diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
index 76d839a07..3724fb123 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
@@ -150,7 +150,7 @@ where
 #[cfg(test)]
 mod tests {
     use crate::{
-        builder::{CurveBuilder, FaceBuilder},
+        builder::{CurveBuilder, CycleBuilder, FaceBuilder},
         partial::{Partial, PartialCurve, PartialFace, PartialObject},
         services::Services,
     };
@@ -188,8 +188,12 @@ mod tests {
         let face = {
             let mut face = PartialFace::default();
             face.exterior.write().surface = surface;
-            face.update_exterior_as_polygon_from_points(exterior);
-            face.add_interior_polygon_from_points(interior);
+            face.exterior
+                .write()
+                .update_as_polygon_from_points(exterior);
+            face.add_interior()
+                .write()
+                .update_as_polygon_from_points(interior);
 
             face.build(&mut services.objects)
         };
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_face.rs b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
index a8ba4668a..499aa0719 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
@@ -69,7 +69,7 @@ mod tests {
 
     use crate::{
         algorithms::intersect::CurveFaceIntersection,
-        builder::{CurveBuilder, FaceBuilder},
+        builder::{CurveBuilder, CycleBuilder},
         insert::Insert,
         partial::{Partial, PartialCurve, PartialFace, PartialObject},
         services::Services,
@@ -95,7 +95,7 @@ mod tests {
         .map(|surface| {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon_from_points(points);
+            face.exterior.write().update_as_polygon_from_points(points);
 
             face.build(&mut services.objects)
         });
@@ -124,7 +124,7 @@ mod tests {
         let [a, b] = surfaces.clone().map(|surface| {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon_from_points(points);
+            face.exterior.write().update_as_polygon_from_points(points);
 
             face.build(&mut services.objects)
         });
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_point.rs b/crates/fj-kernel/src/algorithms/intersect/face_point.rs
index 465f260b2..698999e6c 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_point.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_point.rs
@@ -135,7 +135,7 @@ mod tests {
 
     use crate::{
         algorithms::intersect::{face_point::FacePointIntersection, Intersect},
-        builder::FaceBuilder,
+        builder::CycleBuilder,
         insert::Insert,
         partial::{Partial, PartialFace, PartialObject},
         services::Services,
@@ -148,7 +148,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [1., 1.],
             [0., 2.],
@@ -169,7 +169,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [0., 2.],
@@ -193,7 +193,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [4., 2.],
             [0., 4.],
             [0., 0.],
@@ -217,7 +217,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [3., 0.],
@@ -242,7 +242,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [3., 1.],
@@ -267,7 +267,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [3., 1.],
@@ -293,7 +293,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 0.],
             [0., 1.],
@@ -327,7 +327,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [1., 0.],
             [0., 1.],
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
index 21cb23157..7565a50ef 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
@@ -151,7 +151,7 @@ mod tests {
             },
             transform::TransformObject,
         },
-        builder::FaceBuilder,
+        builder::CycleBuilder,
         insert::Insert,
         partial::{Partial, PartialFace, PartialObject},
         services::Services,
@@ -166,7 +166,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -189,7 +189,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -215,7 +215,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -238,7 +238,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -273,7 +273,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -307,7 +307,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -332,7 +332,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([
+        face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index 65d0f298f..7f54e4ff4 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -87,7 +87,7 @@ mod tests {
 
     use crate::{
         algorithms::{reverse::Reverse, transform::TransformObject},
-        builder::{FaceBuilder, HalfEdgeBuilder, SketchBuilder},
+        builder::{CycleBuilder, HalfEdgeBuilder, SketchBuilder},
         insert::Insert,
         partial::{
             Partial, PartialFace, PartialHalfEdge, PartialObject, PartialSketch,
@@ -116,7 +116,10 @@ mod tests {
 
         let mut bottom = PartialFace::default();
         bottom.exterior.write().surface = Partial::from(surface.clone());
-        bottom.update_exterior_as_polygon_from_points(TRIANGLE);
+        bottom
+            .exterior
+            .write()
+            .update_as_polygon_from_points(TRIANGLE);
         let bottom = bottom
             .build(&mut services.objects)
             .insert(&mut services.objects)
@@ -124,7 +127,7 @@ mod tests {
         let mut top = PartialFace::default();
         top.exterior.write().surface =
             Partial::from(surface.translate(UP, &mut services.objects));
-        top.update_exterior_as_polygon_from_points(TRIANGLE);
+        top.exterior.write().update_as_polygon_from_points(TRIANGLE);
         let top = top
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -169,14 +172,17 @@ mod tests {
         bottom.exterior.write().surface = Partial::from(
             surface.clone().translate(DOWN, &mut services.objects),
         );
-        bottom.update_exterior_as_polygon_from_points(TRIANGLE);
+        bottom
+            .exterior
+            .write()
+            .update_as_polygon_from_points(TRIANGLE);
         let bottom = bottom
             .build(&mut services.objects)
             .insert(&mut services.objects)
             .reverse(&mut services.objects);
         let mut top = PartialFace::default();
         top.exterior.write().surface = Partial::from(surface);
-        top.update_exterior_as_polygon_from_points(TRIANGLE);
+        top.exterior.write().update_as_polygon_from_points(TRIANGLE);
         let top = top
             .build(&mut services.objects)
             .insert(&mut services.objects);
diff --git a/crates/fj-kernel/src/algorithms/triangulate/mod.rs b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
index c252bb7fc..ec841dd1a 100644
--- a/crates/fj-kernel/src/algorithms/triangulate/mod.rs
+++ b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
@@ -77,7 +77,7 @@ mod tests {
 
     use crate::{
         algorithms::approx::{Approx, Tolerance},
-        builder::FaceBuilder,
+        builder::{CycleBuilder, FaceBuilder},
         insert::Insert,
         objects::Face,
         partial::{Partial, PartialFace, PartialObject},
@@ -99,7 +99,9 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon_from_points([a, b, c, d]);
+        face.exterior
+            .write()
+            .update_as_polygon_from_points([a, b, c, d]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -136,8 +138,12 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let mut face = PartialFace::default();
         face.exterior.write().surface = Partial::from(surface.clone());
-        face.update_exterior_as_polygon_from_points([a, b, c, d]);
-        face.add_interior_polygon_from_points([e, f, g, h]);
+        face.exterior
+            .write()
+            .update_as_polygon_from_points([a, b, c, d]);
+        face.add_interior()
+            .write()
+            .update_as_polygon_from_points([e, f, g, h]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -196,7 +202,9 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let mut face = PartialFace::default();
         face.exterior.write().surface = Partial::from(surface.clone());
-        face.update_exterior_as_polygon_from_points([a, b, c, d, e]);
+        face.exterior
+            .write()
+            .update_as_polygon_from_points([a, b, c, d, e]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
diff --git a/crates/fj-kernel/src/builder/face.rs b/crates/fj-kernel/src/builder/face.rs
index fb5e74f61..33ec5f27d 100644
--- a/crates/fj-kernel/src/builder/face.rs
+++ b/crates/fj-kernel/src/builder/face.rs
@@ -1,76 +1,21 @@
-use fj_math::Point;
-
 use crate::{
-    objects::HalfEdge,
+    objects::Cycle,
     partial::{Partial, PartialCycle, PartialFace},
 };
 
-use super::CycleBuilder;
-
 /// Builder API for [`PartialFace`]
 pub trait FaceBuilder {
-    /// Update the face exterior as a polygon from the provided points
-    fn update_exterior_as_polygon_from_points(
-        &mut self,
-        points: impl IntoIterator<Item = impl Into<Point<2>>>,
-    ) -> Vec<Partial<HalfEdge>>;
-
-    /// Update the face exterior as a polygon
-    fn update_exterior_as_polygon(&mut self);
-
-    /// Update the face exterior as a triangle, from 3D points
-    ///
-    /// Uses the three points to infer a plane that is used as the surface.
-    ///
-    /// # Implementation Note
-    ///
-    /// This method is probably just temporary, and will be generalized into a
-    /// "update as polygon from global points" method sooner or later. For now,
-    /// I didn't want to deal with the question of how to infer the surface, and
-    /// how to handle points that don't fit that surface.
-    fn update_exterior_as_triangle_from_global_points(
-        &mut self,
-        points: [impl Into<Point<3>>; 3],
-    ) -> [Partial<HalfEdge>; 3];
-
-    /// Add an interior polygon, from the provided points
-    fn add_interior_polygon_from_points(
-        &mut self,
-        points: impl IntoIterator<Item = impl Into<Point<2>>>,
-    );
+    /// Add an interior cycle
+    fn add_interior(&mut self) -> Partial<Cycle>;
 }
 
 impl FaceBuilder for PartialFace {
-    fn update_exterior_as_polygon_from_points(
-        &mut self,
-        points: impl IntoIterator<Item = impl Into<Point<2>>>,
-    ) -> Vec<Partial<HalfEdge>> {
-        self.exterior.write().update_as_polygon_from_points(points)
-    }
-
-    fn update_exterior_as_polygon(&mut self) {
-        self.exterior.write().update_as_polygon()
-    }
-
-    fn update_exterior_as_triangle_from_global_points(
-        &mut self,
-        points_global: [impl Into<Point<3>>; 3],
-    ) -> [Partial<HalfEdge>; 3] {
-        self.exterior
-            .write()
-            .update_as_triangle_from_global_points(points_global)
-    }
-
-    fn add_interior_polygon_from_points(
-        &mut self,
-        points: impl IntoIterator<Item = impl Into<Point<2>>>,
-    ) {
-        let mut cycle = PartialCycle {
+    fn add_interior(&mut self) -> Partial<Cycle> {
+        let cycle = Partial::from_partial(PartialCycle {
             surface: self.exterior.read().surface.clone(),
             ..Default::default()
-        };
-        cycle.update_as_polygon_from_points(points);
-
-        self.interiors.push(Partial::from_partial(cycle));
+        });
+        self.interiors.push(cycle.clone());
+        cycle
     }
 }
diff --git a/crates/fj-kernel/src/builder/sketch.rs b/crates/fj-kernel/src/builder/sketch.rs
index 57ca036fb..890804b2a 100644
--- a/crates/fj-kernel/src/builder/sketch.rs
+++ b/crates/fj-kernel/src/builder/sketch.rs
@@ -5,7 +5,7 @@ use crate::{
     partial::{Partial, PartialFace, PartialSketch},
 };
 
-use super::FaceBuilder;
+use super::CycleBuilder;
 
 /// Builder API for [`PartialSketch`]
 pub trait SketchBuilder {
@@ -25,7 +25,7 @@ impl SketchBuilder for PartialSketch {
     ) {
         let mut face = PartialFace::default();
         face.exterior.write().surface = surface.into();
-        face.update_exterior_as_polygon_from_points(points);
+        face.exterior.write().update_as_polygon_from_points(points);
 
         self.faces.extend([Partial::from_partial(face)]);
     }
diff --git a/crates/fj-kernel/src/validate/face.rs b/crates/fj-kernel/src/validate/face.rs
index d94bd4325..2aabcda3e 100644
--- a/crates/fj-kernel/src/validate/face.rs
+++ b/crates/fj-kernel/src/validate/face.rs
@@ -122,12 +122,12 @@ mod tests {
         let valid = {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon_from_points([
+            face.exterior.write().update_as_polygon_from_points([
                 [0., 0.],
                 [3., 0.],
                 [0., 3.],
             ]);
-            face.add_interior_polygon_from_points([
+            face.add_interior().write().update_as_polygon_from_points([
                 [1., 1.],
                 [1., 2.],
                 [2., 1.],
@@ -164,12 +164,12 @@ mod tests {
         let valid = {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon_from_points([
+            face.exterior.write().update_as_polygon_from_points([
                 [0., 0.],
                 [3., 0.],
                 [0., 3.],
             ]);
-            face.add_interior_polygon_from_points([
+            face.add_interior().write().update_as_polygon_from_points([
                 [1., 1.],
                 [1., 2.],
                 [2., 1.],