diff --git a/crates/geom/src/curve.rs b/crates/geom/src/curve.rs
index 9d45453a1..89984cea0 100644
--- a/crates/geom/src/curve.rs
+++ b/crates/geom/src/curve.rs
@@ -514,8 +514,31 @@ impl Polyline {
     }
     /// Signed area (CCW positive) of a closed polyline including bulge areas.
     pub fn area(&self) -> f64 {
-        let pts = self.tessellate(1e-4);
-        shoelace(&pts)
+        // Integrate the chords exactly and add the signed circular-segment area
+        // for every bulge. A tessellated shoelace loses precision on large arcs.
+        // Working relative to the first vertex avoids cancellation when a small
+        // shape is located far from the drawing origin.
+        let Some(first) = self.vertices.first() else {
+            return 0.0;
+        };
+        let origin = first.p;
+        let n = self.vertices.len();
+        let mut twice_area = 0.0;
+        for i in 0..n {
+            let (Some(a), Some(b)) = (self.vertices.get(i), self.vertices.get((i + 1) % n)) else {
+                continue;
+            };
+            twice_area += (a.p - origin).cross(b.p - origin);
+            // Open polylines keep their historical implicit straight closing edge.
+            // The final vertex's bulge is only used for an explicit closing segment.
+            if (i + 1 < n || self.closed)
+                && let Some((arc, ccw)) = bulge_to_arc(a.p, b.p, a.bulge)
+            {
+                let sweep = if ccw { arc.sweep() } else { -arc.sweep() };
+                twice_area += arc.radius * arc.radius * (sweep - sweep.sin());
+            }
+        }
+        twice_area / 2.0
     }
     pub fn closest(&self, p: Vec2) -> Option<Vec2> {
         let mut best: Option<(f64, Vec2)> = None;
@@ -540,10 +563,15 @@ pub fn shoelace(pts: &[Vec2]) -> f64 {
     if n < 3 {
         return 0.0;
     }
+    let Some(origin) = pts.first().copied() else {
+        return 0.0;
+    };
+    // Measure about a local origin so a small polygon at a large world
+    // coordinate does not lose its area to cancellation of huge products.
     let mut s = 0.0;
     for i in 0..n {
         if let (Some(a), Some(b)) = (pts.get(i), pts.get((i + 1) % n)) {
-            s += a.cross(*b);
+            s += (*a - origin).cross(*b - origin);
         }
     }
     s / 2.0
diff --git a/crates/geom/src/tests.rs b/crates/geom/src/tests.rs
index a9f5daf6e..1c17fb282 100644
--- a/crates/geom/src/tests.rs
+++ b/crates/geom/src/tests.rs
@@ -236,3 +236,45 @@ fn elliptical_arc_bounds_exclude_extrema_outside_the_sweep() {
     assert!((bounds.min.y - e.at_param(e.end).y).abs() < 1e-12);
     assert!((bounds.max.y - 18.25_f64.sqrt()).abs() < 1e-12);
 }
+
+#[test]
+fn polyline_area_is_exact_for_bulge_segments() {
+    // One semicircular arc from left to right plus a straight closing edge.
+    let mut p = Polyline::from_points(&[Vec2::new(0.0, 0.0), Vec2::new(2.0, 0.0)], true);
+    p.vertices[0].bulge = 1.0;
+    assert!((p.area() - PI / 2.0).abs() < 1e-12, "{}", p.area());
+    p.vertices[0].bulge = -1.0;
+    assert!((p.area() + PI / 2.0).abs() < 1e-12, "{}", p.area());
+}
+
+#[test]
+fn polyline_area_is_stable_far_from_the_origin() {
+    let o = 1.0e12;
+    let p = Polyline::from_points(
+        &[
+            Vec2::new(o, o),
+            Vec2::new(o + 10.0, o),
+            Vec2::new(o + 10.0, o + 10.0),
+            Vec2::new(o, o + 10.0),
+        ],
+        true,
+    );
+    assert!((p.area() - 100.0).abs() < 1e-12, "{}", p.area());
+    // An open polyline's measured area retains the implicit straight closing edge.
+    let mut open = p;
+    open.closed = false;
+    assert!((open.area() - 100.0).abs() < 1e-12);
+}
+
+#[test]
+fn shoelace_preserves_small_polygon_area_at_large_coordinates() {
+    let o = 1.0e12;
+    let corners = [
+        Vec2::new(o, o),
+        Vec2::new(o + 10.0, o),
+        Vec2::new(o + 10.0, o + 10.0),
+        Vec2::new(o, o + 10.0),
+    ];
+    assert_eq!(shoelace(&corners), 100.0);
+    assert_eq!(shoelace(&corners.iter().rev().copied().collect::<Vec<_>>()), -100.0);
+}