Latitude_Longitude_Illustrated_Reading_Final

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SCIENTIFIC INQUIRY WITH AI | ILLUSTRATED READING
Latitude and longitude
Two angles describe a place
Latitude and longitude 1
Your phone reports its location as coordinates. This reading helps you interpret those
numbers, infer the direction of a walk, and estimate short distances in meters. Read pages
1–4 and complete the nine worksheet questions. Allow about 25–30 minutes for both; the
math appendix is optional.
Start with the reference lines
Latitude is angular position north or south of the equator. The equator is 0°; signed decimal
latitude is positive north and negative south. The poles are +90° and −90°.
Longitude is angular position east or west of the zero-longitude reference meridian near
Greenwich. In the signed convention used here, east is positive and west is negative, from
−180° to +180°. All meridians meet at the poles, so longitude is undefined there.
The diagrams use a spherical Earth. Lines of constant latitude run east–west; you cross them
when moving north–south. Longitude meridians run from pole to pole; you cross them when
moving east–west. Seen from above the North Pole, longitude increases toward the east.
Read the order and the units
We write coordinate pairs as (latitude, longitude), in decimal degrees. For example,
(37.36000°, −120.43000°) is north of the equator and west of Greenwich. Some software
reverses the order, so always check the column labels.
A phone reports these coordinates in the WGS 84 reference frame, the same one GPS uses.
Pause and explain: Why can a line of latitude run east–west even though latitude tells you
about north–south position?

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Read coordinates on a grid
Increasing numbers reveal direction
Latitude and longitude 2
On this north-up grid, latitude is the vertical coordinate and longitude is the horizontal
coordinate. Read the axis labels before interpreting a point. This grid is for coordinate
practice; equal steps in degrees need not represent equal distances in meters.
Use final minus initial
A coordinate change is the final value minus the initial value. An increase in latitude means
northward motion; a decrease means southward motion. Locally, an increase in longitude
means eastward motion; a decrease means westward motion.
Negative longitude can increase while it remains negative. Moving from −120.50° to
−120.49° gives Δlongitude = −120.49° − (−120.50°) = +0.01°. That is an eastward change.
“West of Greenwich” describes location; “moving east” describes a change in location.
Combine the two changes
If latitude and longitude both increase, the displacement has northward and eastward
components. If only latitude changes, the idealized motion is north–south. If neither
coordinate changes, the idealized position is constant.
For this lesson, compare nearby points around campus. A route crossing the ±180° longitude
boundary needs special handling; ordinary subtraction can give a misleading large change.
Try it: Read P and Q as (latitude, longitude). If you move from P to Q, what happens to each
coordinate?

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Convert coordinate changes to meters
Use a local scale for each coordinate
Latitude and longitude 3
Degrees measure angles. To estimate a walking distance, multiply a coordinate change by a
local conversion factor. Near Merced, at latitude about 37.36°, use these classroom
approximations consistently:
Northward change (m) ≈ Δlatitude (degrees) × 111,000 m/degree
Eastward change (m) ≈ Δlongitude (degrees) × 88,500 m/degree
Why are the factors different
Longitude meridians meet at the poles. A fixed angular longitude change therefore spans a
smaller surface distance at higher latitudes. One degree of latitude remains roughly 111 km;
one degree of longitude depends strongly on latitude.
A worked example
A student moves from (37.36000°, −120.43000°) to (37.36020°, −120.42970°). Subtract
before converting:
Δlatitude = +0.00020° → +22.2 m north.
Δlongitude = +0.00030° → +26.55 m east.
The approximate straight-line separation is √(22.2² + 26.55²) ≈ 34.6 m. This is the
separation of the endpoints. It is the route length only if the route between them is
approximately straight.
Keep the signs to express direction: a negative northward change means south, and a
negative eastward change means west.
Then report at the precision the data support. These coordinate changes carry two digits, so
the answer is about 22 m north, 27 m east, and a 35 m separation. Carrying extra digits
through the arithmetic protects against round-off; it does not create measurement accuracy.
Optional: pages 5–7 derive these factors and check how accurate they are.

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Use coordinates for a campus walk
Connect the numbers to a route
Latitude and longitude 4
This invented practice route is near Merced in coordinate value. It is a schematic, not a
verified campus walkway. Both axes of the sketch use the same scale. Travel A → B → C → D
→ A.
Waypoint Latitude (°) Longitude (°)
A 37.36000 −120.43000
B 37.36000 −120.42900
C 37.36100 −120.42900
D 37.36100 −120.43000
Notice which coordinate remains constant on each leg. Compute each leg separately, then
add the leg lengths to estimate the total distance traveled. This route ends where it began.
The change from the start to the final point describes displacement; it does not record every
part of the journey.
Keep the measurement in view
A phone reports estimated coordinates. Small coordinate changes while it lies still reflect
measurement variation rather than real walking, and stable digits do not by themselves
prove that the reported location is accurate. Coordinate precision and location accuracy are
different: extra decimal places make smaller numerical changes representable, but they do
not mean the receiver located you that closely.
Most sensor apps also report a horizontal accuracy in meters beside each position. That
column, not the number of decimal places, is the receiver’s own uncertainty estimate, and
for a phone it is usually a few meters. Record it, and use a stationary trial and known route
features as checks during the lab.
Complete the worksheet before asking for full solutions. Preserve your first answers and add
revisions so you can explain what changed in your reasoning.
Coordinate references: NOAA, What is latitude? and What is longitude? Those pages describe latitude and longitude as distances
north–south and east–west; this reading treats them as angles first and converts to meters separately.
https://oceanservice.noaa.gov/facts/latitude.html
https://oceanservice.noaa.gov/facts/longitude.html

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Optional mathematics appendix
From a circle to a latitude factor
Latitude and longitude 5
You can complete the worksheet without this appendix. Here we model Earth as a sphere
with radius R ≈ 6,378,000 m, approximately its equatorial radius. Earth is slightly flattened;
this is a classroom model, not a surveying formula.
Convert degrees to radians
One full turn is 360° or 2π radians. Radians express an angle as arc length divided by radius:
θ in radians = θ in degrees × π/180
Arc length s = Rθ when θ is in radians.
The sketch exaggerates the angle so you can see the curved surface arc and the straight
chord through the sphere.
Identify the approximation
For a small angle u measured in radians, sin(u) ≈ u. The chord length is therefore:
c = 2R sin(θ/2) ≈ 2R(θ/2) = Rθ = s.
The arc relation s = Rθ is exact on a circle. The approximation is replacing the curved arc
with the straight chord, which is the same as treating a short piece of the surface as flat. For
a 100 m arc, θ = 100/6,378,000 ≈ 0.0000157 radians, and the chord falls short of the arc by
about a nanometer.
Calculate the latitude factor
North–south travel follows a meridian circle of radius R. For a 1° change:
R × π/180 ≈ 111,317 m per degree.
This lesson uses 111,000 m per degree of latitude.
The spherical illustration uses geocentric latitude. Precise GPS coordinates use geodetic
latitude on an ellipsoid; page 6 checks what the spherical model and the rounding cost.

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Optional mathematics appendix
Longitude and the limits of a local map
Latitude and longitude 6
At latitude φ (phi), the east–west latitude circle is smaller than the equatorial circle. A right
triangle gives its radius r:
cos(φ) = r/R, so r = R cos(φ). Using this radius in the arc-length equation gives meters per
degree of longitude:
R cos(φ) × π/180.
Near Merced, φ ≈ 37.36° and cos(φ) ≈ 0.79484:
6,378,000 × 0.79484 × π/180 ≈ 88,479 m per degree.
This lesson uses 88,500 m per degree of longitude. Set your calculator to degrees when
evaluating cos(37.36°).
How good are these factors
Keep extra digits until the end. Multiplying the already rounded 111,000 by cos(37.36°) gives
about 88,200, a worse estimate than 88,479.
On the WGS 84 ellipsoid at this latitude, one degree of latitude spans about 110,995 m and
one degree of longitude about 88,590 m. The lesson factors sit within roughly 0.1% of both,
or about 10 cm over a 100 m walk, while a phone position is uncertain by a few meters.
Rounding is therefore not automatically a loss of accuracy: 111,000 is closer to the ellipsoidal
value than the unrounded spherical 111,317 is.
The model is not the largest source of error in this lab. Report that way: quote distances to
the precision your measurement supports, not to the precision the conversion factor could
carry.
Combine short perpendicular changes
For nearby endpoints, use a representative local latitude and estimate:
d ≈ √[(Δnorth)² + (Δeast)²].
This is local endpoint separation. For a route, add the lengths of successive short segments.
Sparse samples can miss turns; noisy samples can add spurious distance.

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Optional mathematics appendix
What a decimal place is worth
Latitude and longitude 7
Read the decimal places
Because the two factors differ, the same coordinate step is worth different distances on the
two axes. Near Merced:
Coordinate step Northward (Δlatitude) Eastward (Δlongitude)
0.1° 11.1 km 8.85 km
0.01° 1.11 km 885 m
0.001° 111 m 88.5 m
0.0001° 11.1 m 8.85 m
0.00001° 1.11 m 0.885 m
The fifth decimal place is worth about a meter, which is already smaller than a phone’s
typical horizontal accuracy. Digits beyond it describe the number, not the location.
Know when to change the model
A local map with fixed Merced factors stays accurate only while the latitude barely changes.
Move 1 km north and the longitude factor shifts by about 0.01%. Move 100 km north, roughly
0.9° of latitude, and it shifts by more than 1%, which is ten times the error in the factors
themselves.
Curvature is not what breaks first. Treating a few hundred meters of surface as flat costs less
than a micrometer. The error you actually carry is in the conversion factors, and it grows
once you travel far enough north or south to change the latitude.
A single flat map with fixed Merced factors cannot accurately represent a journey across the
globe. Great-circle calculations, such as the haversine formula, account for spherical
curvature; ellipsoidal geodesic calculations also account for Earth’s flattened shape. Except
along the equator, a latitude-circle arc is not the shortest global route between two points.
Radius reference: NASA/JPL, Planetary Physical Parameters, which lists Earth’s equatorial radius as 6,378.1366 km and its mean radius
as 6,371.0084 km. The spherical radius used here is a rounded modeling choice.
https://ssd.jpl.nasa.gov/planets/phys_par.html