# What's the maximum area of a sector defined by an inscribed angle?

When $$\theta$$ is in radians, a central angle of $$\theta$$ creates a sector of the circle with area equal to $$\frac12r^2\theta$$. I started wondering what the area formula would be if the vertex of the angle was on the circumference rather than the center. But after searching on this site, I found that same question being asked. The answer was that there is no formula because a single value of $$\theta$$ can lead to different areas (rotate the angle without changing its measure to change sector area). So I have a new question. What is the maximum sector area in a circle, when the inscribed angle is $$\theta$$?

Since the base for a given angle is fixed, we just need to maximize the height

which requires $$AC=BC$$.

According to the following sketch we need to find:

$$A=\operatorname{Area}\left(\triangle{ABC}\right)+ \operatorname{Area}(\newcommand{arc}[1]{\stackrel{\Large\frown}{#1}}\arc{OAB} )-\operatorname{Area}\left(\triangle{OAB}\right)$$

or also

$$A=2\operatorname{Area}\left(\triangle{OBC}\right)+ \operatorname{Area}(\newcommand{arc}[1]{\stackrel{\Large\frown}{#1}}\arc{OAB} )=$$

$$=2\cdot \frac12 \cdot 2r\cos \frac \theta 2 \cdot r \sin \frac \theta 2+\frac12 r^2\cdot 2\theta=\boxed{r^2\left(\sin \theta +\theta\right)}$$

• I think I understand everything except for one thing: how do we know that AC = BC? Commented May 2, 2023 at 7:21
• What is the maximum distance between the chord and the circumference, that is the height?
– user
Commented May 2, 2023 at 7:39
• @MumboJumbo The are for the circular segment $\newcommand{arc}[1]{\stackrel{\Large\frown}{#1}}\arc{AB}$ is fixed. Then we need to maximize the are of the triangle $\triangle{ABC}$ and since AB is fixed we need to maximize its height which is of course obtained when the point $C$ is such that $AC=AB$. It seems trivial but we can also prove this fact if you have some doubt on that.
– user
Commented May 2, 2023 at 16:11
• Is it because perpendicular bisector makes it an isosceles triangle? Commented May 4, 2023 at 13:30
• @MumboJumbo Yes of course the maximum is obtained for the perpendicular bisector!
– user
Commented May 4, 2023 at 13:33

Theoretically you could get $$\theta$$ closer and closer to a straight angle, and the area of the major circular sector (and circular segment) would get closer and closer to the entire circle.

• It seems that the OP is looking for the maximum area for a fixed value for the angle $\theta$.
– user
Commented May 1, 2023 at 10:17