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Math
Practice Problems for High School in Geometry
HSG-69)
Answer: Ö
2 times that of the original coins
Two
coins, both silver dubloons, are melted down and recast as a single coin of the
same thickness. How does the diameter of the new coin compare with the diameter
of one of the original coins? Explain
your answer in detail.
HSG-68)
Answer: intersection is 15 feet above the ground; if 120 feet apart it would
still be 15 feet
Two
poles, 60 feet tall and 20 feet tall, stand on opposite sides of a field. The
poles are 80 feet apart. Support cables are placed from the top of one pole to
the bottom of the opposite pole. How far above the ground is the intersection of
the cables? What if the poles were 120 feet apart?
Explain your
answer in detail.
HSG-67)
Answer: 6x6x6
A
cake in the form of a cube falls into a vat of frosting and comes out frosted on
all six faces. The cake is then cut into smaller cubes, each one inch on an
edge. The cake is cut so that the number of pieces having frosting on three
faces will be one-eighth the number of pieces having no frosting at all. There
are to be exactly enough pieces of cake for everyone. How many people will
receive a piece of cake with frosting on exactly three faces? On exactly two
faces? On exactly one face? On no faces? How large was the original cake?
Explain your
answer in detail.
HSG-66)
Answer: 8D=44.4 inches, D=5.6 inches
There's
an antique bike parade in town. Stuart has a bike that his great grandfather had
given him in which the radius of the front wheel is 8 times the radius of the
rear wheel. When the bike travels 100 feet, the number of rotations made by the
smaller wheel is 60 more than the number of rotations made by the larger wheel.
Find the diameter of each wheel to the nearest tenth of an inch.
Explain your
answer in detail.
HSG-65)
Answer: 4
units; 2 units
The
local recycling plant has just bought a new metal compactor that produces a
smaller cube of scrap iron than does the older machine. Somebody noticed,
however, that the combined volumes of one cube from each compactor was
numerically the same as the combined lengths of all their edges. What are the
dimensions of the cubes, if you consider only integral solutions?
Explain your
answer in detail.
HSG-64)
Answer: There are actually 3 different parallelograms with the given 3 vertices.
The missing vertex not only could have been at (1,8) but also at (-3,0) or at
(5,2)
The
Lenape Valley High School Math Club was out on its field day. The teacher, Mr.
Romeo, assigned Pam and Jason the problem of finding a buried box. He told them
that the box was buried at the fourth vertex of the parallelogram having three
of its vertices at (-1,4), (1,1), and (3,5) on the Cartesian grid that was laid
out on the field. Pam and Jason dug at (1,8), but the box was not there. Why
didn't they find the buried box? What would you do? Defend your decision.
Explain your
answer in detail.
HSG-63)
Answer:
24
Below
is a 4x4x4 cubic block of wood. Suppose all six faces of the cube are painted
red and the cube is then cut into 1x1x1 cubes along the lines shown. How many
1x1x1 cubes will have red paint on just two faces? ?
Explain
your answer in detail.
HSG-62)
Answer:
62
In the figure below, all comer angles are right angles and each number
represents the unit-length of the segment which is nearest to it. How many
square units of area does the figure have? Explain your answer
in detail.
HSG-61)
Answer:
48 units
The figure below is divided into 8 congruent squares as shown. The area of the
figure is 72 square feet. What is the length of the darkened border in feet?
Explain your answer
in detail.
HSG-60)
Answer:
90
Three squares each have sides of length 6 units and overlap each other as shown
below. The points where the sides cross are midpoints. Find the area of the
shaded figure in square units. Explain
your answer in detail.
HSG-59)
Answer:
35
The tower below is made up of five horizontal layers of cubes with no gaps. How
many individual cubes are in the tower? Explain
your answer in detail.
HSG-58)
Answer:
104
A rectangular picture frame is 12 inches wide and 18 inches high. This
includes a 2-inch border (shaded region) around the picture itself.
How many square inches are in the border?
Explain
your answer in detail.
HSG-57)
Answer:
50
Square ABCD has all four of its
vertices on a circle with diameter 10 units in length. Segments AC and BD are
diameters. How many square units of area does square ABCD have? Explain
your answer in detail.
HSG-56)
Answer:
5
The
perimeter of a rectangle is 20 feet and the foot-measure of each side is a whole
number. How many rectangles with different shapes satisfy these conditions?
Explain your answer
in detail.
HSG-55)
Answer:
440
The
front wheel of a vehicle has a circumference of 3 feet, the rear wheel has a
circumference of 4 feet. How many more complete turns will the front wheel make
than the rear wheel in traveling a distance of I mile on a straight road? (I
mile = 5280 feet)
Explain your answer
in detail.
HSG-54)
Answer:
9
ABCD is a rectangle with area equal to 36 square units. Points E, F, and G are
midpoints of the sides on which they are located. How many square units are
there in the area of triangle EFG? Explain
your answer in detail.
HSG-53)
Answer:
5
The perimeter of a rectangle is 22 inches and the inch-measure of each side is a
natural number. How many different areas in square inches can the rectangle
have? Explain your
answer in detail.
HSG-52)
Answer:
38
Each of the boxes in the figure at the right is a square. How many different
squares can be traced using the lines in the figure? Explain your answer
in detail.
HSG-51)
Answer:
11
In the rectangle below, line segment MN separates the rectangle into 2 sections.
What is the largest number of sections into which the rectangle can be separated
when 4 line segments are drawn through the rectangle?
Explain your answer
in detail.
HSG-50)
Answer:
144
Rectangular
cards, 2 inches by 3 inches, are cut from a rectangular sheet 2 feet by 3 feet.
What is the greatest number of cards that can be cut from the sheet? Explain
your answer in detail.
HSG-49)
Answer:
8
Below is a 3 by 3 by 3 cube. Not all of the cubes are visible. Suppose the
entire outside of the cube is painted red including the bottom. How many
different 2 by 2 by 2 cubes- with exactly three red faces can be found in the
shown cube? Explain
your answer in detail.
HSG-48)
Answer:
30 meters
Square ABCD and rectangle AF-FG each have an area of 36 square meters. E is the
midpoint of AB. What is the perimeter of rectangle AEFG?
HSG-47)
Answer:
36
Twenty-four meters of fencing is used to fence a rectangular garden.: Let M
represents the number of square meters in the area of the garden. What is the
largest value that M could have? Explain
your answer in detail.
HSG-46)
Answer:
78
The “staircase” below is 4 units tall and contains 10 unit squares. Suppose
the staircase were extended until it was 12 units tall. How many unit squares
would it then contain all together?
Explain your answer
in detail.
HSG-45)
Answer:
246
A rectangular garden is 14 ft. by 21 ft. and is bordered by a concrete walk 3
ft. wide as shown. How many square feet are in the surface area of just the
concrete walk?
Explain your answer
in detail.
HSG-44)
Answer:
10
ABCD is a rectangle whose sides are 3 units and 2 units long. The length of the
shortest path from
A to C following the lines of the diagram is 5 units. How many different
shortest paths are there from A to C?
Explain your answer
in detail.
HSG-43)
Answer:
150 square units
Rectangle ABCD contains 3 small congruent rectangles. If the smaller dimension
of one of the small rectangles is 5 units, what is the area of rectangle ABCD in
square units? Explain
your answer in detail.
HSG-42)
Answer:
9 square units
Triangle ABC has its vertices A, B, and C on the sides of a rectangle 4 units by
5 units as shown.
What is the area of triangle ABC in square units?
Explain your
answer in detail.
HSG-41
Answer:
6
The length of the shortest trip from A to B along the edges of the cube shown is
the length of 3 edges. How many different 3-edge trips are there from A to B?
Explain your answer
in detail.
HSG-40)
Answer:
6 square meters
ABCD is a square with area 16 sq. meters. E and F are midpoints of sides AB and
BC, respectively. What is the area
of trapezoid AEFC, the shaded region?
Explain your answer
in detail.
HSG-39)
Answer:
100 square centimeters
As shown, ABCD and AFED are squares with a common side AD of length IO cm. Arc
BD and arc DF are quarter-circles. How many square cm. are in the area of the
shaded region?
Explain your answer
in detail.
HSG-38)
Answer:
20
The six faces of a three-inch wooden cube are each painted red. The cube is then
cut into one-inch cubes along the lines shown in the diagram. How many of the
one-inch cubes have red paint on at least two faces?
Explain your answer
in detail.
HSG-37)
Answer:
38
The structure below is made of unit cubes piled on top of each other. Some cubes
are not visible. What is the number of cubes in the structure? Explain your answer
in detail.
HSG-36)
Answer:
32
ABCD is a square with diagonal AC 8 units long. How many square units are in the
area of the square?
Explain your answer
in detail.
HSG-35)
Answer:
720
A tractor wheel is 88 inches in circumference. How many complete turns will the
wheel make in rolling one mile on the ground? (1 mile = 5,280 feet)
Explain your
answer in detail.
HSG-34)
Answer:
8
A wooden block is 4 inches long, 4
inches wide, and I inch high. The block is painted red on all six sides and then
cut into sixteen I inch cubes. How
many of the cubes each have a total number of red faces that is an even number?
Explain your answer in detail.
HSG-33)
Answer:
4
Eight one-inch cubes are put together to fomi the T-figure shown at the right.
The complete outside of the T-figure is painted red and then separated into
one-inch cubes. How many of the cubes have exactly four red faces?
Explain your answer
in detail.
HSG-32)
Answer:
E, A, N in that order
Below are three views of the same cube. What letter is on the face opposite (1)
H, (2) X, and (3) Y? (Give your answer in the same order.)
Explain your
answer in detail.
HSG-31)
Answer:
8
The complete outside including the bottom of a wooden 4 inch cube is painted
red. The painted cube is then cut into 1 inch cubes. How many of the 1 inch
cubes do not have red paint on any face? Explain your answer
in detail.
HSG-30)
Answer:
6
Below, ABCD is a square whose sides are each 2 units long. The length of the
shortest path from A to C following the lines of the diagram is 4 units. How
many different shortest paths are there from A to C? Explain your answer
in detail.
HSG-29)
Answer:
22
Each of the boxes in the figure below is a square, Using the lines of the
figure, how many different
squares can be traced?
Explain your answer
in detail.
HSG-28)
Answer:
10
An acute angle is an angle whose measure is between 0° and 90°. Using the rays
in the diagram, how many different acute angles can be formed?
Explain your
answer in detail.
HSG-27)
Answer:
16
The
figure shown consists of 3 layers of cubes with no gaps. Suppose the complete
exterior of the figure (including the bottom) is painted red and then separated
into individual cubes. How many of these cubes have exactly 3 red faces? Explain your answer
in detail.
HSG-26)
Answer:
20
A square has an area of 144 square inches. Suppose the square is partitioned
into six congruent rectangles below. How many inches are there in the perimeter
of one of the six rectangles?
Explain your answer
in detail.
HSG-25)
Answer:
48
The set of stairs shown below is constructed by placing layers of cubes on top
of each other. What is the total number of cubes contained in the staircase?
Explain your answer
in detail.
HSG-24)
Answer:
24 meters
A square is divided into three congruent rectangles as shown below. Each of the
three rectangles has a perimeter of 16 meters. How many meters are in the
perimeter of the square?
Explain your answer
in detail.
HSG-23)
Answer:
24
A
square piece of paper is folded in half as shown and then cut into two
rectangles along the fold. The perimeter of each of the two rectangles is 18
inches. What is the perimeter of the original square?
Explain your answer
in detail.
HSG-22)
Answer: #1 (almost half a meter)
Imagine
a wire band fitted snugly around the earth at the equator. If you cut the band,
add a piece exactly 10 feet long, reform the band, and then hold it in a
position concentric to the equator, which of the following best describes the
space between the earth's surface and the wire band? 1) You can just crawl under
the band. 2) You can just slide a piece of paper under the band. 3) You can walk
upright under the band. Explain
your answer in detail.
HSG-21)
Answer:
96
A rectangular tile is 2 inches by 3 inches. What is the least number of tiles
that are needed to completely cover a square region 2 feet on each side?
Explain your
answer in detail.
HSG-20)
Answer:
40
The
square at the right is divided into four congruent rectangles. The Perimeter of
each of the four congruent rectangles is 25 units. How many units are there in
the Perimeter of the square?
Explain your answer
in detail.
HSG-19)
Answer:
58
In
the figure below, each number represents the length of the segment which is
nearest it. How many square units are in the area of the figure if there is a
right angle at each comer of the figure? Explain
your answer in detail
HSG-18)
Answer:
36
Each of the small boxes in the figure below is a square. The perimeter of square
ABCD is 36 cm. What is the perimeter of the figure shown with darkened outline?
Explain your answer
in detail.
HSG-17)
Answer:
70
The tower shown below is made of horizontal layers of unit cubes, not all being
visible in the diagram. How many unit cubes are contained in the tower?
HSG-16)
Answer:
11
The tower below has no gaps. Suppose it is painted red on all exterior sides
including the bottom,, and then cut into cubes along the indicated lines. How
many cubes will each have red paint on just three faces? Explain your answer
in detail.
HSG-15)
Answer:
40
Each of the small boxes in the figure below is a square and the area of the
figure is 52 square units. How many units are there in the outer perimeter of
the figure? Explain
your answer in detail.
HSG-14)
Answer: 395,840,674.4 sq. inches
The
Acme Tinplate Corporation has been asked to manufacture one million cans for a
new product about to go to market. Each can is to be cylindrical, with a radius
of 3 inches and a height of 6 inches. How many square feet of tinplate are used
in making the cans? Explain
your answer in detail.
HSG-13)
Answer:
8
The length of AE is 20 cm. B is the midpoint of AC, C is the midpoint of BD, and
D is the midpoint of BE. What is the length of DE in cm? Explain your answer
in detail.
HSG-12)
Answer:
4
Below, there are two large congruent squares with sides 7 units long and four
small congruent squares with sides 3 units long. If the shaded figure is also a
square, what is its area in square units?
Explain your answer
in detail.
HSG-11)
Answer: 4
1/2 inches
An automobile
driveway is 54 feet long by 8 feet wide. It is to be covered with blacktop. If 3
loads of blacktop are available for the job, how deep a layer of blacktop is
possible? (One load =2 cubic yards.) Explain
your answer in detail.
HSG-10)
Answer:
20
The tower shown below is made of unit cubes stacked on top of each other. Some
of the unit cubes are not visible. How many unit cubes are not visible? Explain your answer
in detail
HSG-9)
Answer:
11
What
is the greatest number of points of intersection that can occur when 2 different
circles and 2 different straight lines are drawn on the same piece of paper? Explain
your answer in detail.
HSG-8)
Answer:
1/8
ABCD is a square, and E and F are midpoints of sides AD and AB respectively as
shown. What fractional part of the total area of the square is the area of
triangle AEF? Explain
your answer in detail.
HSG-7)
Answer:
30
Each
of the small boxes in the diagram below is a square and congruent to each of the
others. How many different squares can be traced using the lines of the diagram
as sides? Explain
your answer in detail.
HSG-6)
Answer:
27
The
tower shown below is made by placing congruent cubes on top of each other. Not
all cubes of the tower are visible. How many cubes does the tower contain?
Explain your
answer in detail.
HSG-5)
Answer:
18
The
cube below is constructed of congruent boards, each being of the same size and
shape. How many boards does the cube contain?
Explain your
answer in detail.
HSG-4)
Answer:
64
ABCD
is a square with each side divided into three segments of length 1 unit, 8
units, and 1 unit respectively, as shown in the diagram below. What is the sum
of the areas of the four shaded triangles?
Explain your
answer in detail.
HSG-3)
Answer:
1/4
ABCD
is a square; E, F, G, and H are midpoints of AP, BP, CP, and DP respectively.
What fractional part of the area of square ABCD is the area of square EFGH?
Explain your
answer in detail.
HSG-2)
Answer:
12
ABCD
is a square which contains nine small congruent squares as shown. The area of
square ABCD is 36 square units. What is the area of triangle ACE in square
units? Explain
your answer in detail.
HSG-1)
Answer:
18
Different
rectangles can be traced using the lines in the figure given at the right. How
many different rectangles can be traced? Explain
your answer in detail.
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