Calculating the length of an arc inside a circle

[Aja]

A sad and strange image, I know, but better than none at all. What you see is a stake I'm trying to model using CAD software. With the dimension given (5 inches--in case it's not clear, the distance from the top of the equilateral triangle that encloses this shape to the midpoint of the red arc) I'm willing to bet that I can calculate the length of that arc or other dimensions that will help me model this accurately.

Any ideas?

https://www.physicsforums.com/attachments/4116

 

[Opalg]

A sad and strange image, I know, but better than none at all. What you see is a stake I'm trying to model using CAD software. With the dimension given (5 inches--in case it's not clear, the distance from the top of the equilateral triangle that encloses this shape to the midpoint of the red arc) I'm willing to bet that I can calculate the length of that arc or other dimensions that will help me model this accurately.

Any ideas?

Hi Aja, and welcome to MHB!

It looks as though the shape you are trying to model is a region inside an equilateral triangle, where the boundary of the region consists of arcs of circles. There is a small circle, of radius $r_2$ say, inside each corner of the triangle, and arcs of these circles are connected by arcs of larger circles of radius $r_1$, as in the diagram below.

In order to determine the configuration, you need to know the ratio between those two radii. Alternatively, you could specify the angle of the $r_1$-arcs. In the diagram, I have made this angle $60^\circ$, which makes the calculations a whole lot easier.

In practice, it is probably easiest to start by deciding an arbitrary value for $r_2$. You can then use the ratio $r_1/r_2$ (or alternatively the angle of the $r_1$-arcs) to complete the construction. Then finally you can scale the whole diagram so that the vertical distance from the apex of the triangle to the midpoint of the $r_1$-arc at the base of the diagram becomes $5$ inches.

View attachment 4118​

 

[Aja]

Thanks! I think you've led me to a solution that won't require any calculations or further measurements. How'd you create that nifty diagram?

Just for kicks, how would I go about solving this if I knew the ratio?

One other thing: would you say it's reasonable to assume that the arc angle is 60$^\circ$? That makes intuitive sense to me both given this object's shape and my fuzzy recollection of high school geometry, but I can't articulate why. Give me a nudge?