# How does one calculate the area of the constricted area given below whether the constricted arc converge (not shown) as a circle or not?

How does one calculate the area of the area given below whether the arc converge (not shown) as a circle or not? For your information, the straight line AB is not diameter. It's the constricted area of a table. How do you measure that area in that case?

Is it possible to calculate the area of this shape?

Let $$r$$ be the radius of the circle of which the arc is a part and $$d$$ the length of the chord AB. Observe that the area is just the difference between the circle sector AOB and the isosceles triangle AOB, where,

$$\theta = 2\sin^{-1}\left(\frac d{2r}\right)$$

Thus, the area is

$$A=\frac12 \theta r^2 - \frac12 dh =r^2\sin^{-1}\left(\frac d{2r}\right)-\frac12d \sqrt{r^2 - \frac {d^2}4}$$

If integration can be used:

$$(x-a)^2+y^2 = a^2,\, y=\sqrt{x (2a-x)}$$

Assuming maximum mid-width along x- direction $$=h,$$ the area under circle radius $$a$$ (doubled due to symmetry x-axis) $$\int_0^{h} \sqrt{x (2a-x)} \,dx=$$

$$(h-a) \sqrt{(2a-h)h}+2 a^2\cot^{-1} \sqrt{2a/h-1}\,$$

If half-height is $$c$$, then by circle segment product property the radius can be calculated as:

$$a= \dfrac{c^2/h+h}{2}$$

The graph drawn in this case for $$a=2.$$ (Btw, it is useful to estimate volumes of liquids filled in a prismatic cylinder laid horizontal as a function of filling height $$h).$$

• Suppose you don't know the radius of the circle. Then what should I do please?
– user35885
Jan 16, 2020 at 10:53
• Calculate it from half-height $c$ and maximum width $h$ Jan 16, 2020 at 12:34