The
pentagram is an ancient symbol (http://mathworld.wolfram.com/Pentagram.html)
, recently popularized by Dan Brown (http://www.danbrown.com/the-davinci-code/).
Mathematicians have analysed its
properties since the time of Pythagoras. (www.math.tamu.edu/~dallen/history/pythag/pythag.html)

One such question
is what is the area of the blue star below?

The
exterior angle of a regular pentagon is 360°/5 = 72°. Hence each interior angle
is 180° - 72° = 108°. Angle EAB is one
such angle so <EAB = 108°. Call the
centre of the pentagon K. Segment KA bisects angle EAB. Hence Angle KAB = ½ ×108°
= 54°.

The area of
the pentagon is 5 times the area of the green triangle. The height of the required triangle is 5 tan 54° = 6.88190….
Hence the area of the triangle is ½ (10)( 6.88190….) = 34.40954. The area of
the entire pentagon is 5×34.40954. = 172.047.

Each base angle
of the highlighted isosceles triangle is ½ (180° -108°) = 36°.Hence, the height
of the highlighted triangle is 5 tan 36° = 3.632712. Consequently the area of the triangle is ½ (10)( 3.632712….)
= 18.163563. The area of all five
triangles is 5×34.40954. = 90.81781.

The area of
the star is the area of the area of pentagon (ABCDE) minus the area of the five
triangles; 172.047 - 90.817 = 81.2 units

^{2}to 3 sig figs.
Mary takes
a different approach. She decides the
area of the star is five times the area of the dart KAFB (shaded grey below). A dart is simply an unusual kite.

The area of
a kite is half the product of the diagonals = ½ (AB)(KF). Note that diagonal KF
is inside the polygon and diagonal AB is outside the polygon; nevertheless the
formula still holds. Diagonal AB is given (10 units). Hence, Mary only needs the
length of segment KF = 5 tan 54°- 5 tan 36° = 5(tan 54°- tan 36°)
= 6.88190 - 3.632712 = 3.249188.

Mary
calculates the area of the star = 5×½ (10)( 3.249188)= 25×3.249188 = 81.2 units

^{2 }to 3 sig figs.
Conclusion:
Johnny and Mary both used the five-fold symmetry of the shape, both used the
trigonometric ratio ‘tangent θ’, and both used the same angles (54° and 36°). Hence their algebra is essentially the same. They
only differed in the physical meaning they gave to the algebra. Johnny
calculated the area of two triangles (ABK and ABF) whereas Mary calculated the
area of the dart KAFB.

Can you
find another way to find the area of the star? Is it possible to find the area but
avoid using trigonometry? What about the
mysterious number phi (http://en.wikipedia.org/wiki/Golden_ratio
).

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