diff --git a/crates/kernel/src/mesh_convert.rs b/crates/kernel/src/mesh_convert.rs
index 6ea13565..ebe19af9 100644
--- a/crates/kernel/src/mesh_convert.rs
+++ b/crates/kernel/src/mesh_convert.rs
@@ -18,6 +18,14 @@ fn invalid(message: &str) -> KernelError {
     KernelError::Invalid(format!("mesh conversion: {message}"))
 }
 
+/// A mesh over the triangle budget: how big it is, the limit, and that only constant-section
+/// parts convert, so a smaller freeform mesh would still be refused (#346).
+fn over_budget(triangles: usize) -> KernelError {
+    invalid(&format!(
+        "the mesh has {triangles} triangles, more than the {MAX_TRIANGLES} conversion can take; only closed constant-section parts (one profile extruded straight) can be converted, so a freeform mesh is refused at any size"
+    ))
+}
+
 /// Reconstruct a constant-section mesh as analytic planes and, where certified
 /// within `tolerance` mm, circular cylinders. The source mesh is not modified.
 /// Unsupported shapes are refused instead of silently returning a faceted solid.
@@ -30,7 +38,7 @@ pub fn convert_prismatic_mesh(body: &Body, tolerance: f64) -> Result<Body> {
         let welded = weld(input, tolerance)?;
         let mesh = welded.mesh.as_ref().ok_or_else(|| invalid("empty mesh"))?;
         if mesh.triangles.len() > MAX_TRIANGLES {
-            return Err(invalid("mesh exceeds the 50000-triangle conversion budget"));
+            return Err(over_budget(mesh.triangles.len()));
         }
         let mut edges: BTreeMap<(u32, u32), (usize, i32)> = BTreeMap::new();
         let mut axes: BTreeMap<(i64, i64, i64), (Vec3, f64)> = BTreeMap::new();
@@ -69,8 +77,11 @@ pub fn convert_prismatic_mesh(body: &Body, tolerance: f64) -> Result<Body> {
 }
 
 fn weld(mesh: &crate::meshbody::TriMesh, tolerance: f64) -> Result<Body> {
-    if mesh.triangles.len() > MAX_TRIANGLES || mesh.positions.len() > MAX_TRIANGLES * 3 {
-        return Err(invalid("mesh exceeds the conversion budget"));
+    if mesh.triangles.len() > MAX_TRIANGLES {
+        return Err(over_budget(mesh.triangles.len()));
+    }
+    if mesh.positions.len() > MAX_TRIANGLES * 3 {
+        return Err(invalid(&format!("the mesh has {} vertices, more than the {} conversion can take", mesh.positions.len(), MAX_TRIANGLES * 3)));
     }
     let bounds = mesh.positions.iter().fold(solidcraft_geom::Aabb3::EMPTY, |mut b, p| {
         b.add(*p);
@@ -464,6 +475,22 @@ mod tests {
         }
     }
 
+    #[test]
+    fn a_mesh_over_the_budget_says_how_big_it_is_and_what_converts() {
+        // #346: a freeform mesh just over the limit (a strip of 50001 triangles) is refused with
+        // its size, the limit, and that only constant-section parts convert at any size.
+        let n = MAX_TRIANGLES + 1;
+        let positions: Vec<Vec3> = (0..n + 2).map(|k| Vec3::new((k / 2) as f64, (k % 2) as f64, 0.)).collect();
+        let triangles: Vec<[u32; 3]> = (0..n as u32).map(|i| [i, i + 1, i + 2]).collect();
+        let body = crate::mesh_body(&positions, &triangles).unwrap();
+        let err = convert_prismatic_mesh(&body, 0.05).unwrap_err().to_string();
+        assert!(err.contains("50001 triangles") && err.contains("50000"), "{err}");
+        assert!(err.contains("constant-section") && err.contains("freeform"), "{err}");
+        let mesh = crate::meshbody::TriMesh { positions: vec![Vec3::ZERO; MAX_TRIANGLES * 3 + 1], triangles: vec![] };
+        let err = weld(&mesh, 0.05).unwrap_err().to_string();
+        assert!(err.contains("150001 vertices") && err.contains("150000"), "{err}");
+    }
+
     #[test]
     fn fitting_cannot_open_a_hole_through_a_nearby_wall() {
         let outer = Loop2::polygon(&[Vec2::new(-1., -1.), Vec2::new(1., -1.), Vec2::new(1., 1.), Vec2::new(-1., 1.)]);