Calculating the Radius of a Graduated Circle

  • Context: MHB 
  • Thread starter Thread starter Ragnarok7
  • Start date Start date
  • Tags Tags
    Circle
Click For Summary
SUMMARY

The discussion centers on calculating the radius of a graduated circle using the arc-length formula, specifically for a problem from Loney's Trigonometry (1895). The key parameters are the division value of 5' (5 minutes of arc) and the distance between graduations of 0.1 inches. To find the radius, the formula \(s = r\theta\) is applied, where \(s\) is the arc length and \(\theta\) is the angle in radians. The confusion regarding the measurement of 5' being interpreted as either inches or seconds is clarified, confirming it refers to minutes of arc.

PREREQUISITES
  • Understanding of arc-length formula \(s = r\theta\)
  • Knowledge of angular measurements, specifically minutes of arc
  • Familiarity with basic trigonometry concepts
  • Experience with surveying instruments and graduated circles
NEXT STEPS
  • Research the application of the arc-length formula in surveying
  • Explore the historical context and usage of graduated circles in surveying
  • Learn about angular measurements and conversions between degrees, minutes, and seconds
  • Investigate modern surveying tools that utilize graduated circles
USEFUL FOR

This discussion is beneficial for students of trigonometry, surveying professionals, and anyone interested in the historical applications of geometric principles in practical tools.

Ragnarok7
Messages
50
Reaction score
0
Hello, I was working problems from a very old trigonometry book, Loney's Trigonometry from 1895. There appears here a problem stating:

The value of the divisions on the outer rim of a graduated circle is 5' and the distance between successive graduations is .1 inch. Find the radius of the circle.

I cannot determine what is meant exactly by a graduated circle, other than that it was a surveying instrument. I am also unsure if the 5' measurement is supposed to mean 5 inches or 5 seconds (I'm thinking the latter). Does anyone have any ideas? Thank you!
 
Mathematics news on Phys.org
The expression 5' refers to 5 minutes of arc, or 1/12 of a degree (since there are 60 minutes of arc in a degree). So, I would use the arc-length formula:

$$s=r\theta$$

You are given $s$ and $\theta$, so you can solve for $r$. :D
 
Ah! Okay. I should have said minutes, not seconds. I get it now. Not sure why I was confused.

Thanks!
 

Similar threads

MHB Find angles when circumference is divided into 5 unequal parts
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Calculation for angular excess
  • · Replies 2 ·
Replies
2
Views
2K
Graduate How Do Prime Numbers Influence Circle Packing Patterns?
  • · Replies 23 ·
Replies
23
Views
12K
MHB How to Calculate the Radius of a Circle Tangent to Two Lines?
  • · Replies 4 ·
Replies
4
Views
4K
Carnival Game Probability: Calculating Expected Value
  • · Replies 4 ·
Replies
4
Views
3K
Calculating Arc Length of a Circle with Radius 6 and Center (4,1,5)
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Calculations to prove the non-Euclidean nature of 3-D space
  • · Replies 5 ·
Replies
5
Views
1K
How to Calculate the Mapping of Points from a Circle to a Tangent?
  • · Replies 4 ·
Replies
4
Views
3K
MHB Volume of a torus, in terms of big radius R and little radius r, washer method
  • · Replies 9 ·
Replies
9
Views
12K
How Do You Calculate the E-Field of a Semi-Circle with a Uniformly Charged Line?
  • · Replies 9 ·
Replies
9
Views
5K