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 .) ____________

5) Calculus Exercise: Find the surface area of a sphere of radius R by using 'shells' of revolution as follows:

Use the function and revolve around the x-axis from 0 to R. Arclength is

and the surface area of each shell is 2r h =2 f ds.

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 = ________

A + C = _________

A + D = _________

A + B + C + D = __________

Solve for A (terms of a, b, c, and ): ________________________

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?

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

the angles:

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.

11) Do the same for the 4-sided figure below:

12) We will now show by induction that we can use this new formula for any convex n-gon on the sphere of

radius R. 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:

where and .

Your job is to do the algebra to show that

where , , and