spoiler - behold a square

let totalPoints = 0;
let pointsInside = 0;
const SCALE = 100;
const SHAPE_OFFSET_X = -0.2;
const SHAPE_OFFSET_Y = -0.1;
let theta, smallRadius, largeRadius, smallRadiusScaled, largeRadiusScaled;

function setup() {
  theta = 1 + PI - sqrt(PI * PI + 1);
  smallRadius = 1 / theta - 1;
  largeRadius = smallRadius + 1;
  smallRadiusScaled = smallRadius * SCALE;
  largeRadiusScaled = largeRadius * SCALE;

  createCanvas(600, 600);
  background(220);

  translate(width / 2, height / 2);
  drawShape();
  translate(-width / 2, -height / 2);
}

function draw() {
  for (let i = 0; i < 100; i++) {
    let x = random(
      width / 2 - largeRadiusScaled,
      width / 2 + largeRadiusScaled
    );
    let y = random(
      height / 2 - largeRadiusScaled,
      height / 2 + largeRadiusScaled
    );

    if (isInsideShape(x, y)) {
      stroke(0, 0, 0, 100);
      pointsInside++;
    } else {
      stroke(255, 255, 255, 100);
    }
    point(x, y);
    totalPoints++;
  }

  let sampledArea =
    (pointsInside / totalPoints) * (4 * largeRadiusScaled * largeRadiusScaled);
  let unitArea = sampledArea / (SCALE * SCALE);

  fill(255);
  noStroke();
  rect(10, height - 70, 250, 60);
  fill(0);
  text(`Total points: ${totalPoints}`, 20, height - 50);
  text(`Estimated area (unit scale): ${unitArea.toFixed(6)}`, 20, height - 30);
}

function isInsideShape(x, y) {
  let px = (x - width / 2) / SCALE - SHAPE_OFFSET_X;
  let py = (y - height / 2) / SCALE - SHAPE_OFFSET_Y;

  let r = sqrt(px * px + py * py);
  let angle = atan2(py, px);
  if (angle < 0) angle += TWO_PI;
  let inWedge = angle <= theta && r <= largeRadius;

  let inSmallCircle = px * px + py * py <= smallRadius * smallRadius;

  return inWedge || inSmallCircle;
}

function drawShape() {
  translate(SHAPE_OFFSET_X * SCALE, SHAPE_OFFSET_Y * SCALE);

  noFill();
  noStroke();
  circle(0, 0, smallRadiusScaled * 2);

  arc(0, 0, largeRadiusScaled * 2, largeRadiusScaled * 2, 0, theta);
  line(0, 0, largeRadiusScaled * cos(0), largeRadiusScaled * sin(0));
  line(0, 0, largeRadiusScaled * cos(theta), largeRadiusScaled * sin(theta));
}