# Geometry: Inscribed Angle Area
**URL:** https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765
**Category:** Factual Questions
**Created:** [December 8, 2003, 9:09pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765 "2003-12-08T21:09:13Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)
#### Post date: [December 8, 2003, 9:09pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/1 "2003-12-08T21:09:13Z")
For this question refer to this diagram:
[http://www.1728.com/circsect.htm](http://www.1728.com/circsect.htm)
In Circle One, Angle ACB is called an **inscribed angle**.
What would the _area_ inside Angle ACB be called?
(If this were a central angle (Circle 2) it would be called a sector.
Is there a formula for determining the area of the inscribed angle “sector” ? (For lack of the proper term).
I researched this on Google and consulted several math books but could find nothing so I thought I’d use a great resource - the SDMB.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [December 8, 2003, 9:16pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/2 "2003-12-08T21:16:09Z")
I don’t know of any term, but you could easily derive a formula–the area of triangle ACB (use the sine form) plus the area of the cap (which is the area of the sector AOB minus the area of triangle AOB).
I’ll try to figure it out, but it may be a while.
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### Author:  [@MikeS](https://boards.straightdope.com/u/MikeS)
#### Post date: [December 8, 2003, 9:50pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/3 "2003-12-08T21:50:25Z")
It’s simply the area of the sector AOB plus twice the triangle AOC, or
theta \* r[sup]2[/sup] / 2 + 2 \* (r \* r sin(theta) / 2)
= r[sup]2[/sup] (theta / 2 + sin (theta) )
where r is the radius of the circle and theta is measured in radians. The only tricky part is figuring out the area of AOC, i.e., what to use for “base” and “altitude” and so forth.
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### Author:  [@MikeS](https://boards.straightdope.com/u/MikeS)
#### Post date: [December 8, 2003, 9:55pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/4 "2003-12-08T21:55:27Z")
Ack, dagnabit. My previous post should read:
> [@](#):
>
> It’s simply the area of the sector AOB plus twice the triangle AOC, or
>
> theta \* r[sup]2[/sup] / 2 + 2 \* (r \* r sin(theta/2) / 2)
> = r[sup]2[/sup] (theta / 2 + sin (theta/2) )
>
> where r is the radius of the circle and theta is measured in radians. The only tricky part is figuring out the area of AOC, i.e., what to use for “base” and “altitude” and so forth.
Also, I should note that theta, as I’ve defined it, is just the angle AOB.
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### Author:  [@Orbifold](https://boards.straightdope.com/u/Orbifold)
#### Post date: [December 8, 2003, 10:06pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/5 "2003-12-08T22:06:35Z")
It should probably be pointed out that **MikeS** ’s formula (while correct) assumes that angle AOC equals angle BOC. While this is how it appears in the diagram, if the point C is not directly opposite the midpoint of the chord AB then the area of the region you’re describing would change.
In the general case, **MikeS** ’s formula would become:
angle(BOA) r[sup]2[/sup] / 2 + sin(angle(AOC)) r[sup]2[/sup] / 2 + sin(angle(COB)) r[sup]2[/sup] /2
where all angles are measured in radians, and in a consistent direction (i.e. either all clockwise or all counterclockwise).
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### Author:  [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)
#### Post date: [December 8, 2003, 11:18pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/6 "2003-12-08T23:18:55Z")
Thanks for the help folks.
I am still in the process of writing a calculator for partially filled horizontal cylinders or tanks. Basically I am still going to use the formula for the central angle of a circle.
Still, I was surprised that for an inscribed angle (something relatively common in geometry), there is **no name** for the area it subtends and there is **no formula** to determine it. (At least something that could be easily found on a search engine or in a book).
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### Author:  [@MikeS](https://boards.straightdope.com/u/MikeS)
#### Post date: [December 9, 2003, 2:33pm UTC](https://boards.straightdope.com/t/geometry-inscribed-angle-area/217765/7 "2003-12-09T14:33:16Z")
Touché, **Orbifold**. Yeah, I did assume that the diagram had reflection symmetry.