TI 30Xa Algorithms: Fundamental Horizontal Circle Calculations
Introduction and Formulas
Given the following:
r = radius of the curve
Δ = central angle, or the angular sweep of the curve. The angle is usually given in whole degrees or in surveying, degrees-minutes-seconds.
We can calculate the following:
The length of the tangent line to the central meeting point:
t = r × tan( Δ / 2 )
The linear distance from where the curve begins to the point the curve ends, known as the chord distance:
c = 2 × r × sin( Δ / 2 )
The distance of the curve from the end points, known as the arc length:
s = (Δ° / 180°) × π × r
Note the formula for arc length assumes that the angle is in degrees. If the angle is in radians, the formula for arc length reduces to s = Δ rad × r.
Procedure
The procedure is made for the TI-30Xa, where we can take advantage of the three memory registers it has. You can use another calculator, except most basic scientific calculators only have one memory register; you will just have type in one of the extra variables.
1. Check the angle mode. We should be in degrees mode, which is indicated by the DEG indicator. If not, press the [ DRG ] key until degrees mode is selected.
2. Enter the central degree (Δ).
If the degree is in whole degrees, just enter the angle and store into memory register 1. ( [ STO ] [ 1 ] ).
If the degree is in degrees-minutes-seconds format (known as DMS or sometimes as HMS), enter the angle in DD.MMSSS format. Then press [ 2nd ] [ + ] {DMS>DD} to convert the angle into whole degrees. Then store the result into memory register 1.
3. Enter the radius length and store in memory register 2. ( [ STO ] [ 2 ] ).
Memory registers:
M1 = Δ; M2 = r
4. Calculate the tangent length, t:
[ ( ] [ RCL ] 1 [ ÷ ] 2 [ ) ] [ TAN ] [ × ] [ RCL ] 2 [ = ]
5. Calculate the chord distance, c:
2 [ × ] [ RCL ] 2 [ × ] [ ( ] [ RCL ] 1 [ ÷ ] 2 [ ) ] [ SIN ] [ = ]
6. Calculate the arc length, s:
[ RCL ] 1 [ × ] [ π ] [ × ] [ RCL ] 2 [ ÷ ] 180 [ = ]
Examples
Let’s assume for the following examples Degrees mode is set (DEG).
Example 1:
r = 150 ft (it can be any length unit as long as you keep it consistent)
Δ = 60° (whole degrees)
60 [ STO ] 1
150 [ STO ] 2
t: [ ( ] [ RCL ] 1 [ ÷ ] 2 [ ) ] [ TAN ] [ × ] [ RCL ] 2 [ = ]
t = 86.60254038 ft
c: 2 [ × ] [ RCL ] 2 [ × ] [ ( ] [ RCL ] 1 [ ÷ ] 2 [ ) ] [ SIN ] [ = ]
c = 150 ft
(In this case, the chord length is the same as the radius. Why? There is a geometric reason.)
s: [ RCL ] 1 [ × ] [ π ] [ × ] [ RCL ] 2 [ ÷ ] 180 [ = ]
s = 157.0796327 ft
Example 2:
r = 324 ft
Δ = 92°22’18” (92 degrees, 22 minutes, 18 seconds)
92.2218 [ 2nd ] [ + ] {DMS>DD} [ STO ] 1
324 [ STO ] 2
t: [ ( ] [ RCL ] 1 [ ÷ ] 2 [ ) ] [ TAN ] [ × ] [ RCL ] 2 [ = ]
t = 337.6968953 ft
c: 2 [ × ] [ RCL ] 2 [ × ] [ ( ] [ RCL ] 1 [ ÷ ] 2 [ ) ] [ SIN ] [ = ]
c = 467.5897175 ft
s: [ RCL ] 1 [ × ] [ π ] [ × ] [ RCL ] 2 [ ÷ ] 180 [ = ]
s = 522.3494689 ft
Source
Hewlett-Packard Company. HP-46 sample applications. Loveland, CO. February 1,1975. Part No. 00046-90018. pg. 104-105
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