Need an (approximate) area formula for a ring arc.

Factual Questions
1

For a section of lawn.

Outside ring measurement is about 120 feet. Inside is about 80. The width is about 22. And just a ring arc. Not sure exactly what the complete circumference would be for either.

I can estimate it’s about 2200 taking the average of the inside and outside of the ring arc and multiplying by the width, but is there a more precise formula? Doesn’t have to be too exact.

2

That is in fact the exact formula.

3

Cool, thanks!

4

When I saw this I thought “hold on, that can’t be right.” But then I went ahead & proved it to myself; it’s pretty satisfying how it works out.

(I assume that we’re talking about a shape like this, and the “outer” and “inner” measurements are the arc lengths along the outside & inside.)

5

I proved it to myself by cutting out wedges and turning them inside-out to straighten the curve. But you can do it algebraically, too.

My first thought on reading it was that the “outer” and “inner” measurements were radii, but that’s inconsistent with the stated width.

6

The outside circle encloses an area PIR^2 where R is radius of outside.
Inside, similarly, is PIr^2
So area of donut is PI*(R^2-r^2)
Assuming edges of the segment are radii for the concentric arcs - that is, form 90º corners at each point, then multiply by the degrees of the segment over 360. i.e. a 90º segment is 1/4 of the donut area, a 120º is 1/3, 45 degrees is 1/8 and so on…

7

Question, could you not just straighten out the lines he gave, average out the length of each one as he did and figure it like a rectangle?

8

Depends how precise you want to be. At extremes, no. (consider a 1-foot radius and a 100 foot radius - the pie shape is effectively the area of a pie slice - PIR^2(arc degrees)/360; at the other extreme - say, 1000 feet and 999 feet radii - it’s about the same as the average length of the arc segment times 1 foot.

9

…Which is equal to R * piR(angle/360º), which is equal to the thickness (here equal to the radius) times the arc-length (piR(angle/360º). In other words, yes, you can, in all cases, just straighten out the lines, average them, and figure it as a rectangle, just like I and MikeS said.

10

Known in the inner sanctum of mathematicians as the “macaroni formula”.

Dennis