1) A bug crawls at a rate of 14π units per hour around a circle with a radius
of 3.5 units. How many hours does it take the bug to complete 26
revolutions of the circle? (3) 2004-WU4-10
2) A bike’s front wheel has a 1-foot radius. How many revolutions does the
wheel make in a one-mile trip? (2) 2004-WO2-7
3) A wheel has a circumference of 3 meters. The radius can be expressed
A
as
meters, with relatively prime integers A and B. What is the value of
B
A + B? (3) 2004-WU6-4
4) A circular disk has a radius of two units. A point is marked on the edge of
the disk. The disk rotates about its center, causing the point to travel a
distance of 90 units. How many rotations did the disk make? Round to the
nearest tenth. (2) 2004-WO3-10
5) A greeting card is six inches wide and eight inches tall. Point A is three
inches from the fold, as shown. As the card is opened to an angle of 45º,
how many inches does point A travel? Express your answer in terms of π. (2)
2004-WU8-7
A

Happy
Birthday
6) A circular field circumscribes a plot of land that is 200 meters on each
side, as shown. How many square meters of the circular region are not inside
the square region? (3) 2004-WO4-7
200 m
7) The sum of the areas of the six congruent circles is 150π cm2. What is
the area of the rectangle? (3) 2004-WU11-7
8) The students of Glenview Middle School are planning a carnival. One of
the contests will be a dart throw. The radius of the entire round dartboard
is eight inches. What is the radius of the shaded circle if the area of the
non-shaded region is three times the area of the shaded region? (3) 2004WU12-7
9) A carpenter wants to make the largest possible circular tabletop from a
4-foot by 8-foot sheet of plywood. If the tabletop is constructed from the
two largest congruent semi-circular pieces that can be cut from the sheet,
what is the diameter of the resulting table? (3) 2004-WU16-3
8 feet
10) The radius of circle O is 12, and AB and CD are tangent to the circle at
B and D respectively. AB = 16, and CD = 5. What is the sum of OC and OA?
(2) 1999-WU3-33
D
C
O
A
B
11) Two small circles with radii 2 and 3 are externally tangent. A third circle
is circumscribed about the first two as shown. What is the ratio of the area
of the smallest circle to the area of the shaded region? (2) 1999-WU4-7
12) Two tangent congruent circles are circumscribed by a larger circle. The
diameter of the larger circle is 24. Find the area of the shaded region.
Express your answer in terms of π. (2) 1999-WU12-8
13) Square MNOP is tangent to circle O at N and P. What is the area of the
shaded region? Round to the nearest whole number. (2) 2000-WU1-6
4 cm
14) In the circle, AP = 2 cm, PO = 3 cm, and m<BPO = 90º. What is the length
of BP? (2) 2000-WU12-9
A
P
O
B
15) The circle with center A and right triangle ACB have equal areas. Radius AC
= 5. What is the length of BC ? Round to the nearest tenth. (2) 2000-WO4-4
B
A
C
16) A square is constructed on diameter AC such that the area of the square
is equal to the area of the circle. What percent of AC is BC? Round to
nearest whole number. (3) 2000-WO8-2
A
B
C
17) A wheel on a racing bike has a diameter of 28 inches. How many
complete revolutions will the wheel make in one mile? 2000-WO3-2 (2)
18) The area of this square is 4π square units. What is the area of the
inscribed circle? (3) 2003-WU13-2
19) The circle is inscribed in the larger square. The smaller square is
inscribed in the circle. The radius of the circle is 10 cm. What is the area
of the region between the two squares? (3) 2003-WU14-4
20) A farmer ties his goat to the corner of a 10-foot by 15-foot rectangular
shed in an otherwise empty field. The length of rope from the shed to the
goat is 25 feet. Over how many square feet of the field can his goat roam?
Express in terms of π. (3) 2003-WU13-2
15’
25’
10’
21) A circle is inscribed in a square. What is the ratio of the area of the square
not within the circle to the total area of the square? Leave your answer in
terms of π. (3) 2002-WU16-4
22) The concentric circles are drawn as shown with radii 2, 4 and
6 units. What is the ratio of the area of the smallest circle to
the area of the shaded region? (2) 2002-WU16-5
23) Points A, B, C and D (not shown) are positioned on a circle such that AB = 7,
BC = 24, diameter AC = 25 and CD = 15. What is the length of AD ?
(3) 2002-WO9-5
B
A
C
24) A regular hexagon is inscribed in a circle. If the area of the circle is 36π
cm2, what is the perimeter of the hexagon? (2) 1997-WU6-10
25) A regular hexagon in inscribed in a circle. If the perimeter of the hexagon
is 42 inches, what is the circumference of the circle? Leave your answer in
terms of π. (2) 1997-WU8-7
26) An equilateral triangle is inscribed in a circle. Then, a circle is inscribed in
that equilateral triangle, an equilateral triangle is inscribed in the second circle
and so forth. If the circumference of the largest circle is 12π inches, what is
the circumference of the second inscribed circle? (3) 1997-WU13-6
27) A circle is inscribed in a square. Determine the ratio of the circumference
of the inscribed circle to the perimeter of the square. Express your answer in
terms of π (3) 1997-WO1-7
28) An equilateral triangle is constructed in the interior of a
semicircle as pictured. Point B is equidistant from points A and C.
If the diameter of the semicircle is 9 inches, what is the area
of the triangle? Round your answer to the nearest tenth.
(3) 1997-WO4-9
A
C
B
29) A 3-inch by 4-inch rectangle is rotated about a corner. What is the
maximum number of square inches in the area of the region touched by
some point of the rectangle as it makes a full rotation? Express your
answer in terms of π. (3) 2001-WU7-3
30) What is the ratio of the area of a square
inscribed in a semicircle with radius 10 inches
to the area of a square inscribed in a circle
with radius 10 inches? (3) 2001-WU7-3
31) A goat is attached to an L-shaped rod with a leash
that allows the goat to move a ground distance of 8
meters form the rod on all sides. AB = 10 m, BC = 20 m
and AB is perpendicular to BC. The attached end of the
leash may move along the entire rod and the goat may
roam on all sides of the rod. What is the area of the
region of grass that the goat may reach? Express your
answer in terms of π. (3) 2001-WO1-7
A
B
C
32) Circles A and B are externally tangent and have radii 9 inches and 16 inches,
respectively. What is the length of the common tangent MN? (3) 2001-WO3-8
B
A
M
N
33) A circle with diameter 2 cm in centered at
a vertex of the square and of the equilateral
triangle. What is the area of the union of
the three figures? Round to the nearest
hundredth. (3) 2001-WO6-9
34) AB and CD each measure 12”, the arcs are semicircles, AB is bisected
and CD is trisected. What is the difference between the areas of the
shaded regions of the two figures? (2) 1996-WU11-5
A
B
C
D
35) The diameter of the large semicircle is 8. Find the area of the shaded region.
Leave your answer in terms of π. (2) 1996-WO2-5
36) The area of the circle shown is 8π square inches. The
base of the triangle is a diameter of the circle. Find the
area of the triangle. (3) 1996-WO6-10
37)
The circles below are mutually tangent. The
areas of the circles are 16π, 25π, 100π and 225π.
What is the perimeter of a quadrilateral that
has the centers of the circles as its four
vertices? (2) 1993-WU9-3
38) Each wheel of Sally’s new bicycle has a diameter of 60 cm. How many
revolutions does a wheel make with Sally pedaling a distance of one
kilometer? Use π = 3.14 and round to the nearest revolution. (2) 1993-WU126
39) Pascal’s Pizza Parlor is changing the size of its circular pizza from 12
inches to 16 inches and increasing the number of slices per pizza from 8 to
10 pieces. What is the percent increase in the size of each new slice?
Express to the nearest whole percent. (2) 1993-WO3-10
40) Circle O has a radius of 12 cm and m  AOB
is 60º. Find the area of the shaded region.
Round to the nearest tenth. (3) 1993-WO5-7
A
O
41) In the diagram, mACB  60º and the length
of arc AB equals the length of arc PRQ . C is the center
1
B
A
of arc AB and arc PRQ . Find the ratio of PC to AC.
(3) 1995-WU16-9
B
P
R
C
Q
42) If a semicircle is constructed on the side of
an equilateral triangle, as pictured, what percent
of the area of the semicircle lies outside the triangle?
Round to the nearest whole percent. (3) 1995-WO4-9
687295018
11
Circles – Answer Key
22) 1/5
1) 13 hrs
2)

23) 20
840
24) 36 cm
3) 5
25) 14π in
4) 7.2
3
5)
in
4
6)
 22,832 m2
26) 3π in
27)  /4
28) 11.7
7) 600 cm2
29) 25 
8) 4”
30) 2:5
9) 5 ft
31) 80π + 416 m2
10) 33 in
32) 24 in
11) 1:3
33) 7.56 cm2
12) 72 
34) 3π in2
13) 38 cm2
35) 2π
14) 4
36) 8
15) 31.4
37) 68
16) 89%
38) 531
17) 720
39) 42%
18)  2 u2
40) 13.0 cm2
19) 200 cm2
41) 1:5
20) 550  ft2
42) 12%
21)
4
4
687295018
12
Circles