Construction Pro 5: Geometry Algorithms

Construction Pro 5:  Geometry Algorithms

Before We Begin

Before we begin, some things to note:

1.  The [ Circ ] [ Circ ] sequence gives the area of a circle given by the denominator.  Area = (π * diameter^2) / 4 

2.  The [ Conv ] [ Rcl ] will clear the Construction Master 5's memory register.  So will [ Rcl ] [ Rcl ], the only difference is that the former sequence will not recall the memory's contents.

3.  The [ Rcl ] [ M+ ] sequence will recall the memory register's contents.

4.  The Construction Master 5 operates in Chain mode, like standard four-function calculators.  

5.  The algorithms presented today is one way to approach these calculation, most of them demonstrate the [ Circ ] [ Circ ] and memory features.   

Area:  Donut Driveway

Area = ((2D + I)^2 - I^2) * π/4

Key Sequence:

[ Conv ] [ Rcl ]

2 [ x ] D [ + ] I [ = ] [ Circ ] [ Circ ] [ M+ ]

I [ Circ ] [ Circ ] [ Conv ] ( M- )

[ Rcl ] [ M+ ]

Example:

D = 50 feet, I = 10 feet

[ Conv ] [ Rcl ]

2 [ x ] 50 [ Feet ] [ + ] 10 [ Feet ] [ = ] [ Circ ] [ Circ ] [ M+ ]

10 [ Feet ] [ Circ ] [ Circ ] [ Conv ] ( M- )

[ Rcl ] [ M+ ]

Result:  9,424.778 ft^2

Volume:  One-Hole Concrete Block

Note:  The border length (d) is equal around the entire block.

V = ( ( W + L ) * 2 * d - 4 * d^2 ) * t

Key Sequence:

[ Conv ] [ Rcl ]

W [ + ] L [ x ] 2 [ x ] d [ M+ ]

d [ Conv ] ( x^2 ) [ x ] 4 [ M- ]

[ Rcl ] [ M+ ] [ x ] t [ = ]

Example:

L = 12 in, W = 8 in, d = 1 in, t  10 in

[ Conv ] [ Rcl ]

8 [ Inch ] [ + ] 12 [ Inch ] [ x ] 2 [ x ] 1 [ Inch ] [ M+ ]

1 [ Inch ] [ Conv ] ( x^2 ) [ x ] 4 [ M- ]

[ Rcl ] [ M+ ] [ x ] 10 [ Inch ] [ = ]

Result:  360 in^3

Volume: Right Triangular Prism

V = D * H * B / 2 

Key Sequence:

D [ x ] H [ x ] B [ ÷ ] 2 [ = ]

Example:

D = 325 ft, H = 77 ft, B = 148 ft

325 [ Feet ] [ x ] 77 [ Feet ] [ x ] 148 [ Feet ] [ ÷ ] 2 [ = ]

Result:  1,851,850 ft^3 ≈ 68,587.04 yd^3

Volume:  Sphere Using the Circ Function

V = 4/3 * π * r^3 = π * d^3 / 6 = area_circle * d / 1.5

where area_circle = π * d^2 / 4

Key Sequence:

[ Conv ] [ Rcl ]

D [ M+ ] [ = ] [ Circ ] [ Circ ] [ x ] [ Rcl ] [ M+ ] [ ÷ ] 1.5 [ = ]

Note:  The first equals key "locks" in the value of D on to the register and allows it to be picked up with the Circ function without having to re-type it.  

Example:

D = 5 ft

[ Conv ] [ Rcl ]

5 [ Feet ] [ = ] [ M+ ] [ Circ ] [ Circ ] [ x ] [ Rcl ] [ M+ ] [ ÷ ] 1.5 [ = ]

Result: 65.44985 ft^3

Volume:  Column Using the Circ Function

V = π * D^2 * H / 4 = area_circle * H

where area_circle = π * d^2 / 4

Key Sequence:

D [ Circ ] [ Circ ] [ x ] H [ = ]

Example:

D = 2 ft 2 in, H = 1 ft 8 in

2 [ Feet ] 2 [ Inch ] [ Circ ] [ Circ ] [ x ] 1 [ Feet ] 8 [ Inch ] [ = ]

Result:  6.145013 ft^3

Commas added for readability.  

Eddie

All original content copyright, © 2011-2021.  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. 

Review: Calculated Industries ConcreteCalc Pro (4225)

Review:  Calculated Industries ConcreteCalc Pro (4225)

Quick Facts

Model Number:  4225

Company:  Calculated Industries

Year of Production:  2010 - present

Operating System:  Chain

Batteries: 1 x CR 2016

Memory Register:  1 independent (M), memories M1 - M3

Paperless Tape:  20 steps, arithmetic functions are recorded

Cost:  Retail:  $74.95, but selling prices vary.  I purchased mine used for $30 on eBay.

Product Page:

https://www.calculated.com/mobile/prd104/ConcreteCalc-Pro-4225-Concrete-Calculator.html

Consistent my recent reviews of Calculated Industries calculators, including the Pipe Trades Pro and the ElectricCalc Pro, I am impressed by the keyboard and the feel of the keys, love the Armadillo Gear case cover, and the calculator serves specific purpose and industry.  

Highlighted Features

Here are some, just some, of the features of ConcreteCalc Pro:  

Dimensional Math:  Calculations and conversions with units, both US and Metric, are available on the ConcreteCalc Pro.

Areas and Volumes:   The [ Length ], [ Width ], and [ Height ] keys serve as the entry keys to calculate area and volume.  

VOL = volume of the room

WALL = area of the one of the four walls

ROOM = total surface area of the walls

The results are accessed through repeated presses of the [ Height ] key.  

Geometry:  The ConcreteCalc Pro also works with calculating areas and circumferences of circles, volumes of cylinders and cones, and arc lengths.  Regular polygons are also included.  

Example:  6-sided polygon with radius of 5 in.  

5 [ Inch ] [ Conv ] [ Arc ] (Radius)

6 [ Conv ] [ Length ] (Polygon)

FULL:  120.00°

HALF:  60.00°

SIDE:  6

PER:  30 in (perimeter)

AREA:  64.95191 in^2  (area)

Loads, Blocks, Bags:  Calculate loads and bags of concrete based on volume, and the number of blocks based on area and the size of the blocks.

Stairs:  Use the [ Height ] key to store the total floor-to-floor rise, desired riser height, floor height, and tread width and you can get the run, incline of the stairs, the length of the stringer and more with the [ Stair ] key.  With the Riser Limited option, we can calculate these statistics restricting the desired the riser height as the maximum.  

Rebar calculations 

Eddie

All original content copyright, © 2011-2021.  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. 

Casio fx-CG 50 and HP Prime: Pyramid Constructed by Points

Casio fx-CG 50 and HP Prime:  Pyramid Constructed by Points

The program PYRAPTS calculates the surface area and volume of a pyramid that is defined by its vertex points.  The program assumes all the points of its base lies on the plane z = 0, and there is only one tip point.

All of the base points, B1, B2, B3, B4, etc., are entered as two coordinates, one list for the X coordinates and the other for the corresponding Y coordinates.  Keep in mind that for this program, all of its base points lie on the plane Z=0.

The coordinates for the tip point, T, are prompted separately. 

Casio fx-CG 50 Program PYRAPTS (text file)

'ProgramMode:RUN

"2018-06-24 EWS"

ClrText

Red Locate 1,1,"FOR THE BASE, Z=0"

For 1->J To 1000

Next

"X _List _"?->List 1

"Y _List _"?->List 2

Dim List 1->N

"TIP X COORD="?->X

"TIP Y COORD="?->Y

"TIP Z COORD="?->Z

0->S

List 1[1]*List 2[N]-List 1[N]*List 2[1]->R

For 2->J To N

R+List 1[J]*List 2[J-1]-List 1[J-1]*List 2[J]->R

Sqrt(Z^<2>+(Y-List 2[J])^<2>+(X-List 1[J])^<2>)->A

Sqrt(Z^<2>+(Y-List 2[J-1])^<2>+(X-List 1[J-1])^<2>)->B

Sqrt((List 2[J]-List 2[J-1])^<2>+(List 1[J]-List 1[J-1])^<2>)->C

(A+B+C)/2->H

S+Sqrt(H(H-A)(H-B)(H-C))->S

Next

Abs R/2->R

S+R->S

RZ/3->V

ClrText

Blue Locate 1,1,"SURFACE AREA="

Red Locate 1,2,S

Blue Locate 1,4,"VOLUME="

Red Locate 1,5,V

Notes:

1.  - >  is the storage arrow →

2.  ^<2> is the square key x^2

3.  If you type in the program manually, the line ‘ProgramMode:RUN is not needed

4.  _ represents the space key

HP Prime Program PYRAPTS

EXPORT PYRAPTS()

BEGIN

// 2018-06-25 EWS

// pyramid by points

LOCAL L1,L2,X,Y,Z;

LOCAL S,V,R,J,N;

INPUT({{L1,[[6]]},

{L2,[[6]]}},"Base Points: Z=0",

{"X List:","Y List:"});

N:=SIZE(L1);

INPUT({X,Y,Z},"Tip Coordinates",

{"X: ","Y: ","Z: "});

S:=0;

R:=L1(1)*L2(N)-L1(N)*L2(1);

FOR J FROM 2 TO N DO

R:=R+L1(J)*L2(J-1)-L1(J-1)*L2(J);

A:=√(Z^2+(Y-L2(J))^2+

(X-L1(J))^2);

B:=√(Z^2+(Y-L2(J-1))^2

+(X-L1(J-1))^2);

C:=√((L2(J)-L2(J-1))^2

+(L1(J)-L1(J-1))^2);

H:=(A+B+C)/2;

S:=S+√(H*(H-A)*(H-B)*(H-C));

END;

R:=ABS(R)/2;

S:=S+R;

V:=R*Z/3;

PRINT();

PRINT("Surface Area: "+S);

PRINT("Volume: "+V);

RETURN {S,V};

END;

Example

Base points:  (0,0,0),  (3,4,0),  (-3,4,0)  (z=0 is implied for base points)

Tip point:  (0,1,√2)

Input:

List X:  {0, 3, -3}

List Y:  {0, 4, 4}

Tip X: 0

Tip Y: 1

Tip Z: √2

Results:

Surface Area: 25.7904472451

Volume:  5.65685424947

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

HP Prime: Volume and Surface Area of Platonic Solids

HP Prime:  Volume and Surface Area of Platonic Solids

About Platonic Solids

A Platonic solid is one of five three dimensional objects consisting of regular polygons which all faces are identical and perfect symmetry is achieved at each vertex (corner).  The five Platonic solids are classified by the number of faces they have:  tetrahedron (4), hexahedron (aka cube) (6), octahedron (8), dodecahedron (12), and icosahedron (20).

Aside from being well known geometric shapes, Platonic solids were considered sacred and thought to play a significant role in cosmology.  In the 360 BC dialogue Timaeus, Plato considered right triangles to be sub-atomic particles, which form the Platonic Solids.  Each of the Platonic solids represents an element:  tetrahedron represents fire, hexahedron represents earth, octahedron represents air, dodecahedron represents ether, and icosahedron represents water.  Those elements help build all of the universe.   Plato also believed that the elements are interchangeable, which particles split up into triangles and rearranging themselves.  [3]

In 1596, Johannes Kepler wrote Mysterium Cosmographicum (The Cosmographic Mystery).  Based on the Copernican system (which considered our Sun as the center of the Universe), Kepler attributed to structure of Solar System with Platonic Solids.  Each planet had its own corresponding sphere where its orbit was located.  Each of the Platonic solids were placed so that they were inscribed and circumscribed by the spheres.  The order went like this:  Mercury, Octahedron, Venus, Icosahedron, Earth, Dodecahedron, Mars, Tetrahedron, Jupiter, Hexahedron, Saturn.  (Uranus and Neptune were not discovered at this time). [4]

Below is some basic geometric and some eccentric information for the Platonic solids:

Platonic Solid	# of Faces	# of Vertices	# of Edges	Volume	Surface Area
Tetrahedron	4	4	6	A^3 * √2 / 12	A^2 * √3
Hexahedron	6	8	12	A^3	A^2 * 6
Octahedron	8	6	12	A^3 * √2 / 3	A^2 * 2 * √3
Dodecahedron	12	20	30	A^3 * (15 + 7 * √5)	A^2 * (3 * √( 20 + 10 *√5))
Icosahedron	20	12	30	A^3 * 5 * (3 + √5) /12	A^2 * 5 * √3

(A = length of a side) 

Platonic Solid	Internal Angle	Element [1]	Philosophy [1]	Chakra [2]	Duals [3]
Tetrahedron	90°	Fire	Balance, Stability	3rd	(none)
Hexahedron	120°	Earth	Earth, Nature	1st	Octahedron
Octahedron	135°	Air	Love, Compassion	4th	Hexahedron
Dodecahedron	150°	Universe/Ether	Spirit, Heavens	5th, 6th, 7th	Icosahedron
Icosahedron	162°	Water	Expression, Creativity	2nd	Dodecahedron

Below are programs to calculate the volume and surface area of each of the Platonic solids

Prime Programs – Platonic Solids

Tetrahedron

Volume:

EXPORT VOLTET(A)

BEGIN

√2*A^3/12;

END;

Surface Area:

EXPORT SURTET(A)

BEGIN

A^2*√3;

END;

Hexahedron:

Volume:

EXPORT VOLHEX(A)

BEGIN

A^3;

END;

Surface Area:

EXPORT SURHEX(A)

BEGIN

6*A^2;

END;

Octahedron:

Volume:

EXPORT VOLOCT(A)

BEGIN

A^3*√2/3;

END;

Surface Area:

EXPORT SUROCT(A)

BEGIN

2*√3*A^2;

END;

Dodecahedron:

Volume:

EXPORT VOLDOD(A)

BEGIN

(15+7*√5)*A^3/4;

END;

Surface Area:

EXPORT SURDOD(A)

BEGIN

(3*√(20+10*√5))*A^2;

END;

Icosahedron

Volume:

EXPORT VOLICO(A)

BEGIN

5*(3+√5)*A^3/12;

END;

Surface Area:

EXPORT SURICO(A)

BEGIN

5*√3*A^2;

END;

Sources

[1]  Punctured Artefact “Symbolism.  The Platonic Solids”  October 13, 2013.  Retrieved January 22, 2017.  Link:  https://puncturedartefact.wordpress.com/2013/10/13/symbolism-the-platonic-solids-tattoo-design-and-culture/

[2] Patinkas.  “The Merkaba, Platonic Solids, & Sacred Geometry” 2014.  Retrieved January 22, 2017.  Link:  http://www.patinkas.co.uk/Merkaba_Feature_Article/merkaba_feature_article.html

[3] Mathpages.  “Platonic Solids and Plato’s Theory of Everything”  Retrieved January 20, 2017.  Link:  http://www.mathpages.com/home/kmath096/kmath096.htm

[4] Wikiepdia.  “Mysterium Cosmographicum”  Retrieved Janaury 23, 2017.  Link:  https://en.wikipedia.org/wiki/Mysterium_Cosmographicum#Shapes_and_the_planets

The first month of 2017 is almost in the books.  Until next time,

Eddie

This blog is property of Edward Shore, 2017.

Geometric Relationships: Circle, Sphere, and Equilateral Triangle

Geometric Relationships: Circle, Sphere, and Equilateral Triangle

Circle:  Relationship between Area and Circumference

We know that π is a constant (π ≈ 355/113, but more accurately, π ≈ 3.141592654).  And:

Circumference of a Circle:  C  = 2 * π * r

Area of a Circle: A = π * r^2

Observe that:

C = 2 * π * r

π = C / (2 * r)

And:

A = π * r^2

π = A / r^2

Hence:

C / (2 * r) = A / r^2

(2 * r) / C = r^2 / A

A * 2 * r = r^2 *C

A = C * r / 2

Sphere:  Relationship between Area and Circumference

Volume of a Sphere:  V = 4/3 * r^3 * π

Surface Area of a Sphere:  S = 4 * π * r^2

Solving for π:

V = 4/3 * r^3 * π

3 * V = 4 * r^3 * π

π = (3 * V) / (4 * r^3)

And:

S = 4 * π * r^2

π = S / (4 * r^2)

Then:

(3 * V) / (4 * r^3) = S / (4 * r^2)

Multiply both sides by 4 * r^2:

S = 3 * V / r

Equilateral Triangle:  Relationship between Perimeter and Area

Let a (small a) be the length’s side.   Then the area of the triangle:

A = 2 * (1/2 * a/2 * √3/2 * a) = a^2 * √3/4

With the perimeter:  P = 3 * a,

P = 3 * a

P^2 = 9 * a^2

a^2 = P^2 / 9

And

A = a^2 * √3 / 4

a^2 = 4 * A / √3

P^2 / 9 = 4 * A / √3

A = P^2 * √3 / 36

To summarize:

Circle:  Area and Circumference:  A = C * r / 2

Sphere:   Volume and Surface Area:  S = 3 * V / r

Equilateral Triangle:  Area and Perimeter:  A = P^2 * √3 / 36

The next blog will cover Platonic solids.  At least that’s the plan.  Have a great rest of the weekend.  

Eddie

This blog is property of Edward Shore, 2017.

Roman Arch – Area and Volume Derivation

It’s good to be back. 

This is your basic Roman Arch, with the top part having an elliptical shape, which sits on two pillars.

Dimensions:

S = length of the track

H = height of the pillars

D = depth of the arch, usually small

W = gap between the pillars

R = gap between the bottom of the curve to the point where the arch curves

Surface Area:

A = Total Surface Area

AT = Area of the Elliptical ring (top)

A0 = Outer Elliptical Area

AI = Inner Elliptical Area

AP = Pillars Area (2 pillars)

Area of the Elliptical Ring (top)

AT = AO – AI

AT = π*(S + W/2)*(S + R) – π * W/2 * R

AT = π*((S + W/2)*(S + R) – W/2 * R)

AT = π*(S^2 + R*S + S*W/2)

Area of the Pillars (there are two of them):

AP = 2*H*S

Total Surface Area:

A = AT + AP

A = π*(S^2 + R*S + S*W/2) + 2*H*S

Volume of the Roman Arch:

V = D * A

V = D * (π*(S^2 + R*S + S*W/2) + 2*H*S)

Eddie

This blog is property of Edward Shore.