spreadsheet calculating
triangle side length for several different values of angle ABC.
Calculating Isosceles
Triangle Sides


String Length

12

12

12

12

12

12

Angle between equal
sides a

30

45

60

90

120

150

sine a/2

0.26

0.38

0.5

0.71

0.87

0.97

x (AB or BC)

4.77

4.34

4

3.51

3.22

3.05

Hypotenuse BC
122x

2.47

3.32

4

4.97

5.57

5.90

Technical
Note: In
Excel get sine from sin function and use radian function to translate degrees
to radian. sin(radians(30/2)) is equal
to 0.26.
Intuitive
Note: We
know that an Isosceles triangle with top angle of 60 degrees is an equilateral
triangle. Our calculations reveal that
this is indeed the case.
Followup
problem: Find
the area of all triangles listed in the chart above.
The height of the triangle (the line from
point B to the midpoint of AC) is
H=X*Cos(a/2).
The area is 0.5*H*B where B is the base or in
this case the hypotenuse.
Calculations
are presented below.
Calculating Isosceles
Triangle Sides


String Length

12

12

12

12

12

12

Angle between equal
sides a

30

45

60

90

120

150

sine a/2

0.26

0.38

0.5

0.71

0.87

0.97

x (AB or BC)

4.77

4.34

4

3.51

3.22

3.05

Hypotenuse BC

2.47

3.32

4

4.97

5.57

5.90

Triangle Height

4.60396

4.0091

3.464

2.485

1.6077

0.7899

Area Height * Hypotenuse
* 0.5

5.68

6.66

6.93

6.18

4.48

2.33

Intuitive
Note: Note
that the equilateral triangle has the largest area. This makes sense because the areal of an equilateral
triangle is half the area of square.
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