Determine whether circles intersect and calculate the area if intersect

[Comede_way]

x1,y1 are the coordinates of the centre of the circle, r is the radius and id is the number of the corresponding circle. I want to draw each circle and output the data frame. The data frame contains whether the different circles intersect (T or F) and what is the area of the overlap section between the intersecting circle. Right now I have just drawn the diagram and calculated the area of each circle. I don't know what to do next. Thank you for your help in advance.

x1<-c(10.3,13.4,14.3,16.4,17.4,19.1,19.2,18.5,18.3,16.8,16.7,15.5)
y1<-c(10.1,13.3,13.3,13.7,14.2,11.4,12.2,14.9,16.4,15.8,17.8,16.5)
r<-c(1.00,0.95,0.83,0.88,1.05,0.48,0.93,0.90,0.93,0.53,0.83,0.68)
id<-c(1:12)
library(ggplot2)
library(ggforce)
circles<-data.frame(
  x0=x1,
  y0=y1,
  r=r,
  id=id
)
p<-ggplot(data = circles)+
  geom_point(aes(x=x0,y=y0),color="black",shape=1)+
  geom_circle(aes(x0=x0,y0=y0,r=r),
              data = circles,
              color = "red",
              alpha=0.05,
              linetype="dashed")+
  geom_text(aes(circles$x0,circles$y0,label=circles$id,size=3,angle=0,family="mono"))+
  theme(legend.position = 'none')
p
s<-data.frame(pi*(circles$r)^2)
colnames(s)<-"area"
circles<-cbind(circles,s)
``` r
<sup>Created on 2022-11-15 with [reprex v2.0.2](https://reprex.tidyverse.org)</sup>

[Comede_way]

These pictures just show what i have achieved. The area in picture is every circle's, not the overlapped area between intersected circle. Problem has not been solved. I still don't know how to achieve my thought like post1 said.

[nirgrahamuk]

My first impression is that this is first a math question before an r question; therefore a math resource is most helpful if you are lacking the information.
Calculate the intersection area of two circles • Computer Science and Machine Learning (xarg.org)

[Comede_way]

OK, thank you. Actually, i just don't know how to achieve it in R. Especially, there are many circles as i show in post 2. I am a complete R beginner .

[nirgrahamuk]

well, if you understand the math, the next step would be to write a function in R to do any two circles;
Then you would simply have to apply that function over every overlapping pair you already managed to detect.
The linked article gives code (I think its python) for how to implement their approach; I expect this could be adapted to R.
In the meantime to someone providing such a code; you may like to learn about R functions and writing your own functions in general. And also about iteration in R, particularly the map family of iterators which are part of purrr/tidyverse

[Comede_way]

Good idea! I would try to do it in R. Thank you for your advice.