# Need an (approximate) area formula for a ring arc. **URL:** \ **Category:** Factual Questions\ **Created:** [July 6, 2017, 3:47pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615 "2017-07-06T15:47:13Z")\ **Posts on this page:** 10\ **Page:** 1
**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\ **Post date:** [July 6, 2017, 3:47pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/1 "2017-07-06T15:47:13Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 6, 2017, 4:04pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/2 "2017-07-06T16:04:14Z")
That is in fact the exact formula. ---
**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\ **Post date:** [July 6, 2017, 4:31pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/3 "2017-07-06T16:31:42Z")
Cool, thanks! ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [July 6, 2017, 5:12pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/4 "2017-07-06T17:12:45Z")
> [@Chronos](#): > > That is in fact the exact formula. 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](https://www.first4magnets.com/arc-segment-magnets-c46/20mm-o-r-x-12-5mm-i-r-x-90-degree-x-5mm-thick-n42-neodymium-arc-magnet-5kg-pull-p564#ps_1-567), and the “outer” and “inner” measurements are the arc lengths along the outside & inside.) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 6, 2017, 6:32pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/5 "2017-07-06T18:32:18Z")
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. > [@](#): > > (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.) My first thought on reading it was that the “outer” and “inner” measurements were radii, but that’s inconsistent with the stated width. ---
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [July 6, 2017, 6:37pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/6 "2017-07-06T18:37:17Z")
The outside circle encloses an area PI_R^2 where R is radius of outside. Inside, similarly, is PI_r^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… ---
**Author:** ![HoneyBadgerDC](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/honeybadgerdc/32/1033_2.png) [@HoneyBadgerDC](https://boards.straightdope.com/u/HoneyBadgerDC)\ **Post date:** [July 6, 2017, 6:44pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/7 "2017-07-06T18:44:41Z")
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? ---
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [July 6, 2017, 7:25pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/8 "2017-07-06T19:25:36Z")
> [@HoneyBadgerDC](#): > > 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? 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 - PI_R^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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 6, 2017, 9:09pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/9 "2017-07-06T21:09:11Z")
> [@](#): > > 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 - PI_R^2_(arc degrees)/360; …Which is equal to R \* pi_R_(angle/360º), which is equal to the thickness (here equal to the radius) times the arc-length (pi_R_(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. ---
**Author:** ![mixdenny](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mixdenny/32/2962_2.png) [@mixdenny](https://boards.straightdope.com/u/mixdenny)\ **Post date:** [July 6, 2017, 9:39pm UTC](https://boards.straightdope.com/t/need-an-approximate-area-formula-for-a-ring-arc/790615/10 "2017-07-06T21:39:53Z")
Known in the inner sanctum of mathematicians as the “macaroni formula”. Dennis