SME
Cartography · Problem type
Great-circle distance
Use this for “How far is it from (40° N, 75° W) to (52° N, 1° W) along the shortest route over the globe?”
Diagram
cos c = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos(λ₂ − λ₁), so c = 50.5°, the angle at Earth’s centre.
The great-circle arc (on the rim) is d = 6,371 × 0.881 rad = 5,610 km.
The dashed rhumb line keeps one bearing (straight on a Mercator map) and is longer.
This is an example. Type your own values to start a new problem.
φ₁
Latitude of place 1
example · °
λ₁
Longitude of place 1
example · °
φ₂
Latitude of place 2
example · °
λ₂
Longitude of place 2
example · °
c
Central angle
calculated · °
d
Great-circle distance
calculated · km
Formulas
cos(
c
°)
=
sin(
φ₁
°)
×
sin(
φ₂
°)
+
cos(
φ₁
°)
×
cos(
φ₂
°)
×
cos(
λ₂
°
−
λ₁
°)
cos(50.4671°) = sin(40°) × sin(52°) + cos(40°) × cos(52°) × cos(−1° − (−75)°)
d
=
6,371
×
c
×
π
180
5,611.6891
=
6,371
×
50.4671
×
π
180
Assumptions
•
North latitudes and east longitudes are positive; south and west are negative (75° W is −75°).
•
Earth is a sphere of radius R = 6,371 km, within 0.5 % of the real ellipsoid.
•
The shortest route is an arc of a great circle, not the straight line on a Mercator map.
Step-by-step
WE KNOW
φ₁ = 40°, λ₁ = −75°, φ₂ = 52°, λ₂ = −1°
FIND
central angle (c), great-circle distance (d)
Step 1 · Find central angle (c)
Use cos(c°) = sin(φ₁°) × sin(φ₂°) + cos(φ₁°) × cos(φ₂°) × cos(λ₂° − λ₁°)
The spherical law of cosines gives the cosine of the angle at Earth’s centre between the two places.
c
=
cos⁻¹(sin(
φ₁
°)
×
sin(
φ₂
°)
+
cos(
φ₁
°)
×
cos(
φ₂
°)
×
cos(
λ₂
°
−
λ₁
°))
c
=
cos⁻¹(sin(40°)
×
sin(52°)
+
cos(40°)
×
cos(52°)
×
cos(−1°
−
(−75)°))
λ₂
−
λ₁
=
(−1)
−
(−75)
=
74°
sin(40°) × sin(52°) = 0.5065
cos(40°) × cos(52°) × cos(74°) = 0.13
cos
c
=
0.5065
+
0.13
=
0.6365
c
=
50.46713°
Step 2 · Find great-circle distance (d)
Use d = 6,371 × c × π ÷ 180
An
arc
is
the
radius
times
its
angle
in
radians,
and
×
π
180
turns
degrees
into
radians.
d
=
6,371
×
50.46713
×
π
180
50.4671
×
π
180
=
0.8808
rad
6,371 × 0.8808 = 5,611.6
d
=
5,611.689
km
Check
Put the numbers back in. Both sides should match.
cos(50.46713°) = sin(40°) × sin(52°) + cos(40°) × cos(52°) × cos(−1° − (−75)°) ✓
5,611.689
=
6,371
×
50.46713
×
π
180
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.