For this question refer to this diagram:
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.
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.
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.
Ack, dagnabit. My previous post should read:
Also, I should note that theta, as I’ve defined it, is just the angle AOB.
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).
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).
Touché, Orbifold. Yeah, I did assume that the diagram had reflection symmetry.