Earth's Radius by Latitude

Earth's Radius by Latitude

Introduction: Calculating the Earth’s Radius

In quick, general calculations, we assume that the shape of the Earth approximates the shape of sphere, with the radius to generally be approximated as 6371 km. However, the shape of the Earth is closer to an oblate ellipsoid. The Earth is flattened at the poles and bulges the greatest at the Equator.

There are several calculations to estimate the Earth’s radius given the latitude (degrees North or South from the Equator), this blog is working with the WGS 84 formula (see Source):

R = √[ ((a² * cos(φ))² + (b² * sin(φ))²) ÷ ((a * cos(φ))² + (b * sin(φ))²) ]

Which is simplified from:

R = √[ (a^4 + b^4 * tan²(φ)) ÷ (a² + b² * tan²(φ))

where:

φ: latitude (in degrees)

a: semi-major axis: 6378.137 km

b: semi-minor axis: 6356.7523142 km

R: radius of kilometers

HP 15C Code: Earth’s Radius by Latitude

Step; Key; Key Code

001: LBL E; 42, 21, 15

002: DEG; 43, 7

003: →H; 43, 2

004: TAN; 25

005: x²; 43, 11

006: STO 1; 44, 1

007: 6; 6

008: 3; 3

009: 7; 7

010: 8; 8

011: . ; 48

012: 1; 1

013: 3; 3

014: 7; 7

015: STO 2; 44, 2

016: 4; 4

017: 14; y^x

018: 6; 6

019: 3; 3

020: 5; 5

021: 6; 6

022: . ; 48

023: 7; 7

024: 5; 5

025: 2; 2

026: 3; 3

027: 1; 1

028: 4; 4

029: 2; 2

030: STO 3; 44, 3

031: 4; 4

032: y^x; 14

033: RCL 1; 45, 1

034: ×; 20

035: +; 40

036: RCL 2; 45, 2

037: x²; 43, 11

038: RCL 3; 45, 3

039: x²; 43, 11

040: RCL 1; 45, 1

041: ×; 20

042: +; 40

043: ÷; 10

044: √; 11

045: RTN; 43, 32

TI-60 Code: Earth’s Radius by Latitude

Step; Key; Key Code

Set degrees mode before beginning.

00: DMS-DD; 39

01: TAN; 34

02: x²; 96

03: STO; 61

04: 1; 01

05: ( ; 53

06: 6; 06

07: 3; 03

08: 7; 07

09: 8; 08

10: . ; 93

11: 1; 01

12: 3; 03

13: 7; 07

14: STO; 61

15: 2; 02

16: y^x; 45

17: 4; 04

18: +; 85

19: 6; 06

20: 3; 03

21: 5; 05

22: 6; 06

23: . ; 93

24: 7; 07

25: 5; 05

26: 2; 02

27: 3; 03

28: 1; 01

29: 4; 04

30: 2; 02

31: STO; 61

32: 3; 03

33: y^x; 45

34: 4; 04

35: ×; 25

36: RCL; 71

37: 1; 01

38: ) ; 54

39: ÷; 55

40: ( ; 53

41: RCL; 71

42: 2; 02

43: x²; 96

44: +; 85

45: RCL; 71

46: 3; 03

47: x²; 96

48: ×; 65

49: RCL; 71

50: 1; 01

51: ) ; 54

52: = ; 95

53: √; 86

54: R/S; 13

55: RST; 22

Examples

Latitude: 80° 00’; Radius (km): 6357.402412

Latitude: 57° 24’; Radius (km): 6362.996788

Latitude: 43° 40’; Radius (km): 6367.986902

Latitude: 9° 15’; Radius (km): 6377.588959

Latitude: 2° 56’; Radius (km): 6378.081467

Source

Planetcalc. “Earth Radius by Latitude (WGS 84)” Timur. 2021. https://planetcalc.com/7721/ Retrieved March 22, 2026.

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-60 and HP 65: Distance by Stadia Tacheometry

TI-60 and HP 65: Distance by Stadia Tacheometry

Introduction

The stadia calculation measures the distance from the level or theodolite (measuring device) to a graduated staff. Two readings are taken from the graduated staff, an upper reading (UR) and a lower reading (LR). The word tacheometry means "swift calculation". The distance between the theodolite and the staff is calculated as:

dist = k * s

k = is a factor taken of a radius of focal distance over image distance. Typically, k is set to 100, and will be assumed that k = 100 for these programs.

s = the difference between the upper reading and lower reading on the graduated staff. (UL - RL). The readings are assumed to be in meters.

If the theodolite is tilted at angle α, known as the vertical angle, then the distance becomes:

dist = k * s * cos² α

α is usually given in degrees-minutes-seconds and must be converted to decimal degrees.

TI-60 Program: Stadia Tracheotomy

Store before running: R1: UR (m), R2: LR (m), R3: α (D.MMSS)

Code:

00: 1 ; 01

01: 0 ; 00

02: 0 ; 00

03: × ; 65

04: ( ; 53

05: RCL; 71

06: 1 ; 01

07: - ; 75

08: RCL; 71

09: 2 ; 02

10: ) ; 54

11: × ; 65

12: RCL; 71

13: 3 ; 03

14: DMS-DD; 39

15: cos; 33

16: x²; 96

17: = ; 95

18: R/S; 13

19: RST; 22

HP 65 Program: Stadia Tracheotomy

Input Stack:

Z: vertical angle in degrees-minutes-second (V.MS)

Y: upper reading (UR)

X: lower reading (LR)

Code:

23: LBL

14: D

51: -

02: 2

32: f^-1

08: LOG (10^x)

71: ×

35 07: x<>y

32: f^-1

03: →D.MS (→DD)

31: f

05: COS

32: f^-1

09: √ (x²)

71: ×

24: RTN

Examples

Example 1:

V: 1°05' (Z stack, R3)

UR: 2.014 m (Y stack, R1)

LR: 1.668 m (X stack, R2)

Result: distance ≈ 34.5876 m

Example 2 (from HP 35):

V: 4°17' (Z stack, R3)

UR: 3.144 m (Y stack, R1)

LR: 1.761 m (X stack, R2)

Result: distance ≈ 137.5285 m

Example 3 (from HP 35):

V: -7°21' (Z stack, R3)

UR: 2.817 m (Y stack, R1)

LR: 0.731 m (X stack, R2)

Result: distance ≈ 205.1860 m

Sources

CivilFerba "Measure the distance by stadia method" Video posted on YouTube on December 8, 2018. https://www.youtube.com/watch?v=oon5ayl9DYs Retrieved March 22, 2026

HP-35 Surveying. Hewlett Packard. Buchs, Switzerland. February 1973. pp. 19-20

fx-FD10 Pro. User's Guide. Casio. Tokyo, Japan. 2014. pg. α-16 (alpha-16)

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-60 Geometry: Intersection and Angle Between Two Lines

TI-60 Geometry: Intersection and Angle Between Two Lines

We have two lines in the standard form:

y = R1 * x + R2

y = R3 * x + R4

where R1, R3 are the slopes and R2, R4 are the y-intercepts, which are stored prior to executing the programs.

TI-60: Intersection Point

x = (R4 – R3) / (R1 – R3), y = R1 * x + R2

Step Key Code	Key	Step Key Code	Key
00 53	(	14 54	)
01 71	RCL	15 95	=
02 04	4	16 13	R/S
03 75	-	17 65	×
04 71	RCL	18 71	RCL
05 02	2	19 01	1
06 54	)	20 85	+
07 55	÷	21 71	RCL
08 53	(	22 02	2
09 71	RCL	23 95	=
10 01	1	24 13	R/S
11 75	-	25 22	RST
12 71	RCL
13 03	3

TI-60: Angle Between Two Lines

Θ = arctan(abs((R1 – R3) / (1 + R1 * R3)))

Step Key Code	Key	Step Key Code	Key
00 53	(	13 65	×
01 71	RCL	14 71	RCL
02 01	1	15 03	3
03 75	-	16 54	)
04 71	RCL	17 95	=
05 03	3	18 96	x²
06 54	)	19 86	√x
07 55	÷	20 12	R/S
08 53	(	21 34	RST
09 01	1
10 85	+
11 71	RCL
12 01	1

Examples

	Intersection Point: (x, y)	Angle: Θ (degrees)
y = -2 * x + 3 y = 3 * x + 4	(-0.2, 3.4)	45°
y = x + 8 y = 3 * x – 6	(7, 15)	26.56505118°
y = 4 * x – 6 y = 2 * x + 1	(3.5, 8)	12.52880771°

May I close with this: Happy Birthday, Susan Sarandon!

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

The author does not use AI engines and never will.

HP 15C and TI-60: The Lottery Probability Function

HP 15C and TI-60:  The Lottery Probability Function

The Lottery Probability Function

The expression, of what I call the Lottery Probability Function:

prob =  nCr(k, b) × nCr(n-k, k-b) ÷ nCr(n, k)

where:

*  nCr is the combination function.  nCr(x,y) = x! ÷ (y! × (y - x)!)

*  n = the number of total balls in the lottery

*  k = the number of balls drawn

*  b = the number of balls your ticket matches

*  prob = probability 

Lottery odds are often stated as "1 in number" instead of decimal form.  The number is the statement "1 in number" is 1/p.   For example, "1 in 5" means that the probability is 1/5 or 0.20.  (20%)

HP 15C (Emulator) Program Code

Program Size:

19 steps, 30 bytes

Store before running label A:

R1 = n

R2 = k

R3 = b

Step : Step Code : Key

001 : 42,21,11 : LBL A

002 : 45, 2 : RCL 2

003 : 45, 3 : RCL 3

004 : 43,40 : Cy,x

005 : 44, 0 : STO 0

006 : 45, 1 : RCL 1

007 : 45,30, 2 : RCL- 2

008 : 45, 2 : RCL 2

009 : 45,30, 3 : RCL- 3

010 : 43,40 : Cy,x

011 : 44,20, 0 : STO× 0

012 : 45, 1 : RCL 1

013 : 45, 2 : RCL 2

014 : 43,40 : Cy,x

015 : 44,10, 0 : STO÷ 0

016 : 45, 0 : RCL 0

017 : 15 : 1/x

018 : 44, 0 : STO 0

019 : 43, 32 : RTN

Caution:  HP 15C Limited Edition.  It is a known bug when the combination (Cy,x) and permutation (Py,x) are run multiple times, the display flashes as if a overflow error occurs.  The program otherwise runs fine (with the examples I tested - 2011/04/15 firmware).   

To eliminate unnecessary flashing, use the following key sequence:

[ GTO ] ( A ) [ R/S ]

Full a list of known bugs, as compiled by Katie Wasserman, click here:

https://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=1089

TI-60 (Programmable Scientific/Advanced Scientific) Program Code

57 steps.   No commands are merged.  

For the combination (and permutation) functions, the number must be formatted as the following: nnn.rrr

The command sequence INV EE cancels scientific notation display.  

Step : Step Code : Key

01 : 71 : RCL

02 : 02 :  2

03 : 85 : +

04 : 03 : 3

05 : 55 : ÷

06 : 01 : 1

07 : 42 : EE

08 : 03 : 3

09 : 95 : =

10 : 80 : nCr

11 : 61 : STO 

12 : 00 : 0

13 : 71 : RCL

14 : 01 : 1

15 : 85 : +

16 : 71 : RCL

17 : 02 : 2

18 : 55 : ÷

19 : 01 : 1

20 : 42 : EE

21 : 03 : 3

22 : 95 : =

23 : 80 : nCr

24 : 61 : STO

25 : 55 : ÷

26 : 00 : 0

27 : 71 : RCL

28 : 01 : 1

29 : 75 : -

30 : 71 : RCL

31 : 01 : 1

32 : 85 : +

33 : 53 : (

34 : 71 : RCL

35 : 02 : 2

36 : 75 : -

37 : 71 : RCL

38 : 03 : 3

39 : 54 : )

40 : 55 : ÷

41 : 01 : 1

42 : 42 : EE

43 : 03 : 3

44 : 95 : =

45 : 80 : nCr

46 : 61 : STO

47 : 65 : ×

48 : 00 : 0

49 : 71 : RCL

50 : 00 : 0

51 : 76 : 1/x

52 : 61 : STO

53 : 00 : 0

54 : 12 : INV

55 : 42 : EE

56 : 13 : R/S

57 : 22 : RST

Examples

1.  49 balls and 6 are drawn.    Set up variables:  R1 = n =  49 and R2 = k = 6.

R3 = b = 6:  1 in 13,983,816

R3 = b = 5:  1 in 54,200.83721

R3 = b = 4:  1 in 1,032.3396899

2.  52 balls and 6 are drawn.  Set up variables:  R1 = n = 52 and R2 = k = 6.

R3 = b = 6:  1 in 20,358,520

R3 = b = 5:  1 in 73,762.75362

R3 = b = 4:  1 in 1,311.337842

Note that this function does not take the power ball, which only 1 is drawn in addition to the lottery balls, into account.   If you have a power ball of p balls, we can calculate the odds as follows:

You match the power ball:

prob = 1 ÷ p × nCr(k, b) × nCr(n-k, k-b) ÷ nCr(n, k)

1 in odds:  1/prob

You don't match the powerball:

prob = (p - 1) ÷ p × nCr(k, b) × nCr(n-k, k-b) ÷ nCr(n, k)

1 in odds:  1/prob

Source:

"Lottery mathematics".  Wikipedia.  Last edited July 10, 2023.  Retrieved July 23, 2023.  https://en.wikipedia.org/wiki/Lottery_mathematics

Eddie 

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.