Download Answers to Balloon Two Day

ANSWERS TO:
MA-3329 Modern Geometry
Name: ____________________________
Fun Balloon Day #2
Instructions:
1) Blow up your balloon and again pretend it is a perfect sphere.
2) Any circle around the sphere which divides the sphere into two equal hemispheres is called
a great circle. (The equator is a great circle around the Earth.) On your balloon, draw two different
great circles which don't intersect at 90 degrees. (If they were to intersect at 90 degrees, it would make
this exercise less interesting and harder to do!) Hint: Draw these 'circles' from top to bottom of the balloon.
3) You should have created two kinds of angles by doing this. Call the acute angle a and the obtuse angle a' .
Label the 8 resulting angles on your balloon. The 4 sections you have just drawn are called lunes.
4) What proportion of the entire sphere's surface area is contained in each of the 4 lunes you have created?
(Give your answer in terms of a and  .) ANS: a whole _ surface _ area 
2
5) Calculus Exercise: Find the surface area of a sphere of radius R by using 'shells' of revolution as follows:
Use the function f ( x)  R 2  x 2 and revolve around the x-axis from 0 to R. Arclength is
2
 df 
ds  1    dx
 dx 
and the surface area of each shell is 2  r h =2  f ds.
R
R
2 2f ds  2 2 R  x ds  4  R  x
0
0
R
 4  R 2  x 2 1 
0
2
2
0
R
 1  2x
 df 
1    dx  4  R 2  x 2 1  
2
2
 dx 
0
2 R x
2
R
2
2
x2
dx  4  R 2  x 2
R2  x2
0
R
2

 dx


R2  x2  x2
dx  4  R dx  4 R 2
R2  x2
0
R
6) Now, draw a third great circle on your balloon (not intersecting the others at 90 degrees). You should now
have two new kinds of intersections. Call them b and b' , and c and c' and label them on balloon so that a
convenient triangle has the angles a, b, and c. Label this triangle "A". Use acuteness or obtuseness to help
you figure out which angles are which. Hint: draw half of the circle on the front of the balloon, then rotate
the balloon 90 degrees and hold up to the light to draw some more of the circle, then repeat until done.
You should notice you now have 8 triangles. Based on the angles you have labeled, what do you notice about
the triangles? Label the triangles with A, B, C, and D appropriately so that the triangles having the same
angles have the same labels (because they have the same areas). See my styrofoam sphere if you need a hint!
7) Find the areas of the following lunes (and hemisphere) in terms of a, b, c, and  .
 a 
 b 
 c 
2
2
2
A+B= 
A+C= 
A+D= 
(2nd and 3rd may be reversed)
4 R
4 R
4 R
 2 
 2 
 2 
A + B + C + D = 2 R 2
Solve for A (terms of a, b, c, and  ):
A  (a  b  c   ) R 2
8) If you had a convex 4-sided polygon with angles a, b, c, and d, instead of a triangle on your sphere, how
could you find the area?
A  R 2 (a  b  c  d  2 ) because you split it into two triangles, each of which requires a minus Pi.
9) If you had a convex n-sided polygon instead of a 3 or 4 sided one, what would the area be? Call the sum of
n
the angles:

i 1
i
 n

A  R 2   i  (n  2) 
 i 1

10) If we measure the angles differently, it will be even easier to find the areas of convex polygons. Find
the area of the triangle below in terms of a', b', and c' instead of a, b, and c.
A  R 2 (a  b  c   )  R 2 ((  a' )  (  b' )  (  c' )   )
 R 2 (2  (a'b'c' ))
11) Do the same for the 4-sided figure below:
You will end up with  R 2 (2  (a'b'c'd ' )) (Do the algebra!)
12) We will now show by induction that we can use this new formula for any convex n-gon. Any convex n-gon
can be turned into a convex (n+1)-gon by replacing a side with two new sides. A new triangle is added to
the area. Assume that the area of the n-gon can be computed by the formula:

 n 
AOLD  R 2  2    i   where  n1  a  d and  n  c  g .
 i 1  

Your job is to do the algebra to show that

 n1  
ANEW  R 2  2    i   where  n1  d ,  n  h , and  n1  g
 i 1  

-------------------------------------------------------------------------------------------------------------------

 n 
ANEW  AOLD  R 2 (a  b  c   )  R 2  2    i   (a  b  c   ) 
 i 1 



 n2

 R 2  2     i  a  d  c  g    R 2 (a  b  c   )
 i 1




 n2 
 R 2  2    i   a  d  c  g  a  (  h)  c   
 i 1 






 n2 
 n2

 n1  
 R 2  2    i   d  h  g   R 2  2    i  d  h  g    R 2  2    i  
 i 1 
 i 1

 i 1  



