Numworks (Python): Determining Earth’s Acceleration of Gravity

Numworks (Python): Determining Earth’s Acceleration of Gravity

Introduction

Note: We will only be using SI units on this blog entry and program.

In general, the Earth’s acceleration constant, labeled with a lower case g, is conventionally defined to be 9.80665 m/s². Did you know that the gravitation acceleration on this planet actually varies (albeit slightly) depending on factors? For example, the gravity’s acceleration in Los Angeles, California, is about 9.796 m/s² (see Wikipedia article in the Source section).

On today’s blog, we will estimate the gravity of Earth based with two parameters: the latitude (North/South position on Earth’s grid) and elevation (in meters). This will give us a more accurate estimated on gravity, which in turn will give other physics calculations more accuracy.

For this program, I’m using a mix of formulas to estimate Earth’s gravitational acceleration:

Let Φ = latitude in radian measure:

Φ = (degrees + minutes ÷ 60 + seconds ÷ 3600) * π ÷ 180

For this calculation, consider the absolute value of the latitude. Gravity acceleration is the a given the northern latitude north and the corresponding southern latitude. (Example: gravity is about 9.79325 m/s² at 30° north and 30° south).

Gravity at the surface (0 m) is calculated as:

gs = (a * ga * (cos Φ)^2 + b * gb * (sin Φ)^2) ÷ √((a * cos(Φ))^2 + (b * sin(Φ))^2)

[Bilsin]

Where:

Φ = latitude in radians (see formula above)

a = 6378137 m (Earth’s radius at the equator)

ga = 9.70803253359 m/s² (gravity at the equator)

b = 6356752.3142 m (Earth’s radius at the poles)

gb = 9.8321849378 m/s² (gravity at the poles)

[Bilsin, Table 2]

Gravity at the selected elevation is estimated at:

ge = (100 * gs – 0.3086 * h ÷ 1000) ÷ 100

[Glover]

Where:

h = height in meters (m)

Note: The table on the Desk Ref list gravity as cm/s² and the original formula called for the height in kilometers (km).

To convert feet to meters, multiply by 0.3048.

Numworks Program: earthg.py

https://my.numworks.com/python/ews31415/earthg

Code:

from math import *

# Numworks, EWS, January 2026

# constants

# equator: radius, gravity

a=6378137

ga=9.7803253359

# poles: radius, gravity

b=6356721.3142

gb=9.8321849378

print("Latitude:","\nNorth or South")

d=eval(input("Decimal? "))

m=eval(input("Minutes? "))

s=eval(input("Seconds? "))

phi=radians(d+m/60+s/3600)

print("Height (m)?")

h=eval(input("? "))

gs=a*ga*cos(phi)**2+b*gb*sin(phi)**2

gs/=sqrt((a*cos(phi))**2+(b*sin(phi))**2)

print("Gravity on the surface:")

print(str(gs)," m/s**2")

print("At altitude: ",str(h)," m")

ge=(100*gs-0.3086*h/1000)/100

print(str(ge)," m/s**2")

Examples

Latitude	Elevation	g_surface	g_elevation
30°0’0”	0	9.793247191840559	9.793247191840559
40°0’0”	304.8	9.801696762579571	9.800756149779572
45°0’0”	762	9.806197665936962	9.803846133936961
12°5’13”	1219.2	9.782589588493437	9.778827137293437
34°26’44”	500	9.796866770553882	9.795323770553882

g_surface: gravity at surface level (0 m)

g_elevation: gravity at the selected elevation

Sources

Bislin, Walter. “Earth Gravity Calculator” September 1, 2018. https://walter.bislins.ch/bloge/index.asp?page=Earth+Gravity+Calculator Retrieved January 15, 2026.

Glover, Thomas J. and Richard A. Young Desk Ref Sequoia Publishing, Inc. Anchorage, AK. 4^th Edition, 2022. Soft Cover ISBN 978-1-885071-60-6. pg. 587

“Gravity of Earth” Wikipedia. Last edited January 31, 2026. https://en.wikipedia.org/wiki/Gravity_of_Earth Retrieved February 1, 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: Rafter Multipliers and Roof Area

TI-84 Plus CE: Rafter Multipliers and Roof Area

Introduction

Rafter multipliers are a quick way to calculate the area of a roof. All that is needed are three measurements:

(1) The width of a house

(2) The length of a house, including eaves and lengths of overhangs

(3) The pitch of the roof. The pitch the rise of the roof in inches over a run of 12 inches.

It is assumed that we have a simple roof and the area of the house is rectangular. Even in more complex cases, we can use the multipliers come up with a good approximation.

The area is approximated by:

roof area ≈ length * width * rafter factor

The common values are shown here:

Pitch (n in 12”)	Pitch (decimal – 3 places)	Rafter Factor (decimal -3 places)
2 in 12	0.167	1.014
3 in 12	0.250	1.031
4 in 12	0.333	1.054
5 in 12	0.417	1.083
6 in 12	0.500	1.118
7 in 12	0.583	1.158
8 in 12	0.667	1.202
9 in 12	0.750	1.250*
10 in 12	0.833	1.302
11 in 12	0.917	1.357
12 in 12	1.000	1.413
13 in 12	1.083	1.474
14 in 12	1.167	1.537
15 in 12	1.250	1.601
16 in 12	1.333	1.667
17 in 12	1.417	1.734
18 in 12	1.500	1.803
19 in 12	1.583	1.875
20 in 12	1.667	1.948
21 in 12	1.750	2.010
22 in 12	1.833	2.083
23 in 12	1.917	2.167

* This was erroneously labeled as 1.357 on the Dewalt Construction Math Quick Check guide (see Sources). Several sources correctly has the value of 1.250.

Curve Fitting

What if I can fit a formula to the above data?

I used a TI-84 Plus CE to assist me with this.

First, we’ll use the quadratic regression with the data from the above table:

y = a * x^2 + b * x + c

The resulting coefficients:

a = 0.1867080745

b = 0.2897396951

c = 0.938

r^2 = 0.9992015789

That’s a pretty good fit.

What about the square root fit:

y = √(a * x^2 + b)

If we square both sides, we get:

y^2 = a * x^2 + b

This equation takes somewhat the form of the linear regression.

Let z = x^2 and t = y^2, then:

z = a * t + b

Pitch (n in 12”)	X = Pitch (decimal – 3 places)	Z = X^2 (approximate)	Rafter Factor (decimal -3 places)	T = Y^2 (approximate)
2 in 12	0.167	0.0278	1.014	1.0282
3 in 12	0.250	0.0625	1.031	1.063
4 in 12	0.333	0.1111	1.054	1.1109
5 in 12	0.417	0.1736	1.083	1.1729
6 in 12	0.500	0.25	1.118	1.2499
7 in 12	0.583	0.3403	1.158	1.341
8 in 12	0.667	0.4444	1.202	1.4448
9 in 12	0.750	0.5625	1.250*	1.5625
10 in 12	0.833	0.6944	1.302	1.6952
11 in 12	0.917	0.8403	1.357	1.8414
12 in 12	1.000	1	1.413	1.9966
13 in 12	1.083	1.1736	1.474	2.1727
14 in 12	1.167	1.3611	1.537	2.3624
15 in 12	1.250	1.5625	1.601	2.5632
16 in 12	1.333	1.7778	1.667	2.7789
17 in 12	1.417	2.0069	1.734	3.0068
18 in 12	1.500	2.25	1.803	3.2508
19 in 12	1.583	2.5069	1.875	3.5156
20 in 12	1.667	2.7778	1.948	3.7947
21 in 12	1.750	3.0625	2.010	4.0401
22 in 12	1.833	3.3611	2.083	4.3389
23 in 12	1.917	3.6736	2.167	4.6959

Running the linear regression calculation with (z, t) we get:

a = 1.000120031

b = 1.00008392

r^2 = 0.9999332927

More accurate that the quadratic regression.

z = 1.000120031 * t + 1.00008392

y^2 = 1.000120031 * x^2 + 1.00008392

y = √(1.000120031 * x^2 + 1.00008392)

Analysis

I suspect that the exact formula for the rafter factor is:

rafter factor = √( pitch^2 + 1 ), pitch as a decimal

And the roof area is:

roof area = length * width * √( pitch^2 + 1 ) [ in inches ]

Roof area in square feet = Roof area in square inches / 144

Example

Area: length = 18 feet 6 inches, width = 20 feet, pitch = 7/12

In inches, length = 222 inches, width = 240 inches

roof area = 222 * 240 * √((7/12)^2 + 1) ≈ 61682.45131 in^2 ≈ 428.3503 ft^2

Sources

Prince, Christopher DeWALT Construction Math Quick Check: Extreme Duty Edition. Delmar, Cengage Learning: Cliffon Park, NY. 2010

Elliot. “Roof Area Calculator * Surface Area Multiplied by Pitch”. sktechandcalc.com. October 20, 2017. https://www.calculator.net/roofing-calculator.html?acarea=176&acareaunit=feet&roofpitch=4&angle=25&eaves=0&eavesunit=feet&price=&priceunit=feet&ctype=pitch&tp=ar&x=Calculate

calculator.net. “Roofing Calculator” 2008 – 2024. Retrieved November 4, 2024. https://www.calculator.net/roofing-calculator.html?acarea=176&acareaunit=feet&roofpitch=4&angle=25&eaves=0&eavesunit=feet&price=&priceunit=feet&ctype=pitch&tp=ar&x=Calculate

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.

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