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.

HP 32SII: Volume of a Square Frustum (Three Approaches in RPN)

HP 32SII: Volume of a Square Frustum (Three Approaches in RPN)

The volume of a square frustum is:

v = height ÷ 3 × (s1^2 + s2^2 + √(s1^2 × s2^2))

which is based off the formula for the volume of general frustum:

v = height ÷ 3 × (base_area_1 + base_area_2 + √(base_area_1 × base_area_2))

Here are three approaches, which were all done with a HP 32SII calculator. The beauty of RPN is on display here. The three programs are labeled A, C, and D.

s1, s2: side of the square base h: height

Stack: Z: s1 Y: s2 X: h

Program A: (24 steps, 36 bytes) 

V = h/3 * (s1^2 + s2^2 + √(s1^2 * s2^2))

LBL A

STO H

R↓

x²

x<>y

x²

ENTER

ENTER

CLx

-

R↓

x<>y

R↓

×

√

R↓

+

R↑

+

RCL H

×

3

÷

RTN

Program C: (17 steps, 25.5 bytes)

Note: √(s1^2 * s2^2) = s1 * s2

V = h/3 * (s1^2 + s2^2 + (s1 * s2))

LBL C

ENTER

3

÷

R↓

×

LST x

x<>y

R↓

x²

x<>y

x²

+

R↑

+

×

RTN

Program D: (15 steps, 22.5 bytes) 

Note: s1^2 + s2^2 + s1 * s2 = s1^2 + s2^2 + 2 * s1 * s2 – s1 *s2 = (s1 + s2)^2 - s1*s2 

V = h/3 * ((s1 + s2)^2 - s1 * s2)

LBL D

3

÷

R↓

×

LST x

x<>y

R↓

+

x²

R↑

-

×

RTN

Examples

Example 1:

s1: 24

s2: 30

h: 5.8

Result: 4245.6

Example 2:

s1: 2.54

s2: 12.15

h: 10

Result: ≈ 616.4503

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.

Numworks (Python): Parallelograms Described by Vectors

Numworks (Python): Parallelograms Described by Vectors

Introduction

The script drawpgram.py draws a parallelogram constructed by a pair of two-dimensional vectors. Both vectors original from the origin of the Cartesian plane ([0, 0]). Let the two vectors be labeled as [x1, y1] and [x2, y2].

This script also calculates the area and perimeter of the constructed parallelogram consisting of the end points (0,0), (x1, y1), (x2, y2), and (x1+x2, y1+y2).

Then the perimeter of the parallelogram is calculated as:

perimeter = 2 * (norm([x1, y1] + norm[x2, y2]) = 2 * (√(x1 + y1) * √(x2, y2))

And the area is calculated as:

area = abs(det([[x1, y1] [x2, y2]])) = abs(x1 * y2 – x2 * y1)

Numworks Script: drawpgram.py

Modules used: Math, PyPlot

Script page: https://my.numworks.com/python/ews31415/drawpgram

# Draw a parallelogram with two vectors

# Edward Shore, 2/15/2026

# Numworks

# drawpgram.py

from math import *

from matplotlib.pyplot import *

print("Draw a parallelogram with 2 vectors")

x1=eval(input("x1? "))

y1=eval(input("y1? "))

x2=eval(input("x2? "))

y2=eval(input("y2? "))

# Area

area=abs(x1*y2-x2*y1)

# Perimeter

perim=2*((x1**2+y1**2)**(1/2)+(x2**2+y2**2)**(1/2))

minx=min([x1,x2,x1+x2,0])

maxx=max([x1,x2,x1+x2,0])

miny=min([y1,y2,y1+y2,0])

maxy=max([y1,y2,y1+y2,0])

axis((minx-1,maxx+1,miny-1,maxy+5))

plot([0,x1],[0,y1],'blue')

plot([0,x2],[0,y2],'purple')

plot([x1,x1+x2],[y1,y1+y2],'purple')

plot([x2,x1+x2],[y2,y1+y2],'blue')

str1="area = {0:.6f}".format(area)

text(minx-1,maxy+4.5,str1)

str2="perim = {0:.6f}".format(perim)

text((minx+maxx)/2,maxy+4.5,str2)

show()

Notes:

1. I use eval(input( “prompt” )) and importing the Math module to allow for expressions to be entered, such as expressions involving pi (π). If I use float(input( “prompt” )) instead, only numbers would be allowed.

2. For the purposes of display, the results are rounded off to six decimal places ({0:.6f}) and displayed in a line. The full precision results of area and perimeter are stored in the variables area and perim, respectively.

Examples

Example 1: [2, 4], [5, 2]

Example 2: [-3, 1], [5, 3]

Example 3: [-6, 0], [0, -4]

Source

Margalit, Dan and Joseph Rabinoff. 2025. “Determinants and Volumes” Interactive Linear Algebra LibreTexts. Accessed February 8, 2026. https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/04%3A_Determinants/4.03%3A_Determinants_and_Volumes

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.

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.