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 20S: The July 2026 Program Collection

HP 20S: The July 2026 Program Collection

Triangulation

d = l * sin(α) * sin(ß) ÷ sin(α + ß)

Store in the following registers before calculation:

R1 = measure of angle A

R2 = measure of angle B

R3 = length l from point A to B

Solve:

R4: distance

Code:

01: LBL A; 61, 41 ,A

02: DEG; 61, 23

03: RCL 1; 22, 1

04: SIN; 23

05: ×; 55

06: RCL 2; 22, 2

07: SIN; 23

08: ÷; 45

09: (; 33

10: RCL 1; 22, 1

11: +; 75

12: RCL 2; 22, 2

13: ); 34

14: SIN; 23

15: ×; 55

16: RCL 3; 22, 3

17: =; 74

18: STO 4; 21, 4

19: RTN; 61, 26

Examples

Example 1:

Inputs: R1 = 60°, R2 = 50°, R3 = 10

Output: R4: 7.05990377592

Example 2:

Input: R1 = 30°, R2 = 80°, R3 = 27.5

Output: R4: 14.4101446626

Source:

“Triangulation (surveying)” Wikipedia. https://en.wikipedia.org/wiki/Triangulation_(surveying) (last edited October 24, 2025). Retried March 17, 2026

Payment: Continuous Compounding

PMT = PV * (e^r – 1) ÷ (1 – e^(-r * t))

Store in the following registers before calculation:

R1 = t: number of payments

R2 = r: periodic interest rate (as a decimal)

R3 = PV: present value

Solve:

R4: PMT: present

Code:

01: GTO B; 61, 41, b

02: RCL 3; 22, 3

03: ×; 55

04: (; 33

05: RCL 2; 22, 2

06: e^x; 12

07: -; 65

08: 1; 1

09: ); 34

10: ÷; 45

11: (; 33

12: 1; 1

13: -; 65

14: (; 33

15: RCL 1; 22, 1

16: ×; 55

17: RCL 2; 22, 2

18: ); 34

19: +/-; 32

20: e^x; 12

21: ); 34

22: =; 74

23: STO 4; 21, 4

24: RTN; 61, 26

Examples

Example 1:

Input: t: 36, r: 0.10 ÷ 12, PV: 5,000.00

Output: PMT: 161.434037378

Example 2:

Input: t: 60, r: 0.05 ÷ 12; PV: 26,349.56

Output: PMT: 497.374636585

Electrical Engineering: System Temperature to Noise Figure

When an amplifier is activated, two ways to express noise are the noise temperature (in Kelvin) and the noise figure (in decibels, dB). Using a reference temperature of 290 K, when the noise temperature (T) is known, the noise figure (F) can be calculated by:

F = 10 * log((T + 290) ÷ 290)

If temperature is given in degrees Celsius (°C), then the formula for noise figure becomes:

F = 10 * log((T°C + 563.15) ÷ 290)

since T = T°C + 273.15.

The following program assumes that temperature is given in degrees Celsius.

Store in the following registers before calculation:

R1 = T°C; temperature in degrees Celsius

Solve:

R2: F: noise figure

Code:

01: LBL C; 61, 41, C

02: RCL 1; 21, 1

03: +; 75

04: 5; 5

05: 6; 6

06: 3; 3

07 . ; 73

08: 1; 1

09: 5; 5

10: =; 74

11: ÷; 45

12: 2; 2

13: 9; 9

14: 0; 0

15: =; 74

16: LOG; 51, 13

17: ×; 55

18: 1; 1

19: 0; 0

20: =; 74

21: STO 2; 21, 2

22: RTN; 61, 26

Examples

Example 1:

Input: T = -160 °C (store in R1)

Output: F: 1.43068666235 dB

Example 2:

Input: T = 15°C

Output: F: 2.99642532081 dB

Source:

Ball, John A. Algorithms for RPN Calculators John Wiley & Sons: New York. 1978. ISBN 0-47-03070-8. pp. 266-267

Moderate Exercise: Target Heart Rate Range

The following equations calculate the target heart range for a typical person engaging in moderate exercise. According to the particle by Jenna Fletcher (see Source), the American Heart Society (AHA) states the person is engaged in moderate exercise when their heart rate is 50% to 70% of their maximum heart rate. The maximum heart rate is 220 minus the person’s age.

Disclaimer: This is not medical advice, any questions should be discussed with a doctor or health professional.

The moderate range is determined by:

high = (220 – age) * 0.7 = low * 1.4

low = (220 – age) * 0.5 = high * 5/7

For high intensity, use the range 70% to 85%.

Code:

01: LBL D; 61, 41, d

02: +/-; 32

03: +; 75

04: 2; 2

05: 2; 2

06: 0; 0

07: =; 74

08: ÷; 2

09: 2; 2

10: =; 74

11: R/S; 26

12: ×; 55

13: 1; 1

14: . ; 73

15: 4 ; 4

16: =; 74

17: RTN; 61, 26

Examples

Example 1:

Input: Age 49 (my age at the time of this blog)

Output: low: 85.5, high: 119.7

Example 2:

Input: Age 25

Output: low: 97.5, high: 136.5

Source:

Fletcher, Jenna. “What a target heart rate is and how to calculate it”. Medically Reviewed by Debra Sullivan Ph. D. Medical News Today. January 22, 2024. https://www.medicalnewstoday.com/articles/target-heart-rate-calculator March 16, 2026.

Percentile in a Range

The program calculates the percentile of x in the range [a, b].

percentile = (x – a) ÷ (b – a) * 100%

Store before calculating:

R1 = a, R2 = b, R3 = x

Code:

01: LBL E; 61, 42, E

02: ( ; 33

03: RCL 3; 22, 3

04: - ; 65

05 RCL 1; 22, 1

06: ) ; 34

07: ÷; 45

08: ( ; 33

09: RCL 2; 22, 2

10: - ; 65

11: RCL 1; 22, 1

12: ) ; 34

13: ×; 55

14: 2; 2

15: 10^x; 51, 12

16: =; 74

17: RTN; 61, 26

Examples

Example 1:

Input: R1 = 10, R2 = 50, R3 = 30 (30 in [10, 50])

Output: 50 (50%)

Example 2:

Input: R1 = 85.5, R2 = 117.5, R3 = 110 (110 in [85.5, 117.5])

Output: 76.5625 (76.5625%)

Programming all five of these programs will fill the program space of the HP 20S entirely (99 steps)!

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.

Spotlight: AccuMath 400B Slide Rule

Spotlight: AccuMath 400B Slide Rule

Introduction

I purchased an AccuMath 400B slide rule at Antique Station, an antique store in Oro Grande just north of Victorville, CA.

I am impressed of the larger markings on the slide rule. The slide rule very easy to read, yet has a lot of scales for calculations including powers, roots, trigonometry, and logarithms.

The Scales

The slide rules have the following scales:

Top Frame (stationary):

S: Sine of angles in degrees. A = sin(S°) ÷ 100, S° = arcsin(A × 100)

K: Cube and cube root scale associated with scale D. K = D³, D = ³√K

A: Square and square root scale associated with scale D. A = D², D = √A

Slide:

B: Square and square root scale. Scale is from 1 to 100 and it is the same scale as Scale A.

CI: Reciprocal of scale C.

C: Multiplication and division scale from 1 to 10. Same as scale D.

Bottom Frame (stationary):

D: Multiplication and division scale from 1 to 10.

L: Logarithmic and exponential scale associated with D. L = log D, D = 10^L. (base 10 logs)

T: Tangent of angles in degrees. D = tan(T°) ÷ 10, T° = arctan(D × 10)

Note: A lot of slide rules associate the S scale with the C/D scales, but for this particular design, the A scale is used instead.

The Back Side

U.S./Metric Conversions, Equivalents, and Settings (i.e. 1 in mercury = 1.133 ft water)

Fractions up to 64ths and their decimal equivalents (i.e. 47/64 = 0.734375)

Trigonometric Identities, Right Triangles, Law of Sines and Cosines

Example Calculations

Keep your exponents in mind! N = mantissa * 10^exponent

Hairline: the plastic cursor you move around, Slide: the center piece of the slide rule you move around

4 × 2 = 8

Slide C right to match C: 1, D: 4

Move cursor right to C = 2

Read down to D: 8

3 × 9 = 27

Rearrange to 9 × 3. Slide C left to match C:1, D: 9

Move cursor left to C = 3

Read down to D: 2.7

Multiply by 10 since we slide C left: 2.7 × 10 = 27

12 × 33 = 396 (Approximate method)

Rearrange to 33 × 12. Slide B left to match: B: 1, A: 33

Move cursor left to B = 12

Read on A: the cursor is very close to 4.

Multiply by 100 since we moved B left. Result: approx 400.

54 ÷ 9 = 6

Move Slide B to match: A = 54, B = 9

Move cursor left to B = 1

Read A = 6

√81 = 9, 9^3 = 729 (approximate)

On A, slide cursor to 81.

Read on D: 9 (square root)

Read on K: in between 720 and 730. (so ≈725?)

Note that the scales are limited.

√144 = 12 (√(144 ÷ 100 × 100) = √1.44 × √100 = √1.44 × 10)

On A, slide cursor to 1.44.

Read on D: 1.2

Multiply by 10. Result: 12.

Logarithms: D and L scales. L = log D, D = 10^L

log 3 ≈ 0.47712

On D, slide cursor to 3.

Read on L: about 0.47

10^0.55 ≈ 3.54814

On L, slide cursor to 0.55

Read on D: about 3.54

Tangent of Angles in degrees: D and T scales. D = tan(T°) ÷ 10, T° = arctan(10 × D)

Tan(25°) ≈ 0.46631 (approximate)

On T, slide cursor to 25.

Read on D: about 4.65. Divide by 10 for a result of approximately 0.465

arctan(0.7) ≈ 34.99202 (approximate)

On D, slide to 7 (0.7 × 10)

Read on T: close to 35. Result: approximately 35°

Sine of Angles in degrees: A and S scales (for this model!) A = sin(S°)÷100, S° = arcsin(A×100)

sin(30°) = 0.5

On S, slide cursor to 30.

Read on A: 50. Divide by 100 for a final answer of 0.5

arcsin(0.2) ≈ 11.53700°

On A, slide to 20. (0.2 × 100)

Read on S: slightly after 11.5, for an approximation of 11.5°.

Reciprocal: CI and D scales. When the slide is at its home position, that is C = 1 and D = 1 are aligned:

1/CI = D

Find 1 / 4 = 0.25

Slide D to 4.

Read CI at 2.5 and divide by 10. Result: 0.25

Source

Instruction Manual for the AccuMath 400B.

Page 1: https://www.sliderule.ca/stermf.jpg

Page 2: https://www.sliderule.ca/stermb.jpg

Eric’s Slide Rule Page, last updated December 1, 2002. Accessed June 27, 2026.

Best,

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.