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Originally published in Mathematics in School, 28(5), Nov. 1999, p. 30, but with slight alterations here.

All rights reserved by the Mathematical Association, http://www.m-a.org.uk

180° ! "A

( )

+ 180° ! "B

( )

+ 180° ! "C

( )

+ 180° ! "D

( )

+ 180° ! "E

( )

= 720°

# "A + "B + "C + "D + "E = 5 • 180° ! 4 • 180° = 180°

Stars: A Second Look

Star polygons as presented by Winicki-Landman (1999) certainly provide an excellent

opportunity for students for investigating, conjecturing, refuting and explaining

(proving). However, it could also be insightful to alternatively explain (prove) the

results in terms of the exterior angles of the star polygons. It is also likely that

students who have had a strong experiential background in LOGO (Turtle Geometry)

would find this approach quite natural and easy (see Activity 1 in De Villiers, 2011).

For example, consider the star pentagon shown below. Imagine that one is a

turtle (like in turtle geometry) starting from A, then walking along the perimeter from

A to B, turning through the exterior angle b, then from B to C, turning through exterior

angle c, etc. When one returns to A, turning through the exterior angle a, one again

faces in the same direction one started off from. The total turning undergone is

therefore two full revolutions (use a pen or pencil and consider the sum of the

clockwise turns at each of the vertices), therefore:

a + b + c + d + e = 720°

.

Since

a = 180° ! "A

, etc., the interior angle sum can now easily be determined as

follows:

The value of this approach is that it is almost immediately generalizable to any closed

polygon (of which star polygons are only a special case) as follows. In general, after

walking completely around the perimeter of the polygon, one is facing in the same

Originally published in Mathematics in School, 28(5), Nov. 1999, p. 30, but with slight alterations here.

All rights reserved by the Mathematical Association, http://www.m-a.org.uk

direction one has started off from, and therefore the total turning (sum of all the

turning angles) must be a multiple of 360º, i.e. k.360º where k = 0; 1; 2; 3; etc. (This

intuitively obvious result is called the Turtle Closed Path Theorem by Abelson &

DiSessa, 1986, who also give a formal proof). The sum of the interior angles is now

simply the difference between n.180º and the sum of the turning angles where n is the

number of vertices, for example:

Note that k represents the total number of full revolutions of 360º one undergoes as

one walks around the figure and turning at each vertex. Using this formula one can

now easily determine the interior angle sums of the septagon and octagon shown

below. In the first case, the value of k is 2 and in the second it is 3. The respective

interior angle sums are therefore 540º and 360º.

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

This formula, however, becomes even more useful if one is working with more

complicated closed polygons, such as a crossed quadrilateral as shown below. In this

case, the total number of full revolutions k = 0, since one first undergoes two

clockwise turns at B and C, but these are cancelled out by the two anti-clockwise turns

at D and A respectively. The interior angle sum of the crossed quadrilateral is

therefore 180º x 4 = 720º, and two of the "interior" angles are now reflexive and lie

S = n • 180° ! k • 360°

= 180°(n ! 2k)

m arc BAD( )+ m arc CBA( )+ m arc ADC( )+ m arc BCD( ) = 720.00°

m arc BAD = 295.6°

m arc ADC = 55.6°

m arc BCD = 50.1°

m arc CBA = 318.8°

Hide arc angles

D

D

B

A

Originally published in Mathematics in School, 28(5), Nov. 1999, p. 30, but with slight alterations here.

All rights reserved by the Mathematical Association, http://www.m-a.org.uk

"outside"! This is usually very surprising to children and adults alike, and often at first

want to reject the notion. It is therefore a good example of the Lakatosian heuristic of

‘refutation’ and ‘monster-barring’ at an accessible level at school. (This is discussed,

illustrated and developed in more detail as a learning activity in De Villiers

(1999/2003) where arc measurements are used in Sketchpad for the crossed

quadrilateral - see last figure above – and as also in De Villiers (2010), 'interior

angles' are formally defined in terms of the concept of 'directed angles' so as to

extend it in a consistent way to the interior angle sum of a crossed quadrilateral, and

crossed polygons in general.)

Michael de Villiers

Mathematics Education

University of Durban-Westville (now University of KawZulu-Natal)

4000 Durban

South Africa

profmd@mweb.co.za

http://mzone.mweb.co.za/residents/profmd/homepage.html

References

Abelson, H. & DiSessa, A. (1986). Turtle Geometry: The computer as a medium for

exploring mathematics. Boston: MIT Press.

De Villiers, M. (1999/2003). Rethinking Proof with Geometer's Sketchpad. Berkeley,

CA: Key Curriculum Press. (Available from Key Curriculum Press:

http://www.keypress.com/x5588.xml )

De Villiers, M. (2010). Some Adventures in Euclidean Geometry. Lulu Publishers, pp.

49-50, 135-136. (Available from http://www.lulu.com/content/7622884 )

De Villiers, M. (2011). The interior angle sum of polygons: A general formula. An

online investigation for students at:

http://math.kennesaw.edu/~mdevilli/angle-sum-student-explore.html

Winicki-Landman, Greisy. (1999). 'Stars as a Source of Surprise', Mathematics in

School, 28(1), March, 22-27.