Hexagons/Circles Stretch
Hexagons appear in beehives, tile patterns, quilts, games and elsewhere. They have many unique
properties, some of which are explored here. For the problems in this stretch we will be using
regular hexagon ABCDEF, shown in problem #1, the sides of which are each one unit long.
A
diags How many diagonals does hexagon ABCDEF have?
1._ _________
1
B
F
C
E
D
units The three main diagonals of hexagon ABCDEF meet at point G, the
2.__________
center of the hexagon. What is the length of segment GB?
A
B
F
C
G
E
degrees What is the measure of each interior angle of hexagon ABCDEF?
3.__________
D
lines Segment BE is a line of symmetry for hexagon ABCDEF. How many additional lines
4.__________
of symmetry does the hexagon have?
A
sq units What is the area of hexagon ABCDEF? Express your answer as a
5.__________
common fraction in simplest radical form.
A
units
6.__________
C
G
D
A circle with center G is inscribed in hexagon ABCDEF. What is
the circumference of this circle? Express your answer in simplest
C
radical form in terms of π.
G
E
F
E
B
F
B
D
sq units A circle with center G is circumscribed about hexagon ABCDEF.
7.__________
What is the area of this circle? Express your answer in terms
of π.
A
F
B
C
G
E
D
8.__________ Sides AB and DC are extended to meet at point H. What is the ratio of the area
of triangle BCG to the area of triangle BCH? Express your answer in simplest form.
A
F
B
C
G
E
MATHCOUNTS 2006-2007
H
D
71
sq units The extension of other pairs of sides, in the manner used
9.__________
in the previous problem, results in a six-point star, as
shown. What is the total area of the shaded regions?
Express your answer as a common fraction in simplest
radical form.
M
A
L
B
F
K
H
C
G
E
D
I
J
units
10._ ________
H
G
A circle with radius GH is drawn to circumscribe the
six-point star. How long is the radius of this new circle?
Express your answer in simplest radical form.
sq units What is the difference between the area of the circle circumscribed about the
11.__________
six-point star and the area of the circle inscribed in the original hexagon? Express
your answer as a common fraction in terms of π.
U
sq units Segment GB is extended to point V such that GB = BV.
12._ ________
Likewise, GC, GD, GE, etc. are extended such that GC = CW; GD = DX; GE = EY, etc. What is the area of Z
trapezoid BCWV? Express your answer as a common
fraction in simplest radical form.
V
A
F
B
C
G
E
W
D
Y
X
13._ ________ What is the ratio of the length of a side of hexagon ABCDEF to the length of a
side of hexagon UVWXYZ? Express your answer as a common fraction.
14._ ________ What is the ratio of the area of hexagon ABCDEF to the area of hexagon
UVWXYZ? Express your answer as a common fraction.
deg If point G is held fixed and hexagon ABCDEF is rotated 60 degrees clockwise
15._ ________
(shown below in four steps) such that point A moves to the previous location
of point B, the resulting hexagon has the exact same orientation. What is the
maximum number of degrees less than 500 degrees that this figure can be rotated
clockwise and maintain its exact same orientation?
B
B
G
C
60
A
A
C
D
D
B
F
G
B
A
E
E
B
C
D
C
A
B
G
F
A
A
E
BB
D
D
F
A
G
72
C
G
E
A
E
F
B
F
A
MATHCOUNTS 2006-2007
Reading-a-Table Stretch
5 to 13 14 to 17 18 to 24 16 years 18 years 15 to 44 45 to 64 65 years 85 years
years
years
years
or over or over
years
years
or over or over
20,071,268
16,831,078
228,621,674
125,841,694
36,293,985
49,758
92,603
45,868
73,960 489,770 467,206 290,899 168,404
41,887
3502
2,633,972 4,821,851
3,596,126
26,297,336
8,085,096 3,822,957 514,013
53,781
95,151
44,574
83,684 658,857 636,858 354,853 206,081 108,961
13,259
35,029
52,906
21,612
57,899 454,029 443,976 264,188 128,068
67,171
9,288
Under 5
years
The following are estimates of the Resident Population by Selected Age Groups for some of the states/regions in
the U.S.: July 1, 2004. The category “5 to 13 years” refers to the number of people in that state/region who are 5 years old
through 13 years old, inclusive. Some numbers have been left out of the table due to space or the following questions.
Area
Total
United States
293,655,404
Alaska
655,435
California
35,893,799
Delaware
830,364
D.C.
ppl How many people who live in Alaska are 65 years to 84 years old?
1._ ________
ppl What is the total population of D.C.?
2._________
ppl How many people in Delaware are 14 years old?
3._________
ppl Using the answer to the previous question, how many people in Delaware are 15 years old?
4._________
5._________ Of the four states/regions shown, which one has the highest percent of its population in the “18 to 24 years”
group? What is the percent, expressed to the nearest tenth?
% 6.__________
73
MATHCOUNTS 2006-2007
Area
Total
United States
293,655,404
Alaska
655,435
California
35,893,799
Delaware
830,364
D.C.
Under 5
years
5 to 13 14 to 17 18 to 24 16 years 18 years 15 to 44 45 to 64 65 years 85 years
years
years
years
or over or over
years
years
or over or over
20,071,268
16,831,078
228,621,674
125,841,694
36,293,985
49,758
92,603
45,868
73,960 489,770 467,206 290,899 168,404
41,887
3502
2,633,972 4,821,851
3,596,126
26,297,336
8,085,096 3,822,957 514,013
53,781
95,151
44,574
83,684 658,857 636,858 354,853 206,081 108,961
13,259
35,029
52,906
21,612
57,899 454,029 443,976 264,188 128,068
67,171
9,288
ppl What is the median number of people in the group “5 to 13 years” for these four states/regions?
7.__________
ppl How many people in Delaware are 25 years or older?
8.__________
% What percent of the people in California are under the age of 14? Express your answer to the nearest tenth.
9.__________
% Calculate the ratio of “65 years or over” to “45 to 64 years” for Alaska and then for California. What is the ratio for each of
10._ AK:
________
these two states, and why do you think that might be the case? Express the ratios as percents to the nearest tenth.
CA:
%
11. _________
12.
ppl Assuming there are the same number of people of each of the ages in the category “18 to 24 years,” how many people in Alaska
13._ ________
are 19 years old? Express your answer as a whole number.
% When considering California, Delaware and D.C., what is the greatest difference between the percent of population from the 14._ ________
“65 years or over” category to the “85 years or over” category for these three states/regions? Express your answer to the
nearest tenth.
% California has what percent of the country’s kids under the age of 5 years? Express your answer to the nearest tenth.
15._ ________
MATHCOUNTS 2006-2007
74
Hexagons/Circles Stretch
Answers
1. 9
6. π 3
11.
9π
4
2. 1
7. π
12.
3 3
4
3. 120
8. 1
13.
1
2
4. 5
9.
14.
1
4
5. 3 3
2
10. 3
3 3
2
15. 480
Reading-a-Table Stretch
Answers
1. 38,385
7. 93,877
2. 553,523
8. 553,174
3. 11,537
9. 20.8
12. Possibly Alaska is a great
place to work, but when
retiring, more people are
inclined to move due to climate.
4. 11,038
10. 24.9
13. 10,566
5. Alaska
11. 47.3
14. 11.5
6. 11.3
MATHCOUNTS 2006-2007
15. 13.1
75