Basic vs. Python: Circle Inscribed in Circle [HP 71B, Casio fx-CG 100]

Basic vs. Python: Circle Inscribed in Circle

Calculators Used:

Basic: HP 71B

Python: Casio fx-CG 100

Introduction

Let a rectangle with sides A and B be inscribed in a circle. The circle has a radius R. Let the line segments with the length A have a corresponding angle Θ. If we are given R and Θ, we can use the chord length formula to calculate A and B.

The angle corresponding with side B can be determined as:

φ: angle corresponding with side B

φ + Θ + φ + Θ = 360°

2 × (φ + Θ) = 360°

φ + Θ = 180°

φ = 180° - Θ

Chord length:

A = 2 × R × sin(Θ/2)

B = 2 × R × sin(φ/2) = 2 × R × sin((180° - Θ)/2)

Knowing A and B, we can calculate the following:

Area of the rectangle = A × B

Area of the shaded area (circle – rectangle) = π × R^2 – A × B

Basic: HP 71B – INSCRECT

10 DESTROY R,A,B,C,D,T

15 DEGREES @ FIX 5

20 DISP “RECT. INSCR. CIRCLE” @ PAUSED

25 DISP “DEGREES MODE” @ WAIT 0.5

30 INPUT “RADIUS? “; R

35 INPUT “ANGLE-SIDE A? “; A

50 A = 2 * R * SIN(T/2)

55 B = 2 * R * SIN((180 – T)/2)

60 C = A * B

65 D = R^2 * PI – C

80 DISP “SIDE A = “, A @ PAUSE

85 DISP “SIDE B = “, B @ PAUSE

90 DISP “RECT ANGLE = “, C @ PAUSE

95 DISP “CIRC-RECT = “, D @ PAUSE

98 STD

Notes:

* FIX 5: Fix 5 display

* STD: Standard mode, floating point display

* DEGREES: Sets the HP 71B in degrees mode

Python: Casio fx-CG 100, insecrect.py

from math import *

print(“Rectangle Inscribed in a Circle”)

print(“Math module imported”)

r=eval(input(“Radius? “))

print(“Enter angle in degrees”)

t=eval(input(“Angle- A? “))

u=t*pi/180

a=2*r*sin(u/2)

b=2*r*sin((pi-u)/2)

c=a*b

d=r**2*pi-c

print(“A: {0.5f}”.format(a))

print(“B: {0.5f}”.format(b))

print(“Rect. Area: {0.5f}”.format(c))

print(“CIRC-AREA: {0:.5f}”.format(d))

Notes:

* Since the math module is used, this script can be used in any calculator with Python.

* Python’s angle mode is always in radians.

Examples

Example 1:

Inputs:

r = 10

Θ = 30°

Outputs:

a ≈ 5.17638

b ≈ 19.31852

c = 100

d ≈ 214.15927

Example 2:

Inputs:

r = 20

Θ = 80°

Outputs:

a ≈ 25.71150

b ≈ 30.64178

c ≈ 787.84620

d ≈ 468.79086

Example 3:

Inputs:

r = 20

Θ = 90°

Outputs:

a ≈ 35.35534

b ≈ 35.35534

c = 1250

d ≈ 713.49541

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 71B Programs: September 2025

HP 71B Programs: September 2025

One of my favorite calculators/pocket calculators of all time is the HP 71B.

ADDMOD: (a + b) mod n = a mod n + b mod n

10 DESTROY A, B, N

20 DISP “(A+B) MOD N” @ WAIT .5

30 INPUT “A? “; A

40 INPUT “B? “; B

50 INPUT “C? “; C

60 S = MOD(A, N) + MOD(B, N)

70 S = MOD(S, N)

80 DISP “SUM = “; S

MULTMOD: (a * b) mod n = a mod n * b mod n

10 DESTROY A, B, N

20 DISP “(A*B) MOD N” @ WAIT .5

30 INPUT “A? “; A

40 INPUT “B? “; B

50 INPUT “N? “; N

60 P = MOD(A, N) * MOD(B, N)

70 P = MOD(P, N)

80 DISP “PRODUCT = “; P

Examples:

A	630	48	15
B	320	99	47
N	700	15	7
SUM:	250	12	6
PRODUCT:	0	12	5

GABLE: Area of a Gable Roof

10 DESTROY P, S, L

20 DEGREES

30 INPUT “PITCH (IN.)? “; P

40 INPUT “SPAN (FT.)? “; S

50 INPUT “LENGTH (FT.)? “; L

60 A = 2 * S * L / COS(ATAN(P/12))

70 DISP “AREA = “; A; “ FT^2”

Input: Pitch (P)	6”	3”	4”
Input: Span (S)	110” (110/12 ft)	5’ 8” (5 + 8/12 ft)	15 ft
Input: Length (L)	100” (100/12 ft)	10’	10 ft
Output: Area (A)	170.81074828 ft^2	116.821326059 ft^2	316.227766017 ft^2

INTENSE: Light Intensity of a Spherical or a Cylindrical Light Source

intensity = power / surface area

intensity (W/m^2), power (W), surface area (m^2)

Surface area: sphere = 4 * π * r^2, cylinder = 2 * π * (r * h + r^2)

10 DESTROY P, K$, R, H

20 DISP “INTENSITY OF LIGHT!” @ WAIT .5

30 INPUT “POWER (W)? “; P

40 DISP “1. SPHERE 2. CYLINDER”

50 DELAY 0,0

60 K$ = KEY$

70 IF K$ = “1” OR K$ = “S” THEN GOTO 100

80 IF K$ = “2” OR K$ = “C” THEN GOTO 200

90 GOTO 40

100 INPUT “RADIUS (M)? “; R

110 A = 4 * PI * R^2

120 GOTO 300

200 INPUT “RADIUS (M)? “; R

210 INPUT “HEIGHT (M)? “; H

220 A = 2 * PI * (R * H + R^2)

230 GOTO 300

300 I = P / A

310 DISP I; “W/M^2”

Examples:

Sphere: r = 4 m, Power = 60 W: Intensity = .298415518297 W/M^2

Cylinder: r = 1.25 m, h = 0.75 m, Power = 70 W; Intensity = 4.45633840656 W/M^2

SIMPLE: Simple Interest – Banker’s Rule

interest = amount * rate% / 100 * #days / 360

final = principal + interest (final, maturity value)

10 DESTROY P, R, I, M

20 DISP “SIMPLE INTEREST” @ WAIT .5

30 INPUT “AMT? “; P

40 INPUT “RATE%? “; R

50 INPUT “# DAYS? “; D

60 I = P * R * D / 360 / 100

70 M = P + I

80 IMAGE K, M10D.2D

90 DISP USING 80; “INT.=”, I @ PAUSE

100 DISP USING 80; “FINAL=”, M

Notes:

* When the HP 71B is in pause, press [ f ] [ + ] to continue (CONT).

* To stop execution, press [ ON ] (ATTN).

* IMAGE K, M10D.2D: display “prompt string”, number rounded to 2 decimal places. The number is right-justified, make the prompt string 7 characters or less to fit both the string and number on the screen.

Examples

Input: Amount	1500	2000	2800
Input: Rate %	4.00%	8.00%	6.65%
Input: # Days	180	90	60
Output: Interest	30	40	31.03
Output: Final	1530	2040	2831.03

AIRSOUND: Speed of Sound in Air

S = 331.5 + .606 * T

S = speed of sound in air, m/s

T = temperature in °C

10 DESTROY K$, S, T

20 DISP “SPEED OF SOUND IN AIR” @ WAIT .5

30 DISP “SOLVE FOR…” @ WAIT .5

40 DISP “1. SPEED 2. TEMP”

60 K$ = KEY$

70 IF K$=”1” OR K$=”S” THEN GOTO 100

80 IF K$=”2” OR K$=”T” THEN GOTO 200

90 GOTO 40

100 INPUT “TEMP (°C)? “; T

110 S = 331.5 + .606 * T

120 DISP “S=”; S; “ M/S”

130 END

200 INPUT “SPEED (M/S)? “; S

210 T = (S- 331.5) / .606

220 DISP “T=”; T ; “°C”

230 END

Examples:

Input: T = 74 °F = (32 + 1/3) °C Output: S = 345.64 m/s	Input: S = 350 m/s Output: T = 30.5280528053 °C
Input: T = -8.5 °C Output: S = 326.349 m/s	Input: S = 382 m/s Output: T = 83.3333333333 °C

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 71B and HP Prime Python: Gaussian Quadrature

HP 71B and HP Prime Python: Gaussian Quadrature

A Very Brief Introduction

This is a top-level overview of the Gaussian Quadrature method. For more details, please check out the Sources section.

The design of GQ is to calculate:

∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)

Where do xi and wi come from?

* The xi values come from find the roots of the polynomials P’(x)= 0

* Pn’(x) are the derivatives of the Legendre polynomials Pn(x)

* After getting the roots, xi, the weights, w are calculated by the formula:

wi = 2 / ((1 -xi)^2 * Pn’(xi)^2)

Weights and Points of orders 3 and 5:

Among 3 points:

Roots (x) come from the derivative of the Legendre polynomial of the 3^rd order.

Pn’(x) = 3/2 * (5*x^2 – 1)

Point xi	Weight wi
-√(3/5) ≈ -0.7745 9669 2 = -√0.6	5/9
0	8/9
√(3/5) ≈ 0.7745 9669 2 = √0.6	5/9

Among 5 points:

Roots (x) come from the derivative of the Legendre polynomial of the 5^th order.

Pn’(x) = 15/8 * (21*x^4 – 14*x^2 + 1)

Point xi	Weight wi
-0.90617 98459	0.23692 68851
-0.53846 93101	0.47862 86705
0	128/225 ≈ 0.56888 88889
0.53846 93101	0.47862 86705
0.90617 98459	0.23692 68851

We can fit any interval [a, b] by this conversion:

∫( f(x) dx, a, b) = (b – a) / 2 * ∫( f( (b – a) / 2* x + (b + a) / 2 dx, -1, 1)

= (b – a) / 2 * Σ( wi * f( (b – a) / 2 * xi + (b + a) / 2 ), i = 1, n)

However, the programs on this blog entry will focus on the basic version of the Gaussian Quadrature.

The Code

The following code works with the following integrals:

∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)

HP 71B Basic

HP 71B: GQ3 (Gaussian Quadrature – 3 Point: Integrals from -1 to 1)

10 DEF FNF(X) = insert function of X here

15 S = 0

20 RADIANS

25 FOR I = 1 TO 3

30 READ X, W

31 DATA -SQR(0.6),5/9

32 DATA 0,8/9

33 DATA SQR(0.6),5/9

40 S = S + W * FNF(X)

55 NEXT I

50 DISP “ INTEGRAL= ”; S

HP 71B: GQ5 (Gaussian Quadrature – 5 Point: Integrals from -1 to 1)

10 DEF FNF(X) = insert function of X here

15 S = 0

20 RADIANS

25 FOR I = 1 TO 5

30 READ X, W

31 DATA -.9061798459, .2369268851

32 DATA -.5384693101, .4786286705

33 DATA 0, 128/255

34 DATA .5384693101, .4786286705

35 DATA .9061798459, .2369268851

40 S = S + W * FNF(X)

45 NEXT I

50 DISP “ INTEGRAL= ”; S

Integral Test: ∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)

Function f(x)	Actual (approx)	Gaussian 3 point (GQ3)	Gaussian 5 point (GQ5)
(x-3)*(x+4)*(x-5)	352/3 ≈ 117.333333333	117.333333333	117.333333334
-x^2 + 9*x + 10	58/3 ≈ 19.33333333	19.3333333334	19.3333333333
1.5 * cos(x)	2.524412954	2.52450532159	2.52441295561
exp(-x)	2.350402387	2.35033692868	2.35040238646
exp(sin(x))	2.283194521	2.28303911433	2.28319665353
sin(x)	0	0	0
1/(x - 2)	-1.09861228867	-1.09803921569	-1.09860924181

HP Prime Python App

HP Prime Python: gq3.py

# Gaussian Quadrature 3 point

from math import *

def f(x):

  # insert Function here

  return 1/(x-2)

s=0

x=[-sqrt(0.6),0,sqrt(0.6)]

w=[5/9,8/9,5/9]

for i in range(3):

  s=s+w[i]*f(x[i])

print("Integral: ",s)

HP Prime Python: gq5.py

# Gaussian Quadrature 5 point

from math import *

def f(x):

  # insert Function here

  return sin(x)

s=0

x=[0,-.5384693101056831,.5384693101056831,-.9061798459386640, .9061798459386640]

w=[.5688888888889,.4786286704993665,.4786286704993665,.2369268850561891,.2369268850561891]

for i in range(5):

  s=s+w[i]*f(x[i])

print("Integral: ",s)

Function f(x)	Actual (approx)	Gaussian 3 point (GQ3)	Gaussian 5 point (GQ5)
(x-3)*(x+4)*(x-5)	352/3 ≈ 117.333333333	117.333333333	117.333333333
-x^2 + 9*x + 10	58/3 ≈ 19.33333333	19.3333333333	19.3333333334
1.5 * cos(x)	2.524412954	2.52450532159038	2.52441295561081
exp(-x)	2.350402387	2.35033692868001	2.35040238646284
exp(sin(x))	2.283194521	2.2830391143268	2.28319665352905
sin(x)	0	0.0	0.0
1/(x - 2)	-1.09861228867	-1.09803921568627	-1.09860924181248

Sources

Kamermans, Mike “Pomax”. “Gaussian Quadrature Weights and Abscissae” https://pomax.github.io/bezierinfo/legendre-gauss.html 2011. Retrieved January 19, 2025.

Wikipedia. “Gauss-Legendre quadrature” https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature Last edited January 19, 2025. Retrieved January 19, 2025.

Wikipedia. “Legendre Polynomials” https://en.wikipedia.org/wiki/Legendre_polynomials Last edited December 4, 2024. Retrieved January 19, 2025.

When it comes to evaluating integrals numerically, is Gaussian Quadrature better than Simpson’s Method?

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 content on this blog is 100% generated by humans. The author does not use AI engines and never will.