Construction Master 5: Right Triangles

Construction Master 5:   Right Triangles

Introduction

The Calculated Industries Construction Master 5 does not have trigonometric keys, but it has four keys to dedicated to solving problems with right triangles, which appear a lot in construction.  Those four keys are:

[ Pitch ]:  there are several ways to enter the pitch:

If the amount is in inches, then the measurement is put as a pitch (x inches per 12 inches).  Example:  3 [ Inch ] [ Pitch]  inputs the angle as a 3 inch pitch (approximately 14.04°)

If the amount is in percent, the angle is entered as a percent grade.   Example: To enter 2% grade, press 2 [ % ] [ Pitch ] (approximately 1.15°)

If the amount is not attached to a unit, the amount is entered as degrees.  Example: To enter 30°, press 30 [ Pitch ].

[ Rise ]:  rise - opposite side.  Amounts can be entered with or without units.   

[ Run ]:  run - adjacent side.  Amounts can be entered with or without units.

[ Diag ]:  diagonal - hypotenuse.  Amounts can be entered with or without units.  Solving for the diagonal is known as a squaring-up calculation. 

To solve for a variable, just press the key without entering an amount. 

After all the variables have been calculated, you can calculate perimeter and area by:

Perimeter:  [ Rcl ] [ Rise ] [ + ] [ Rcl ] [ Run ] [ + ] [ Rcl ] [ Diag ]

Area:  [ Rcl ] [ Rise ] [ × ] [ Rcl ] [ Run ] [ ÷ ] 2

Examples 

Example 1:

Given:  Diagonal:  17.4 feet, Pitch:  20°

Results:
Rise:  5.95115 feet;  Run:  16.35065
Perimeter:  39.7018 feet;  Area:  48.65259 square feet

Example 2:

Given: Rise:  50 feet, Run:  40 feet

Results:
Diagonal:  64 feet 3/8 inch;  Pitch:  51.34°
Perimeter:  154 feet 3/8 inch feet;  Area:  1000 square feet

Example 3:

Given:  Grade: 3%, Feet: 17.5 Feet

Note, to enter the grade:  3 [ % ] [ Pitch ]

Results:
Run:  583.3333 ft, Diag:  583.5958 ft
Angle:  1.72°
Perimeter:  1184.429; Area: 5104.167 square feet

Eddie

All original content copyright, © 2011-2020.  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.

Using Pitch in Right Triangle Calculations

Using Pitch in Right Triangle Calculations

Using Pitch to Solve Right Triangle Lengths

In construction problems, we sometimes are working with roots and other structures that are shaped as right triangles.  In roofing applications, we are often working with the pitch of the roof.  The pitch is similar to the standard angle of right triangle.

The pitch is defined in 1 unit of rise over 12 units of run.  In the United States, the units are typically either feet or yards. 

pitch = 1 unit of rise / 12 units of run

With the right triangle, we can use similar triangles to determine that:

pitch /12 = rise / run

Knowing either one of the variables, we can use ratio calculations to determine the other.

Example 1:

Pitch: 3/12, Run:  48

Rise: 
12/48 = 3/x
x = 3 * 48 / 12
x = 12

Hypotenuse:
√(48^2 + 12^2) ≈ 49.4773

Example 2:

Pitch: 5/12, Rise: 30

Run:
5/12 = 30/x
x = 30 * 12 / 5
x = 72

Hypotenuse:
√(30^2 + 72^2) = 78

Approximating Angle with Pitch

To find the angle using pitch:

θ = atan(pitch / 12)

If you do not have a scientific calculator, you can approximate the angle by using any of the approximations (these are not the only approximation equations).   They were found using the curve fitting features of a Casio fx-9860gii.

θ ≈ 3.72674504 * p  + 2.78886014     (r^2 = 0.99084258)
θ ≈ 4.93070418 * p^0.91971679    (r^2 = 0.997149)
θ ≈ -0.1158911 * p^2 + 5.17538414 * p - 0.3498579   (r^2 = 0.9999772)

Data used:  θ to four decimal places are used. 

Eddie

All original content copyright, © 2011-2020.  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: Staircases

TI-84 Plus:  Staircases

  

Given the rise (height from lower floor to upper floor), run (length of the staircase), and desired riser height (how high each stair is), the program STAIRS calculates the number risers needed, along with the width of each stair, the incline, and finally illustrates the staircase. STAIRS is designed with inches and feet (US units) in mind. 

Formulas

Given:  Rise, Run, Desired Rise Height (DRH)

Number of Risers:

n = rise/DRH, rounded to the nearest integer

Tread width:

TW = run/(n – 1)

Adjusted Riser Height (ARH)

ARH = rise/n, rounded to the nearest 1/16^th

One way to approach this: 

ARH = round(16*frac(rise/n),0)*16 + int(rise/n)

Incline:

θ = atan(RH/TW)

Other Calculations:

Stringer:

S = (n – 1)*√(ARH^2 + TW^2)

Number of Stairs:

N_stairs = n - 1

TI-84 Plus Program: STAIRS

Input:  Rise, Run, Desired Riser Height.  Keep the units consistent.  (12 inches = 1 foot)

Output:  Number of Risers (R), Tread width of each stair (T), Adjusted Riser Height (H), Angle of Incline (θ)

* Adjusted Riser Height is rounded to the nearest 1/16^th (of an inch).  This is accomplished by the line iPart(H)+round(16*fPart(H),0)/16. 

The graph screen shows the staircase.  A stat plot shows where each stair ends with X (L1) representing the position and Y representing the height (L2). 

The program sets the TI-84 Plus to Degrees mode.

"EWS 2016-12-03"

Degree

Input "RISE:",B

Input "RUN:",A

Input "DESIRED RISER HEIGHT:",H

round(B/H,0)→N

A/(N-1)→T

B/N→H

"ROUND H TO 1/16"

iPart(H)+round(16*fPart(H),0)/16→H

tan^-1(H/T)→θ

√(H²+T²)*(N-1)→S

Disp "NUMBER OF RISERS:",N

Disp "TREAD WIDTH:",T

Pause

Disp "ADJ. RISER HEIGHT:",H

Disp "ANGLE:",θ

Pause

{0}→L1:{0}→L2

­.5→Xmin:A+.5→Xmax

­.5→Ymin:B+.5→Ymax

ClrDraw

For(I,1,N-1)

augment(L1,{I*T})→L1

augment(L2,{I*H})→L2

End

PlotsOff

PlotsOn 1

Plot1(xyLine,L1,L2)

Line(0,0,A,0)

Line(A,0,A,B)

For(I,0,N-1)

Line(T*I,H*I,T*I,H*(I+1))

Line(T*I,H*(I+1),T*(I+1),H*(I+1))

End

DispGraph

Examples

All amounts are in inches.

Example 1:  Rise = 35 in, Run = 84 in, Desired Riser Height = 7 in

Results:  Number of Risers: 5, Tread width: 21 in, Adjusted Riser Height:  7 in, θ ≈ 18.43495°

Example 1 is shown in the screen shots above.

Example 2:  Rise = 40 in, Run = 90 in, Desired Riser Height = 7 in

Results:  Number of Risers: 6, Tread width: 18 in, Adjusted Riser Height:  6.6875 in,

θ ≈ 20.38143°

Example 3:  Rise = 56 in, Run = 50 in, Desired Riser Height = 6.5 in

Results:  Number of Risers: 9, Tread width: 6.25 in, Adjusted Riser Height:  6.25 in,

θ ≈ 45°

This program was inspired by the Calculated Industries Construction Master 5 calculator

Eddie

This blog is property of Edward Shore, 2016