Circle intersection area
JavaScript (ES6), 72 bytes
Alternate formula suggested by @ceilingcat
Takes input as 5 distinct parameters (x0, y0, x1, y1, r).
with(Math)f=(x,y,X,Y,r)=&gt;-(sin(d=2*acos(hypot(x-X,y-Y)/r/2))-d)*r*r*2&gt;&gt;1

Try it online!

JavaScript (ES7), 81 80 77 bytes
Saved 3 bytes thanks to @Neil
Takes input as 5 distinct parameters (x0, y0, x1, y1, r).
(x,y,X,Y,r,d=Math.hypot(x-X,y-Y))=&gt;(r*=2)*r*Math.acos(d/r)-d*(r*r-d*d)**.5&gt;&gt;1

Try it online!
How?
This is based on a generic formula from MathWorld for non-congruent circles:
A = r².arccos((d² + r² - R²) / 2dr) +
    R².arccos((d² + R² - r²) / 2dR) -
    sqrt((-d + r + R)(d + r - R)(d -r + R)(d + r + R)) / 2

where d is the distance between the two centers and r and R are the radii.
With R = r, this is simplified to:
A = 2r².arccos(d / 2r) + d.sqrt((2r - d) * (2r + d)) / 2

And with r' = 2r:
A = (r'².arccos(d / r') + d.sqrt(r'² - d²)) / 2

Note: If d is greater than 2r, Math.acos() will return NaN, which is coerced to 0 when the right-shift is applied. This is the expected result, because d &gt; 2r means that there's no intersection at all.