Activity Lab
Activity Lab
How Far Can
You See?
How Far Can You See?
In these activities students
apply their knowledge of circles,
tangents to circles, and the
Pythagorean Theorem.
Applying the Pythagorean Theorem Imagine that you’re standing on an
ocean beach looking out across the water. The deep blue sky is clearer than
you’ve ever seen it, and it seems as though you can see forever! Well, you
know that isn’t really possible on Earth. The extent of your vision is limited by
Earth’s curvature. You can see to the horizon—and perhaps slightly beyond.
Activating Prior Knowledge
Have students brainstorm how
artists, architects, and engineers
might have used mathematics to
design and build the buildings
shown here.
The Empire State Building in
New York City is 1250 ft tall,
not including its mast. 7000
people visit the building
each day.
Guided Instruction
After students read the first
paragraph, have them relate
their own experiences of seeing
“endless” distances from a ship, a
beach, or a mountain. Encourage
them to estimate how far they
actually could see from that point.
The CN Tower
in Toronto is
1815 ft tall.
English Language Learners ELL
Relate the word curvature to
the word curve. Point out that
curvature refers to the spherical
shape of Earth.
The KTHI-TV
tower in North
Dakota rises to
2063 ft.
The Bank of
China Building
in Hong Kong
is 1033 ft tall.
The Chrysler
Building in
New York City
is 1046 ft tall.
The Eiffel
Tower in Paris
is 1052 ft tall,
which is
16 times
as tall as a
four-story
town house.
A chain of
8000 paper
clips dangled
from the top
floor of
1 Canada
Square in
London
would reach
the ground,
797 ft below.
Engineering Connection
Tell students that the invention
of the elevator helped make tall
buildings feasible. People would
have been unwilling to climb
many flights of stairs each day
to reach their homes or places
of work.
Four-story
town house,
66 ft tall
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The Saturn V
rocket is
364 ft tall.
All photographs © Dorling Kindersley Limited unless otherwise credited on acknowledgments page
Activity 1
Answers depend on structure
chosen.
KTHI-TV tower: about 55.7 mi
CN Tower: about 52.2 mi
Empire State Building:
about 43.3 mi
Chrysler Building: about 39.6 mi
Eiffel Tower: about 39.7 mi
Bank of China building:
about 39.4 mi
1 Canada Square: about 34.6 mi
Saturn V rocket: about 23.4 mi
St. Peter’s Basilica:
about 26.0 mi
Great Pyramid: about 26.9 mi
Cologne Cathedral:
about 27.8 mi
Leaning Tower of Pisa:
about 16.4 mi
Four-story town house:
about 10.0 mi
Teaching Tip
Crow’s nest, lookout
post for land and ships
Students can do Activity 1 for
all the buildings by writing
and graphing an equation on a
graphing calculator. For Activity 2,
students will need to use the right
angle method twice, once for the
crow’s nest and once for the tree.
Flag indicates
ship’s origin.
Activity 1
Choose one of the structures on these
pages. Imagine climbing to the very
top to get a good view of the horizon.
Assume that Earth is spherical and
has a radius of 3963 mi. Also assume
that you see a smooth horizon such
as that of an ocean or desert. Find the
distance from the top of the structure
to the horizon. (Hint: Use a calculator
and the Pythagorean Theorem.)
x
3963 mi
l
Galleon
Activity 1
Galleons were fighting
ships, with 40 to 50
cannons on board.
Materials paper and pencil
Teaching Tip
Activity 2
If necessary, remind students that
a tangent is perpendicular to the
radius containing the point of
tangency.
Sailors used to climb into the
crow’s nest on a ship’s mast
so they could spot land and
other ships at a greater
distance than was possible on deck. Imagine that you are
on watch in a crow’s nest so that your eyes are 40 ft above
the water.
Activity 2
a. Determine the farthest distance from which you could
spot the top of a 50-ft tree on a sea-level island.
Materials paper and pencil
b. Determine the farthest distance from which you could
spot the tree if you were standing on deck with your eyes
15 ft above the water.
Visual Learners
It is essential that students begin
their work by drawing an accurate
representation of the situation.
Students may want to start by
discussing how to interpret the
given information in light of
what they learned in Activity 1.
c. Compare your answers from
parts (a) and (b).
PHSchool.com
With a spyglass, distant
objects appear closer.
For: Information about buildings
Web Code: aue-0853
The dome of
St. Peter's Basilica
in Rome rises to a
height of 451 ft.
The Great Pyramid at Giza
is 481 ft tall. 10,000 people
visit it each day.
Scoring Rubric
The Cologne Cathedral
in Germany has two
identical spires that
taper to 513-ft heights.
The Leaning
Tower in Pisa,
Italy, is about
185 ft high.
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Activity 2
This scoring rubric applies to
both activities. Share this scoring
rubric with students before they
begin work.
4 Calculations are accurate.
Drawings are neat and
accurate and clearly reflect
the problem situations.
Explanations are thorough.
3 Calculations are mostly
accurate. Drawings are
neat and mostly accurate.
Explanations lack detail or
are not completely accurate.
2 Calculations are often
inaccurate. Explanations lack
clarity. Drawings, if included,
are not accurate.
1 Calculations are inaccurate.
No work is shown.
a. about 16.4 mi
b. about 13.4 mi
c. From the crow’s nest, you can spot the top
of the tree from about 3 miles farther away.
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