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-84 Plus CE, HP 15C, and HP 12C: Decoding the Gradematic 100

TI-84 Plus CE, HP 15C, and HP 12C: Decoding the Gradematic 100

GPA as a function of grade

Last December, we have gave a spotlight on the Calculated Industries 100 from 1983. The Gradematic 100 was a specialty calculator that determines the GPA average for a student or a bunch of students in a class. We can either use numerical grades, where the maximum total score and the minimum passing grade (better than E (or F)) are set, or letter grades, where the letters are given an approximated. To see the review, click the link below:

https://edspi31415.blogspot.com/2025/12/spotlight-calculated-industries.html

Using the standard grade scale for a single assignment, with a perfect score being 100 and the minimum passing grade is 60, the following scores are given the GPA:

GRADE (0 – 100)	GPA	Letter Grade Given by Gradematic 100
0	0.00	E (can stand for F)
10	0.08	E (can stand for F)
20	0.16	E (can stand for F)
30	0.25	E (can stand for F)
40	0.33	E (can stand for F)
50	0.41	E (can stand for F)
55	0.45	E (can stand for F)
60	0.50	D-
65	1.00	D
70	1.50	C-
75	2.00	C
80	2.50	B-
85	3.00	B
90	3.50	A-
95	4.00	A
100	4.50	A+

Plot of values (using a TI-84 Plus CE):

As we can see, the plot consists of two line segments: one where grades value from 0 to 60, and one where grades value from 60 and higher. It is apparent that that the two parts makes a piece-wise function consisting of two lines.

Note: the graphs and statistics were done with the TI-84 Plus CE Python (will work with any TI-84 CE family). The piecewise function is from the math-math menu.

The Gradematic 100 distributes the GPA as:

E: 0.00 (or F)	D+: 1.33	C+: 2.33	B+: 3.33	A+: 4.33
D-: 0.66	C-: 1.66	B-: 2.66	A-: 3.66
D: 1.00	C: 2.00	B: 3.00	A: 4.00

Note: The distributed GPA scales will vary among the school districts and systems. However, we will assume the system that matches the default 60/100 system.

HP 15C and HP 12C: Find the GPA given numeric grade

HP 15C Code:

LBL C	001	42, 21, 13
6	002	6
0	003	0
x≤y	004	43, 10
GTO 1	005	22, 1
2	006	2
×	007	20
÷	008	10
RTN	009	43, 32
LBL 1	010	42, 21, 1
CL x	011	43, 35
1	012	1
0	013	0
÷	014	10
5	015	5
.	016	48
5	017	5
-	018	30
RTN	019	43, 32

HP 12C Code:

6	01	6
0	02	0
x≤y	03	43, 34
GTO 09	04	43, 33, 09
2	05	2
×	06	20
÷	07	10
GTO 00	08	43, 33, 00
CL x	09	35
1	10	1
0	11	0
÷	12	10
5	13	5
.	14	48
5	15	5
-	16	30
GTO 00	17	43, 33, 00

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.

HP 15C: Distance and Slope Between Two Points Using Polar Conversion and the Stack

HP 15C: Distance and Slope Between Two Points Using Polar Conversion and the Stack

HP 15C Program: Distance and Slope

This short program calculates the slope and distance between two Cartesian points (x1, y1) and (x2, y2) using the four level stack and rectangular-polar conversion. The code can be adopted to other Hewlett Packard, Swiss Micros, and other RPN with four-stacks. RPL will need a short adjustment.

Input Stack:

T: y2

Z: x2

Y: y1

X: x1

Code:

LBL A	001	42, 21, 11	Program start
X<>Y	002	34
R↓	003	33
-	004	30
R↓	005	33
-	006	30
CHS	007	16
R↑	008	43, 33	Y: Δy, X: Δx
→P	009	43, 1	Rectangular to polar conversion; calculate distance
X<>Y	010	34
TAN	011	25	Calculate slope
X<>Y	012	34
RTN	013	43, 32	Program end

Reference formulas

Distance = √((x2^2 – x1^2) + (y2^2 – y1^2))

Slope = (y2 – y1) ÷ (x2 – x1) = tan(Θ)

Derivation:

Let y’ = y2 – y1 and x’ = x2 – x1

Then by rectangular to polar function, angle:

Θ = arctan( y’ / x’ )

tan Θ = y’ / x’

tan Θ = (y2 – y1) ÷ (x2 – x1)

Examples

Example 1: (-3, 8) to (11, 16)

Stack:

T: 16

Z: 11

Y: 8

X: -3

Result:

Y: slope ≈ 0.5714

X: distance ≈ 16.1245

Example 2: (5, 6) to (7, 9)

Stack:

T: 9

Z: 7

Y: 6

X: 5

Result:

Y: slope = 1.5000

X: distance ≈ 3.6056

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.