diff --git a/crates/fj-core/src/operations/build/edge.rs b/crates/fj-core/src/operations/build/edge.rs
index dc6efe436..0fdda9d55 100644
--- a/crates/fj-core/src/operations/build/edge.rs
+++ b/crates/fj-core/src/operations/build/edge.rs
@@ -3,7 +3,7 @@ use fj_math::{Arc, Point, Scalar};
 
 use crate::{
     geometry::curve::Curve,
-    objects::{GlobalEdge, HalfEdge, Surface, Vertex},
+    objects::{GlobalEdge, HalfEdge, Vertex},
     operations::Insert,
     services::Services,
 };
@@ -75,18 +75,6 @@ pub trait BuildHalfEdge {
 
         HalfEdge::unjoined(curve, boundary, services)
     }
-
-    /// Create a line segment from global points
-    fn line_segment_from_global_points(
-        points_global: [impl Into<Point<3>>; 2],
-        surface: &Surface,
-        boundary: Option<[Point<1>; 2]>,
-        services: &mut Services,
-    ) -> HalfEdge {
-        let points_surface = points_global
-            .map(|point| surface.geometry().project_global_point(point));
-        HalfEdge::line_segment(points_surface, boundary, services)
-    }
 }
 
 impl BuildHalfEdge for HalfEdge {}
diff --git a/crates/fj-core/src/operations/build/face.rs b/crates/fj-core/src/operations/build/face.rs
index 91b02bfbc..5c8f0c810 100644
--- a/crates/fj-core/src/operations/build/face.rs
+++ b/crates/fj-core/src/operations/build/face.rs
@@ -1,11 +1,12 @@
+use std::array;
+
 use fj_interop::ext::ArrayExt;
 use fj_math::Point;
 
 use crate::{
     objects::{Cycle, Face, HalfEdge, Region, Surface, Vertex},
     operations::{
-        BuildCycle, BuildHalfEdge, BuildSurface, Insert, IsInserted,
-        IsInsertedNo,
+        BuildCycle, BuildRegion, BuildSurface, Insert, IsInserted, IsInsertedNo,
     },
     services::Services,
     storage::Handle,
@@ -25,29 +26,24 @@ pub trait BuildFace {
         points: [impl Into<Point<3>>; 3],
         services: &mut Services,
     ) -> Polygon<3> {
-        let [a, b, c] = points.map(Into::into);
-
-        let surface = Surface::plane_from_points([a, b, c]).insert(services);
-        let (exterior, edges, vertices) = {
-            let half_edges = [[a, b], [b, c], [c, a]].map(|points| {
-                let half_edge = HalfEdge::line_segment_from_global_points(
-                    points, &surface, None, services,
-                );
+        let (surface, points_surface) = Surface::plane_from_points(points);
+        let surface = surface.insert(services);
 
-                half_edge.insert(services)
-            });
-            let vertices = half_edges
-                .each_ref_ext()
-                .map(|half_edge| half_edge.start_vertex().clone());
+        let face = Face::polygon(surface, points_surface, services);
 
-            let cycle = Cycle::new(half_edges.clone()).insert(services);
+        let edges = {
+            let mut half_edges = face.region().exterior().half_edges().cloned();
+            assert_eq!(half_edges.clone().count(), 3);
 
-            (cycle, half_edges, vertices)
+            array::from_fn(|_| half_edges.next()).map(|half_edge| {
+                half_edge
+                    .expect("Just asserted that there are three half-edges")
+            })
         };
-
-        let region = Region::new(exterior, [], None).insert(services);
-
-        let face = Face::new(surface, region);
+        let vertices =
+            edges.each_ref_ext().map(|half_edge: &Handle<HalfEdge>| {
+                half_edge.start_vertex().clone()
+            });
 
         Polygon {
             face,
@@ -55,6 +51,21 @@ pub trait BuildFace {
             vertices,
         }
     }
+
+    /// Build a polygon
+    fn polygon<P, Ps>(
+        surface: Handle<Surface>,
+        points: Ps,
+        services: &mut Services,
+    ) -> Face
+    where
+        P: Into<Point<2>>,
+        Ps: IntoIterator<Item = P>,
+        Ps::IntoIter: Clone + ExactSizeIterator,
+    {
+        let region = Region::polygon(points, services).insert(services);
+        Face::new(surface, region)
+    }
 }
 
 impl BuildFace for Face {}
diff --git a/crates/fj-core/src/operations/build/surface.rs b/crates/fj-core/src/operations/build/surface.rs
index bd7bab739..9e74f0fd2 100644
--- a/crates/fj-core/src/operations/build/surface.rs
+++ b/crates/fj-core/src/operations/build/surface.rs
@@ -1,4 +1,4 @@
-use fj_math::Point;
+use fj_math::{Point, Scalar};
 
 use crate::{
     geometry::{curve::GlobalPath, surface::SurfaceGeometry},
@@ -8,15 +8,26 @@ use crate::{
 /// Build a [`Surface`]
 pub trait BuildSurface {
     /// Build a plane from the provided points
-    fn plane_from_points(points: [impl Into<Point<3>>; 3]) -> Surface {
+    fn plane_from_points(
+        points: [impl Into<Point<3>>; 3],
+    ) -> (Surface, [Point<2>; 3]) {
         let [a, b, c] = points.map(Into::into);
 
-        let geometry = SurfaceGeometry {
-            u: GlobalPath::line_from_points([a, b]).0,
-            v: c - a,
+        let (u, u_line) = GlobalPath::line_from_points([a, b]);
+        let v = c - a;
+
+        let geometry = SurfaceGeometry { u, v };
+        let surface = Surface::new(geometry);
+
+        let points_surface = {
+            let [a, b] =
+                u_line.map(|point| Point::from([point.t, Scalar::ZERO]));
+            let c = Point::from([a.u, Scalar::ONE]);
+
+            [a, b, c]
         };
 
-        Surface::new(geometry)
+        (surface, points_surface)
     }
 }
 
diff --git a/crates/fj-core/src/validate/mod.rs b/crates/fj-core/src/validate/mod.rs
index ea8ea3c45..27c8f601c 100644
--- a/crates/fj-core/src/validate/mod.rs
+++ b/crates/fj-core/src/validate/mod.rs
@@ -99,23 +99,23 @@ impl Default for ValidationConfig {
 #[derive(Clone, Debug, thiserror::Error)]
 pub enum ValidationError {
     /// `Cycle` validation error
-    #[error("`Cycle` validation error:\n    {0}")]
+    #[error("`Cycle` validation error")]
     Cycle(#[from] CycleValidationError),
 
     /// `Face` validation error
-    #[error("`Face` validation error\n    {0}")]
+    #[error("`Face` validation error")]
     Face(#[from] FaceValidationError),
 
     /// `HalfEdge` validation error
-    #[error("`HalfEdge` validation error\n    {0}")]
+    #[error("`HalfEdge` validation error")]
     HalfEdge(#[from] HalfEdgeValidationError),
 
     /// `Shell` validation error
-    #[error("`Shell` validation error\n    {0}")]
+    #[error("`Shell` validation error")]
     Shell(#[from] ShellValidationError),
 
     /// `Solid` validation error
-    #[error("`Solid` validation error\n    {0}")]
+    #[error("`Solid` validation error")]
     Solid(#[from] SolidValidationError),
 }