Lines & Angles

1. If the sum of the interior angles of a regular polygon measures up to 1440 degrees, how many
sides does the polygon have?
A. 10 sides B. 8 sides C. 12 sides D. 9 sides E. None of these

2. Given that ABCDE is a regular pentagon, what is the measure of ∠ACE?

(A) 24° (B) 30° (C) 36° (D) 40° (E) 45°

3.

In the figure above, if z = 50, then x + y =

2. A rectangular garden is to be twice as long as it is wide. If 360 yards of fencing, including the gate,
will completely enclose the garden, what will be the length of the garden, in yards?
(A) 120 (B) 140 (C) 160 (D) 180 (E) 200

3. The area of trapezoid ABCD is equal to the area of parallelogram WXYZ. What is m?

A. 4 B. 4.5 C. 5 D. 5.5 E. 6

4. Ramon wants to cut a rectangular board into identical square pieces. If the board is 18 inches by
30 inches, what is the least number of square pieces he can cut without wasting any of the board?
A. 4 B. 6 C. 9 D. 12 E. 15

5. If the area of the outer square is 4x^2, then the area of the inner square is

A. (x2)/2 B. x2 C. √2x2 D. 2x2 E. 2√2*x2

6. ABCD is a rhombus (see figure). ABE is a right triangle. AB is 10 meters. All lengths (triangle and
rhombus) are integral values. The ratio of the length of CE to the length of EB is 2 to 3. What is the
area of trapezoid AECD?
A. 84 B. 76 C. 56 D. 54 E. 64

7. A rectangular garden is surrounded by a 3 ft. wide concrete sidewalk. If the length of the garden is
4 ft more than its width, and if the area of the sidewalk is 60 sq ft more than the area of the garden,
then what is the length of the garden?
A. 6 B. 8 C. 10 D. 12 E. 14

8. For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes
arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each
remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of
the layer has only 1 box, how many boxes are in display?
A. 236 B. 260 C. 269 D. 276 E. 285

9. A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter
were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox,
what would be the capacity, in cubic feet, of the second sandbox?
(A) 20 (B) 40 (C) 60 (D) 80 (E) 100

10. The length, the breadth and the height of a rectangular solid are in the ratio 1:2:3. If the length,
breadth and height are increased by 100%, 200% and 200% respectively, then the increase in the
volume of the rectangular solid is
A) 5 times B) 6 times C) 12 times D) 17 times E) 20 times

11. Eight solid cubes of gold, each with an edge of 2 centimeters long, are melted together and
poured into a level, rectangular-shaped bar. The dimensions of the bar are 5 centimeters by 6
centimeters, and the bar is 4 centimeters deep. How many additional cubic centimeters of liquid
gold must be added to the bar so that it is completely full?
A. 26 B. 30 C. 56 D. 64 E. 120

12. A large cube consists of 125 identical small cubes. How many of them are exposed to air?
(A) 64 (B) 72 (C) 98 (D) 100 (E) 116
 Triangles
1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by
joining 3 of the points A, B, C, D, E and F?
A. 10 B. 15 C. 20 D. 25 E. 30

2. A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the
base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder
reach?
(A) 35 (B) 42 (C) 35√3 (D) 7+35√3 (E) 7+42√3

3.

In the figure above, triangle ABC is equilateral, and point P is equidistant from vertices A, B, and C. If
triangle ABC is rotated clockwise about point P, what is the minimum number of degrees the triangle
must be rotated so that point B will be in the position where point A is now?
A. 60 B. 120 C. 180 D. 240 E. 270

4.

In the diagram above, ABC is an equilateral triangle. D is the midpoint of AC. BD is a diameter of the
circle. If AD = 4, what is the area of the circle?
A. 8√3π B. 12π C. 16π D. 16√3π E. 48π
5. In the figure below, both triangles are right triangles. The area of the shaded region is

A. ½ B. 2/3 C. 7/8 D. 3/2 E. 5/2

6. In the figure below, ∆PST is an isosceles right triangle, and PS = 2. What is the area of the shaded
region URST?

A. 4 B. 2 C. 5 / 4 D. 5 / 6 E. 1 / 2

7. In the figure above, ABC is an isosceles right triangle. Line segment AC defines semicircle S1 and
legs AB and BC define semicircles S2 and S3 respectively. Two ants walk from A to C, The red ant
walks along S1 and the black ant walks along S2 and then along S3. What is the ratio, by distance, of
the path of the black ant to the path of the red ant?
A. 1:2 B. 2:1 C. √2:1 D. 1:√2 E. √2:π
 Circles & Cylinders
1.

If O is the center of the circle above, what fraction of the circular region is shaded?
(A) 1/12 (B) 1/9 (C) 1/6 (D) ¼ (E) 1/3

2.

The figure above shows a circular flower bed, with its center at O, surrounded by a circular path that
is 3 feet wide. What is the area of the path, in square feet?
(A) 25π (B) 38π (C) 55π (D) 57π (E) 64π

3. A point on the edge of a fan blade that is rotating in a plane is 10 centimeters from the center of
the fan. What is the distance traveled, in centimeters, by this point in 15 seconds when the fan runs
at the rate of 300 revolutions per minutes?
A. 750π B. 1500π C. 1875π D. 3000π E. 7500π

4. A circular clock has a minute hand measuring 12 inches long. How far, in inches, does the
outermost tip of the minute hand travel in 20 minutes?
A. 2π B. 4π C. 8π D. 12π E. 24π

5. A right circular cylinder having the radius of its base as 2 centimeters is filled with water upto a
height of 2 centimeters. This water is then poured into an empty rectangular container the
dimensions of whose base are 2π by 3 centimeters. If the volume of water in the rectangular
container is increased by 50 percent by adding extra water, what is the final height, in centimeters,
of the water level in centimeters in the rectangular container?

A) 0.5 B) 1 C) 1.5 D) 2 E) 2.5

6.

If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is
A. 12 √3 B. 8π C. 18 √2 D. 24 √3 E. 48

7. A container in the shape of a right circular cylinder is 1/2 full of water. If the volume of water in
the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of
the base of the cylinder, in inches?
(A) 16/9π (B) 4/π (C) 12/π (D) √(2/π) (E) 4*√(2/π)

8. When a cylindrical tank is filled with water at a rate of 22 cubic meters per hour, the level of water
in the tank rises at a rate of 0.7 meters per hour. Which of the following best approximates the
radius of the tank in meters?
A. √10/2 B. √10 C. 4 D. 5 E. 10