Calculating the Earth's Circumference at Any Date from Any Location

latitude

ϕ

1

(°)

44.13

longitude

λ

1

(°)

1.25

Standard Time Zone

1

DST

latitude

ϕ

2

(°)

29.

longitude

λ

2

(°)

77.7

Standard Time Zone

5.5

DST

year

2011

month

2

day of month

10

zoom

February 10 2011

declination δ = - 14° 51' 12"

solar time (noon)

12: 0' 0"

location

1

latitude

ϕ

1

= 44° 7' 48"

longitude

λ

1

= 1° 15' 0"

zenith at noon,

θ

1

= 58° 59' 0

local time 13: 0' 26"

location

2

latitude

ϕ

2

= 29° 0' 0"

longitude

λ

2

= 77° 42' 0"

zenith at noon,

θ

2

= 43° 51' 12

local time 12: 24' 38"

We verify that

θ

1

=

ϕ

1

- δ

θ

2

=

ϕ

2

- δ

ϕ

1

-

ϕ

2

=

θ

1

-

θ

2

= 15° 7' 48

The Earth's circumference is determined from

θ

1

,

θ

2

, and

D

12

.

circumference =

D

12

360

θ

1

-

θ

2

= 1682 × (360 / 15.13) = 40030 km

This Demonstration generalizes Eratosthenes's calculation of the Earth's circumference. The solar zenith angles

θ

1

and

θ

2

are measured at noon, on the same day, at two different locations. Knowing the distance

D

12

between the parallels of the two locations, the Earth's circumference is given by:

D

12

360



θ

1

-

θ

2



or

D

12

360

θ

1

+

θ

2

,according to whether or not the locations belong to the same hemisphere.

The Earth is, of course, assumed to be a perfect sphere. Actually, it is slightly oblate, with the equatorial radius approximately 21 km greater than the polar radius, but this contributes a negligible correction.