North Carolina Testing Program
EOG Reading Grade 5 Sample Items
This selection describes one way in which geometry—a branch of mathematics—has uses in
everyday life. Read the selection and answer the questions that follow.
Covering Up
by Ivars Peterson
Have you ever wondered why the
cover of a manhole is nearly always round?
Why couldn’t it be oval or square?
The usual answer is that a circular lid,
unlike a square or an oval cover, won’t fall
through the opening. There’s no way to
position a round lid so that it would slip
through a slightly smaller hole of the same
shape. That’s because the circle has a
constant width–it is the same width all the
way around.
In contrast, an oval is longer than it is
wide. You can always find a way to slip an
oval lid through a hole of the same shape.
That’s also true of a square or a six-sided, or
hexagonal, cover.
Amazingly, the circle isn’t the only
shape that would work safely as a
manhole cover. Another possibility is the
Reuleaux triangle, named after engineer
Franz Reuleaux, who was a teacher in
Berlin, Germany, more than a hundred years
ago. An example of a Reuleaux triangle can
be found in your medicine cabinet. If you
turn a bottle of NyQuil or Pepto Bismol
upside down, the shape you see looks like a
Reuleaux triangle.
One way to draw a Reuleaux triangle
is to start with an equilateral triangle, which
has three sides of equal length. Place the
pointed end of a pair of compasses at one
corner of the triangle and stretch the arms
until the pencil reaches another corner.
Then draw an arc between two corners of the
triangle. Draw two more arcs centered on
the triangle’s other corners.
This “curved triangle,” as Reuleaux
called it, has a constant width–just like a
circle. It would certainly work as a manhole
cover.
In fact, you can make a manhole cover
out of any shape with an odd number of
sides. Beginning with a five-sided shape
called a pentagon, for example, you can
construct a rounded pentagonal shape that
has a constant width.
Imagine walking down the street and
finding differently shaped manhole covers on
every block!
“Covering Up” by Ivars Peterson in Muse, July 1999. Copyright © 1999, published by Carus Publishing Company.
Page 1
Published April 2005. May reproduce for instructional and
educational purposes only, not for personal or financial gain.
North Carolina Testing Program
1.
EOG Reading Grade 5 Sample Items
What is the second step in drawing a Reuleaux triangle?
Draw an
equilateral
triangle.
1
2.
3.
Page 2
Stretch the
compass arms
until the pencil
reaches a corner.
Draw an arc
between two
corners of the
equilateral triangle.
3
4
2
A
Measure the sides of the triangle to be sure they are equal.
B
Draw three arcs that cover the triangle’s three sides.
C
Identify the side of the triangle that is the longest.
D
Place the pointed end of a compass at a corner of the equilateral triangle.
Why are manhole covers round?
A
A round cover cannot fall through
the opening.
B
A round cover lasts longer.
C
D
4.
Which of the following would be a good
use for something the shape of a
Reuleaux triangle?
A
plugging the drain hole of a sink
or bathtub
A round cover does not bend
easily.
B
plugging the coin slot of a drink
machine
A round cover is the cheapest to
make.
C
stopping cold air from entering
the house under a door
D
stopping water from flowing over
the top of a bathtub
According to the selection, what is the
most likely place to find a Reuleaux
triangle?
End of Set
A
toy chest
B
car trunk
C
medicine cabinet
D
tool box
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Published April 2005. May reproduce for instructional and
educational purposes only, not for personal or financial gain.
Answers to
Grade 5 Reading Comprehension Sample Items
Selection Title
Question
Number
Correct
Answer
Category
Thinking Skill
Objective
Number
Covering Up
1
D
Cognition
Organizing
2.02
Covering Up
2
A
Cognition
Knowledge
2.02
Covering Up
3
C
Cognition
Knowledge
2.02
Covering Up
4
A
Connections
Analyzing
2.02