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Name:_______________________________________________ Date:________________________
Mr. Carman / Barnett
Geometry RSH: 2011-2012 Problem Set III
Have fun solving the following problems. Each of them can be extended to make a great presentation!
Diagrams are obviously not drawn to scale. Be sure to read each problem carefully!
When applicable, leave all solutions in terms of  .
---------------------------------------------------------------------------------------------------------------------------0) Calculate the individual areas of the first 4 eclipses using 32 as the original radius.
32 units
Investigation #1:
Continuing in this fashion, find the area of the nth eclipse (Use r as the original radius).
Investigation #2:
What if each subsequent radius was one third of the previous radius? Still using r as the original
radius, find the area of the nth eclipse.
Investigation #3:
What if each subsequent radius was
1
of the previous radius? Still using r as the original radius, find
k
the area of the nth eclipse.
NOTE: The problem on this page is a sample. It may not be used as your project.
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1) The diagram shows a large circle (radius of 12 units) tangent to a small circle (radius of 3 units).
The line is tangent to the large circle and the small circle at points A and B respectively. Find the
length of segment AB .
A
B
Investigation #1: Find AB in terms of R and r.
Investigation #2: Find AB in terms of R,
r, and d.
A
A
B
B
R
R
R
d
r
r
r
Investigation #3: Find in terms of R, r, and d.
A
B
R
R
d
r
Investigation #4: What happens when d = 2r in Investigation #3?
What happens when d gets very small in Investigation #3? What
happens when d gets very large in Investigation #2?
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2) The diagram shows a circle inscribed in an equilateral triangle (side length 8 units). Find the area of
the shaded region.
Investigation #1: Using s as the side length of the triangle, find the shaded region in terms of s and  .
Investigation #2: Repeat the process for the “stage 2” shaded region, in terms of s and  .
Investigation #3: Repeat the problem using any regular convex polygon with n sides, in terms of s, n,
and  .
Investigation #4: Use your solution from Investigation #3 to verify your solution to Investigation #1.
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3) The diagram shows a regular 5 point star. If the star has a side length of 8 units, find the area of the
shaded pentagon.
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Investigation #1: Repeat the problem using s as the side length.
Investigation #2: Find the area of the “stage 2” shaded region, in terms of s.
Investigation #3: Express the area of the “stage n” shaded region as a geometric sequence.
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4) The diagram below shows 4 quarter circles, and square ABCD. Find the area of the shaded region.
A
B
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D
C
Investigation #1: Repeat the problem using l as the side length of the outer square.
Investigation #2: Find the area of the “stage 2” shaded region shown below (just the inner-most shaded
region):
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5) The diagrams below show a square with a side length of 8 units. Find the area of the shaded region
in each instance.
Investigation #1: Repeat this problem using side length s for the original square, and using an n x n
array.
Investigation #2: Repeat this problem for any rectangular array of squares, where the length and width
of the array remain in the ratio of a:b.
Investigation #3: Repeat the problem using shaded circles instead of squares.
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6) The diagram below shows an equilateral triangle with side length of 8 units. In each instance, find
the area of the shaded region.
Investigation #1: Repeat the problem using a triangle with side length s.
Investigation #2: Given n, the number of circles in the bottom row, find the area of the shaded region.
Investigation #3: What happens when n gets very large?
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7) Given this square with side length 8 units, find the length of one side of the regular inscribed
octagon.
Investigation #1: Repeat the problem using s as the side length of the square.
Investigation #2: Now find the side length of the “next generation” square, still in terms of s.
Investigation #3: Find the area of the shaded region, in terms of s.
Investigation #4: Can you model the area of the “nth generation” area using a geometric sequence?
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8) The sides of a regular octagon (side length 8 units) are bisected, and a “2nd” generation” octagon is
formed. Find the area of the shaded region, which we will call a “crown”.
Investigation #1: Repeat the problem using s as the side length of the larger octagon.
Investigation #2: Find the area of the “2nd generation” crown.
Investigation #3: Find the area of the “nth generation” crown.
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9) The radius of the larger circle is 8. Each of the three smaller circles is tangent to each other, and
tangent to the larger circle. Find the area of the shaded region in terms of 
Investigation #1: Repeat the problem using R as the original radius.
Investigation #2: Find a formula for the area of the shaded region, when there are n circles in the top
row.
Investigation #3: As you add more and more circles to the top row, what value does the area approach?
How does that compare to the area of the shaded region below?
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10) The square below has a side length of 8 units. We keep surrounding the sides of the square with
more and more isosceles right triangles. Find the area of each figure.
Investigation #1: Repeat the problem, using s as the original square’s side length.
Investigation #2: Repeat the problem, but this time the isosceles triangles do not have to be right
triangles (solution will be in terms of  ).

Investigation #3: See if your formula from Investigation #2 verifies your answer to Investigation #1.
Investigation #4: Repeat the problem using a cube surrounded by square pyramids.
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11) Given a square with side length 8 units, these arrays are created using congruent circles drawn
tangent to each other [and the square]. Find the total “perimeter” and area of each circular array.
Investigation #1: Repeat this problem using s as the square’s side length.
Investigation #2: Given n circles in the top row of the array, find the total perimeter and area of the
array.
Investigation #3: What happens to the perimeter and area of the array as n gets very large?
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12) The diagram shows a regular hexagon with side length s. Find the area of the shaded region.
(1/3)s
(2/3)s
Investigation #1: Repeat the problem using the subdivision of (1/4)s.
Investigation #2: Repeat the problem using the subdivision of (1/5)s.
Investigation #3: Repeat the problem using the subdivision of (1/k)s. What happens as k gets very
large?
Investigation #4: What if the original shape is NOT a hexagon? Can you redo Investigation #3 for
ANY even sided regular convex polygon?
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13) The circle in the center of this figure has a radius of 8 units. It is surrounded by congruent circles,
each tangent to one another, and also tangent to the center circle. Find the radius of each of the
surrounding circles.
Investigation #1: Repeat this problem using R as the center circle’s radius.
Investigation #2: Repeat this problem if the number of the surrounding circles is n.
Investigation #3: What happens as n gets very large?
Investigation #4 (Notes from #25) Is it possible for r to equal R?
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14) The rhombus pictured below has a side length of 8 units. If   40 , find the radius of the
inscribed circle.

Investigation #1: Repeat this problem using s as the side length, and using  as the angle.
Investigation #2: Still using s and  , find the side length of the “2nd generation” rhombus.


Investigation #3: Still using s and  , find the radius of the “2nd generation” circle.


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15) The thickness of each of the rings in this bull’s-eye is R (which is also the radius of the center dot).
Find the total area of the shaded region.
R
Investigation #1: Find the total area of the shaded region for a bull’s-eye with n rings (we don’t count
the center dot as a ring).
Investigation #2: Repeat Investigation #1 for this special bull’s-eye, where the thickness of each ring
gets cut in half as you move outward…
R/16
R/8
R/4
R/2
R
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16) Given an equilateral triangle, find the value x of the angle that will cause the shaded region to have
an area equal to 1/3 of the area of the entire triangle.
x
Investigation #1: Now find the value x of the angle that will cause the shaded region to have an area
equal to 1/4 of the area of the entire triangle.
Investigation #2: Now find the value x of the angle that will cause the shaded region to have an area
equal to 1/n of the area of the entire triangle. Use your solution to verify your first 2 answers.
Investigation #3: What happens to the angle x as n gets very large? What about when n gets very
small?
Investigation #4 Repeat Investigation #2, but this time the outer triangle can be any isosceles triangle.
Does your solution verify your answer to Investigation #2?
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17) Find the side length of a regular hexagon whose height is 8 units.
Investigation #1: Find the side length of a regular octagon whose height is 8 units.
Investigation #2: Find the side length of a regular pentagon whose height is 8 units.
Investigation #3: Find the side length of a regular n-gon whose height is h.
***HINT: You will need to deal with even-sided and odd-sided polygons separately.
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18) Take a regular hexagon with side length 8, and link every other vertex, as shown. Find the side
length of the “2nd generation” hexagon.
x
Investigation #1: Repeat this problem, but use s as the side length of the original hexagon.
Investigation #2: Repeat Investigation #1, but use an octagon instead of a hexagon.
Investigation #3: You can actually do this with any regular convex polygon, not just the hexagon. Find
a general solution for a polygon with n sides, each of length s.
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19) The volume of a pyramid is equal to V 
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1
(Area of the base)(Height).
3
The base of the pyramid shown is a square with side length s. If the height of the pyramid is also s,
what is the volume of the pyramid?
We can imitate the shape of a square pyramid by stacking cubes on top of each other. Here is a top
view of 3 layers of cubes stacked on top of each other to form a pyramid:
Investigation #1: If the pyramid will have n rows, how many cubes will be needed to build it?
Investigation #2: If the height of the pyramid is s, and there are n rows of cubes, what is the volume of
each cube?
Investigation #3: Find a formula to express the volume of a pyramid with height s and n rows of cubes.
Investigation #4: What happens when n gets very large? How does this compare to the formula for the
volume of the square pyramid in the top picture?
Investigation #5: Will this also work by stacking rows of spheres?
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20) The large outer circle has a radius of 8 units. Find the radius of each of the inner congruent circles,
which are tangent to each other, as well as tangent to the outer circle.
Investigation #1: Repeat the problem using R as the large radius.
Investigation #2: Repeat the problem if there are 4 circles, as shown (large radius is still R).
Investigation #3: Solve this problem again, but now there are n circles.
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21) The “petals” of these “flowers” are squares. Find the total area of the petals in each figure.
8
8
(Regular hexagon with radius of 8 units)
(Regular pentagon with radius of 8 units)
Investigation #1: Repeat the problem using r as the length of the radius.
Investigation #2: Can you solve the problem for any radius r, and any number of petals n?
Investigation #3: Find the perimeter of any flower given its radius r, and its number of petals n.
Investigation #4: Repeat the problem using semicircles as petals, as shown below. Find the area of the
petals, and the perimeter of the flower for any r, and any n.
r
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22) The first two iterations of a pattern are shown below. Find the area of the shaded region in each.
8
8
Investigation #1: Find the area of the shaded regions of the next two iterations.
Investigation #2: Find the area of the shaded region of the nth iteration.
Investigation #3: Repeat Investigation #2, but use s as the original square’s side length.
Investigation #4: The first two iterations of a very similar pattern are shown below. Find the area of the
shaded region of the nth iteration (in terms of R and n).
R
Investigation #5: Find the TOTAL shaded area of the first n iterations using a geometric series. (Do
this for both the square and circle patterns)
R
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23) We place 6 congruent circles in the corners of a regular hexagon. Each circle is tangent to two
sides of the hexagon. We create a new hexagon by joining the centers of the 6 circles. Find the side
length of the new hexagon.
1
8
Investigation #1: Repeat this problem, but use s as the side length of the hexagon, and use r as the
radius of each circle.
Investigation #2: Repeat this problem for a regular octagon (still using s and r).
Investigation #3: Repeat this problem for a regular polygon with n sides (still using s and r).
Investigation #4: In investigation #3, what is the largest size that r is allowed to be?
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24) In both diagrams, the radius of the large circle is 8 units. Find the radius of each of the congruent
little circles in either diagram.
Investigation #1: Repeat this problem using R as the radius of the large circle.
Investigation #2: What if there were 16 little circles? What about 25? Can you generalize Investigation
#1 when there are n x n little circles?
Investigation #3: Given a large circle with radius R, and given an array of n x n circles, find the total
area of the little circles.
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25) The petals of this flower are congruent semicircles. Find the area of the shaded region.
8
1
Investigation #1: Repeat this problem, but use r1 as the radius of each petal, and use r2 as the radius of
the pollen center.
Investigation #2: Repeat investigation #1 for a flower with n petals.
Investigation #3: What is the largest that r2 is allowed to be?
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26) The diagram shows an equilateral triangle inscribed in a circle. The 3 little circles are drawn
tangent to the triangle’s sides at its midpoints. The 3 little circles are also tangent to the larger circle.
Find the radius of each of the little circles.
8
Investigation #1: Hmm… was that a coincidence? Repeat this problem, but use R as the radius of the
larger circle.
Investigation #2: Given R, and given that 0    90 , find r.

r
What happens to r when  gets very close to 0? What happens to r when  gets very close to 90 ?
Investigation #3: Given R, and given that 90    180 , find r.

r
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27) Find the area of the shaded quadrilateral (in terms of s).
(1/2)s
(1/2)s
(1/3)s
(4/5)s
(2/3)s
(1/5)s
(3/4)s
(1/4)s
Investigation #1: Repeat this process for a regular pentagon with side length s.
Investigation #2: Given a regular polygon with n sides, each of length s, find the general formula for
the area of the shaded region.
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28) Find the total volume of this three dimensional figure, which is a cube formation consisting of 27
congruent spheres tangent to each other:
8 units
Investigation #1: Repeat this problem using s as the side length of the cube.
Investigation #2: Keeping s as the side length of the cube, repeat this problem using 4 x 4 x 4 spheres.
Investigation #3: Keeping s as the side length of the cube, repeat this problem using n x n x n spheres.
Investigation #4: The formation does not have to be a cube. Try repeating the problem using a
rectangular prism formation (n x m x k).
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29) These diagrams show top views of “pyramids” made from stacks of rows of congruent spheres,
each tangent to one another. Find the total volume of each “pyramid”.
8 units
(2 rows of spheres)
8 units
8 units
(3 rows of spheres)
(4 rows of spheres)
Investigation #1: Repeat the problem, but use s as the length of the base of each pyramid.
Investigation #2: Keeping s as the side length, find the volume of a pyramid with n rows of spheres.
Investigation #3: What happens to the volume of the pyramid as n gets very large?
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30) How many toothpicks would be required to make 10 rows?
(1 Row)
(2 Rows)
(3 Rows)
Investigation #1: How many toothpicks would be required to make n rows?
Investigation #2: Arrive upon your answer to Investigation #1 using a different approach (finite
difference, for example).
Investigation #3: Repeat this problem using a different polygon as the “host”.
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31) The diagram below shows a square with 4 congruent circles tangent to each other, and tangent to
the sides of the square. Find the radius of each circle if the square has a side length of 8 units.
Investigation #1: Repeat the problem, but use l as the side length of the square.
Investigation #2: Keeping l as the side length of the original square, find the side length of the “second
generation” square.
Investigation #3: Keeping l as the side length of the original square, find the length of the radius of the
“second generation” circles.
Investigation #3: Use a geometric sequence to model the side length of the nth generation square.
Investigation #4: Use a geometric sequence to model the radius of the nth generation circle.
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32) First we extend the sides of a regular hexagon to twice their length (figure 1)
Next, we connect the endpoints of the sides to form a new hexagon (figure 2)
Find the side length of this new hexagon.
x
8
8
Investigation #1: Repeat this problem, but use l as the side length of the original hexagon.
Investigation #2: Keeping l as the side length, repeat this problem using a regular octagon.
Investigation #3: Keeping l as a side length, repeat this problem using a regular n-gon.
Investigation #4: What happens to x in Investigation #3 as n gets very large?
Investigation #5: You could also find the area of each of the little triangles.
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33) You may want to become familiar with the properties of the “incenter” of a triangle…
Find the length of the radius of this isosceles triangle’s inscribed circle.
35
r
8 units
Investigation #1: Repeat this problem, but use  as the angle, and use s as the side length.
Investigation #2: The diagram below shows 6 congruent circles tangent to one another, and tangent to
the sides of a regular hexagon. Find the radius of each of these 6 circles. (HINT: Try dividing the
hexagon up into 6 isosceles triangles.)
s
Investigation #3: Find the side length of the “2nd generation” hexagon, shown below:
s
Investigation #4: Repeat Investigations 2 and 3 for
Any regular n sided polygon with side length s.
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34) The diagram below shows a regular octagon with side length 8 units. Find the side length of the
square formed by joining every other vertex, as shown.
x
s
Investigation #1: Repeat this problem, but use s as the side length.
Investigation #2: Keeping s as the side length, repeat this problem for an n sided regular polygon (n
must be even).
Investigation #3: What happens to the side length of the “dashed” polygon as n gets very large?
Investigation #4: Reverse this problem. Now find s in terms of x.
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35) How many diagonals does a hexagon have? What about a heptagon? An octagon?
Investigation #1: Find the number of diagonals d in a convex polygon with n sides.
Investigation #2: You just found d in terms of n. Find n in terms of d. Are there any special
requirements for d?
Investigation #3: Research “complete graphs”, which is a topic in “graph theory”. Can you find
confirmation for your formula from investigation #1?
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36) The diagram shows a larger square with side length s units. In each case, find the area of the
shaded square. (The sub divisions get smaller and smaller each time)
(1/3)s
(1/4)s
(1/2)s
(2/3)s
(1/2)s
Investigation #1: Find a general formula for the area A of the shaded square using the sub division
(1/n)s.
Investigation #2: What happens to A as n gets very large?
Investigation #3: Repeat Investigations 1 and 2 by starting with an equilateral triangle instead of a
square.
(3/4)s
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37) When you draw a line down a piece of paper, you create two partitions. When you draw another
line intersecting the first line, you get 4 partitions. When you draw a third line that intersects each of
the first two lines (at different points), you get 7 partitions. Find the number of partitions after drawing
n lines.
Investigation #1: Repeat the process, but this time find a formula for the number of “bounded” regions.
Investigation #2: Find a formula for the number of “unbounded” regions.
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38) Start with a square of side length s. Find the length of each staircase.
s
(Using halves)
(Using thirds)
(Using quarters)
Investigation #1: Find the length of a staircase with n steps.
Investigation #2: Repeat this process using a different angles (not 90o). Find the length of a staircase
with n steps in terms of  , n, and s.


s

Investigation #3: Use your answer to #2 to verify your answer to #1
Investigation #4: What happens when n gets very large?
Investigation #5: Can you think of a way to repeat this process using curved steps?
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