diff --git a/schemas/mathematicalShape/centroidalPyramid.schema.tpl.json b/schemas/mathematicalShape/centroidalPyramid.schema.tpl.json
new file mode 100644
index 000000000..5f063b708
--- /dev/null
+++ b/schemas/mathematicalShape/centroidalPyramid.schema.tpl.json
@@ -0,0 +1,25 @@
+{
+  "_type": "sands:CentroidalPyramid",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "apexBaseDistance",
+    "baseShape"
+  ],
+  "properties": {
+    "apexBaseDistance": {
+      "_instruction": "Enter the perpendicular distance from the centroid of the base plane to the apex of this centroidal pyramid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "baseShape": {
+      "_instruction": "Enter the two-dimensional base shape of this centroidal pyramid.",
+      "_embeddedCategories": [
+        "simplePolygon"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/circle.schema.tpl.json b/schemas/mathematicalShape/circle.schema.tpl.json
index f6751aa91..7ad06a7d1 100644
--- a/schemas/mathematicalShape/circle.schema.tpl.json
+++ b/schemas/mathematicalShape/circle.schema.tpl.json
@@ -1,7 +1,8 @@
 {
   "_type": "sands:Circle",
   "_categories": [
-    "mathematicalShape"
+    "mathematicalShape",
+    "mathematical2DShape"
   ],
   "required": [
     "radius"
diff --git a/schemas/mathematicalShape/circularSector.schema.tpl.json b/schemas/mathematicalShape/circularSector.schema.tpl.json
new file mode 100644
index 000000000..1a9d8a244
--- /dev/null
+++ b/schemas/mathematicalShape/circularSector.schema.tpl.json
@@ -0,0 +1,25 @@
+{
+  "_type": "sands:CircularSector",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape"
+  ],
+  "required": [
+    "centralAngle",
+    "radius"
+  ],
+  "properties": {
+    "centralAngle": {
+      "_instruction": "Enter the central angle of this circular sector.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "radius": {
+      "_instruction": "Enter the radius of this circular sector.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/cube.schema.tpl.json b/schemas/mathematicalShape/cube.schema.tpl.json
new file mode 100644
index 000000000..a62acd121
--- /dev/null
+++ b/schemas/mathematicalShape/cube.schema.tpl.json
@@ -0,0 +1,18 @@
+{
+  "_type": "sands:Cube",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "sideLength"
+  ],
+  "properties": {
+    "sideLength": {
+      "_instruction": "Enter the common length of the sides of this cube.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/ellipse.schema.tpl.json b/schemas/mathematicalShape/ellipse.schema.tpl.json
index e76a81bba..de537f6da 100644
--- a/schemas/mathematicalShape/ellipse.schema.tpl.json
+++ b/schemas/mathematicalShape/ellipse.schema.tpl.json
@@ -1,21 +1,18 @@
 {
   "_type": "sands:Ellipse",
   "_categories": [
-    "mathematicalShape"
+    "mathematicalShape",
+    "mathematical2DShape"
   ],
   "required": [
-    "semiMinorAxis",
-    "semiMajorAxis"
+    "diameters"
   ],
   "properties": {
-    "semiMinorAxis": {
-      "_instruction": "Enter the length of the semi-major axis of this ellipse.",
-      "_embeddedTypes": [
-        "core:QuantitativeValue"
-      ]
-    },
-    "semiMajorAxis": {
-      "_instruction": "Enter the length of the semi-minor axis of this ellipse.",
+    "diameters": {
+      "type": "array",      
+      "minItems": 2,
+      "maxItems": 2,
+      "_instruction": "Enter the lengths of the major and minor diameters of this ellipse.",
       "_embeddedTypes": [
         "core:QuantitativeValue"
       ]
diff --git a/schemas/mathematicalShape/ellipsoid.schema.tpl.json b/schemas/mathematicalShape/ellipsoid.schema.tpl.json
new file mode 100644
index 000000000..dfdf049c4
--- /dev/null
+++ b/schemas/mathematicalShape/ellipsoid.schema.tpl.json
@@ -0,0 +1,32 @@
+{
+  "_type": "sands:Ellipsoid",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "intermediateDiameter",
+    "majorDiameter",
+    "minorDiameter"
+  ],
+  "properties": {
+    "intermediateDiameter": {
+      "_instruction": "Enter the intermediate diameter of this ellipsoid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "majorDiameter": {
+      "_instruction": "Enter the major diameter of this ellipsoid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "minorDiameter": {
+      "_instruction": "Enter the minor diameter of this ellipsoid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/equilateralTriangle.schema.tpl.json b/schemas/mathematicalShape/equilateralTriangle.schema.tpl.json
new file mode 100644
index 000000000..8a8891bcb
--- /dev/null
+++ b/schemas/mathematicalShape/equilateralTriangle.schema.tpl.json
@@ -0,0 +1,19 @@
+{
+  "_type": "sands:EquilateralTriangle",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "sideLength"
+  ],
+  "properties": {
+    "sideLength": {
+      "_instruction": "Enter the common length of the sides of this equilateral triangle.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/frustum.schema.tpl.json b/schemas/mathematicalShape/frustum.schema.tpl.json
new file mode 100644
index 000000000..95442ce04
--- /dev/null
+++ b/schemas/mathematicalShape/frustum.schema.tpl.json
@@ -0,0 +1,32 @@
+{
+  "_type": "sands:Frustum",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "baseDistance",
+    "majorBaseShape",
+    "minorBaseScale"
+  ],
+  "properties": {
+    "baseDistance": {
+      "_instruction": "Enter the perpendicular distance between the centered major and minor base planes of this frustum.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "majorBaseShape": {
+      "_instruction": "Enter the major two-dimensional base shape of this frustum.",
+      "_embeddedCategories": [
+        "mathematical2DShape"
+      ]
+    },
+    "minorBaseScale": {
+      "type": "float",      
+      "exclusiveMinimum": 0,
+      "exclusiveMaximum": 1,
+      "_instruction": "Enter the ratio of the smaller to the larger base size of this frustum."
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/isoscelesTriangle.schema.tpl.json b/schemas/mathematicalShape/isoscelesTriangle.schema.tpl.json
new file mode 100644
index 000000000..b664ded50
--- /dev/null
+++ b/schemas/mathematicalShape/isoscelesTriangle.schema.tpl.json
@@ -0,0 +1,26 @@
+{
+  "_type": "sands:IsoscelesTriangle",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "apexAngle",
+    "legLength"
+  ],
+  "properties": {
+    "apexAngle": {
+      "_instruction": "Enter the angle at the apex between the two equal sides of this isosceles triangle.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "legLength": {
+      "_instruction": "Enter the common length of the two equal sides meeting at the apex of this isosceles triangle.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/kite.schema.tpl.json b/schemas/mathematicalShape/kite.schema.tpl.json
new file mode 100644
index 000000000..5a68d47b5
--- /dev/null
+++ b/schemas/mathematicalShape/kite.schema.tpl.json
@@ -0,0 +1,29 @@
+{
+  "_type": "sands:Kite",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "legLengths",
+    "symmetryDiagonalLength"
+  ],
+  "properties": {
+    "legLengths": {
+      "type": "array",      
+      "minItems": 2,
+      "maxItems": 2,
+      "_instruction": "Enter the two common leg lengths (for the shorter and longer leg pairs) of this kite.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "symmetryDiagonalLength": {
+      "_instruction": "Enter the length of the symmetry diagonal of this kite.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    } 
+  }
+}
diff --git a/schemas/mathematicalShape/parallelogram.schema.tpl.json b/schemas/mathematicalShape/parallelogram.schema.tpl.json
new file mode 100644
index 000000000..3eaab6c72
--- /dev/null
+++ b/schemas/mathematicalShape/parallelogram.schema.tpl.json
@@ -0,0 +1,34 @@
+{
+  "_type": "sands:Parallelogram",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "baseDistance",
+    "baseLength",
+    "interiorAngle"
+  ],
+  "properties": {
+    "baseDistance": {
+      "_instruction": "Enter the perpendicular distance between the parallel lines of the base of this parallelogram.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "baseLength": {
+      "_instruction": "Enter the uniform length of the bottom and top sides (the base) of this parallelogram.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+
+    "interiorAngle": {
+      "_instruction": "Enter an interior angle between the base and an adjacent side of this parallelogram.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/rectangle.schema.tpl.json b/schemas/mathematicalShape/rectangle.schema.tpl.json
index 39862e077..454eeb431 100644
--- a/schemas/mathematicalShape/rectangle.schema.tpl.json
+++ b/schemas/mathematicalShape/rectangle.schema.tpl.json
@@ -1,21 +1,23 @@
 {
   "_type": "sands:Rectangle",
   "_categories": [
-    "mathematicalShape"
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
   ],
   "required": [
-    "length",
-    "width"
+    "majorSideLength",
+    "minorSideLength"
   ],
   "properties": {
-    "length": {
-      "_instruction": "Enter the length of this rectangle.",
+    "majorSideLength": {
+      "_instruction": "Enter the common length of the major sides of this rectangle.",
       "_embeddedTypes": [
         "core:QuantitativeValue"
       ]
     },
-    "width": {
-      "_instruction": "Enter the width of this rectangle.",
+    "minorSideLength": {
+      "_instruction": "Enter the common length of the minor sides of this rectangle.",
       "_embeddedTypes": [
         "core:QuantitativeValue"
       ]
diff --git a/schemas/mathematicalShape/regularPolygon.schema.tpl.json b/schemas/mathematicalShape/regularPolygon.schema.tpl.json
new file mode 100644
index 000000000..e2ebbe8dd
--- /dev/null
+++ b/schemas/mathematicalShape/regularPolygon.schema.tpl.json
@@ -0,0 +1,25 @@
+{
+  "_type": "sands:RegularPolygon",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "circumradius",
+    "numberOfSides"
+  ],
+  "properties": {
+    "circumradius": {
+      "_instruction": "Enter the common distance from the center to a vertex of this regular polygon.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "numberOfSides": {
+      "type": "integer",  
+      "minimum": 3,
+      "_instruction": "Enter the number of sides of this regular polygon."
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/rhombus.schema.tpl.json b/schemas/mathematicalShape/rhombus.schema.tpl.json
new file mode 100644
index 000000000..08ed6f076
--- /dev/null
+++ b/schemas/mathematicalShape/rhombus.schema.tpl.json
@@ -0,0 +1,22 @@
+{
+  "_type": "sands:Rhombus",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "diagonalLengths"
+  ],
+  "properties": {
+    "diagonalLengths": {
+      "type": "array",      
+      "minItems": 2,
+      "maxItems": 2,
+      "_instruction": "Enter the lengths of the two perpendicular diagonals of this rhombus.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/rightCone.schema.tpl.json b/schemas/mathematicalShape/rightCone.schema.tpl.json
new file mode 100644
index 000000000..fd0f05f76
--- /dev/null
+++ b/schemas/mathematicalShape/rightCone.schema.tpl.json
@@ -0,0 +1,26 @@
+{
+  "_type": "sands:RightCone",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "apexBaseDistance",
+    "baseShape"
+  ],
+  "properties": {
+    "apexBaseDistance": {
+      "_instruction": "Enter the perpendicular distance from the center of the base plane to the apex of this right cone.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "baseShape": {
+      "_instruction": "Enter the two-dimensional elliptic base shape of this right cone.",
+      "_embeddedTypes": [
+        "sands:Circle",
+        "sands:Ellipse"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/rightCylinder.schema.tpl.json b/schemas/mathematicalShape/rightCylinder.schema.tpl.json
new file mode 100644
index 000000000..601e9c1e6
--- /dev/null
+++ b/schemas/mathematicalShape/rightCylinder.schema.tpl.json
@@ -0,0 +1,26 @@
+{
+  "_type": "sands:RightCylinder",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "baseDistance",
+    "baseShape"
+  ],
+  "properties": {
+    "baseDistance": {
+      "_instruction": "Enter the perpendicular distance between the two base planes of this right cylinder.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "baseShape": {
+      "_instruction": "Enter the two-dimensional elliptic base shape of the two planes of this right cylinder.",
+      "_embeddedTypes": [
+        "sands:Circle",
+        "sands:Ellipse"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/rightPrism.schema.tpl.json b/schemas/mathematicalShape/rightPrism.schema.tpl.json
new file mode 100644
index 000000000..291c17217
--- /dev/null
+++ b/schemas/mathematicalShape/rightPrism.schema.tpl.json
@@ -0,0 +1,25 @@
+{
+  "_type": "sands:RightPrism",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "baseDistance",
+    "baseShape"
+  ],
+  "properties": {
+    "baseDistance": {
+      "_instruction": "Enter the perpendicular distance between the two base planes of this right prism.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "baseShape": {
+      "_instruction": "Enter the two-dimensional base shape of the two planes of this right prism.",
+      "_embeddedCategories": [
+        "simplePolygon"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/rightTriangle.schema.tpl.json b/schemas/mathematicalShape/rightTriangle.schema.tpl.json
new file mode 100644
index 000000000..96a7c18b7
--- /dev/null
+++ b/schemas/mathematicalShape/rightTriangle.schema.tpl.json
@@ -0,0 +1,22 @@
+{
+  "_type": "sands:RightTriangle",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "perpendicularLegLengths"
+  ],
+  "properties": {
+    "perpendicularLegLengths": {
+      "type": "array",      
+      "minItems": 2,
+      "maxItems": 2,
+      "_instruction": "Enter the lengths of the two perpendicular legs of this right triangle.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/sphere.schema.tpl.json b/schemas/mathematicalShape/sphere.schema.tpl.json
new file mode 100644
index 000000000..ff7055494
--- /dev/null
+++ b/schemas/mathematicalShape/sphere.schema.tpl.json
@@ -0,0 +1,18 @@
+{
+  "_type": "sands:Sphere",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "radius"
+  ],
+  "properties": {
+    "radius": {
+      "_instruction": "Enter the radius of this sphere.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/spheroid.schema.tpl.json b/schemas/mathematicalShape/spheroid.schema.tpl.json
new file mode 100644
index 000000000..39ff569b1
--- /dev/null
+++ b/schemas/mathematicalShape/spheroid.schema.tpl.json
@@ -0,0 +1,25 @@
+{
+  "_type": "sands:Spheroid",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical3DShape"
+  ],
+  "required": [
+    "equatorialDiameter",
+    "polarDiameter"
+  ],
+  "properties": {
+    "equatorialDiameter": {
+      "_instruction": "Enter the equatorial diameters of this spheroid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "polarDiameter": {
+      "_instruction": "Enter the polar diameter of this spheroid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/square.schema.tpl.json b/schemas/mathematicalShape/square.schema.tpl.json
new file mode 100644
index 000000000..2936baf25
--- /dev/null
+++ b/schemas/mathematicalShape/square.schema.tpl.json
@@ -0,0 +1,19 @@
+{
+  "_type": "sands:Square",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "sideLength"
+  ],
+  "properties": {
+    "sideLength": {
+      "_instruction": "Enter the common length of the sides of this square.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/trapezoid.schema.tpl.json b/schemas/mathematicalShape/trapezoid.schema.tpl.json
new file mode 100644
index 000000000..ea74c6c22
--- /dev/null
+++ b/schemas/mathematicalShape/trapezoid.schema.tpl.json
@@ -0,0 +1,29 @@
+{
+  "_type": "sands:Trapezoid",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "baseDistance",
+    "baseLengths"
+  ],
+  "properties": {
+    "baseDistance": {
+      "_instruction": "Enter the perpendicular distance between the parallel lines of the major and minor bases of this trapezoid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "baseLengths": {
+      "type": "array",      
+      "minItems": 2,
+      "maxItems": 2,
+      "_instruction": "Enter the lengths of the major and minor bases of this trapezoid.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}
diff --git a/schemas/mathematicalShape/triangle.schema.tpl.json b/schemas/mathematicalShape/triangle.schema.tpl.json
new file mode 100644
index 000000000..aa0081b6c
--- /dev/null
+++ b/schemas/mathematicalShape/triangle.schema.tpl.json
@@ -0,0 +1,29 @@
+{
+  "_type": "sands:Triangle",
+  "_categories": [
+    "mathematicalShape",
+    "mathematical2DShape",
+    "simplePolygon"
+  ],
+  "required": [
+    "apexAngle",
+    "legLengths"
+  ],
+  "properties": {
+    "apexAngle": {
+      "_instruction": "Enter the angle at the apex of this triangle.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    },
+    "legLengths": {
+      "type": "array",      
+      "minItems": 2,
+      "maxItems": 2,
+      "_instruction": "Enter the lengths of the two sides meeting at the apex of this triangle.",
+      "_embeddedTypes": [
+        "core:QuantitativeValue"
+      ]
+    }
+  }
+}