Math III · G-GMD.4

Identifying Cross-Sections and Solids Generated by Rotating Two-Dimensional Objects

Cross-sections and rotations teach students to see three-dimensional structure from slices and motion, a core skill in design, engineering, medicine, and visualization.

Concept Geometry

Domain Geometric Measurement and Dimension

Read time 5 minutes

What this learning objective is really asking you to learn

This objective asks students to visualize two related geometric ideas: cross-sections of three-dimensional objects and solids generated by rotating two-dimensional objects.

A cross-section is the shape made when a plane slices through a three-dimensional solid. If you slice a cylinder horizontally, the cross-section is a circle. If you slice it vertically through its center, the cross-section is a rectangle. If you slice a cone parallel to its base, the cross-section is a circle. If you slice a cone at an angle, the cross-section may be an ellipse, parabola, or hyperbola depending on the slice. If you slice a rectangular prism parallel to one face, you get a rectangle. If you slice it diagonally, the cross-section can be a different polygon.

The second idea is solids generated by rotation. If a two-dimensional shape is rotated around an axis, it sweeps out a three-dimensional solid. Rotating a rectangle around one side creates a cylinder. Rotating a right triangle around one leg creates a cone. Rotating a semicircle around its diameter creates a sphere. Rotating a circle around an external axis can create a torus-like shape.

This objective is about spatial reasoning. Students must imagine how a flat slice intersects a solid or how a flat shape moves through space to create a solid. It is not primarily about memorizing volume formulas. It is about seeing structure.

These skills are difficult because the page or screen is two-dimensional while the object is three-dimensional. Students need diagrams, dynamic rotations, physical models, or interactive tools to build the mental image.

Why students should learn this math

Students should learn cross-sections and rotations because three-dimensional reasoning matters in many real fields. Engineers design parts by thinking about slices, surfaces, and rotations. Architects interpret floor plans and elevations as cross-sections of buildings. Doctors read CT scans and MRIs as cross-sectional images of the body. Manufacturers create objects using lathes, molds, and rotational symmetry. Computer graphics render 3D objects from geometric rules. Chefs slice foods and see cross-sections. Geologists interpret layers of earth through sectional views.

Cross-sections are a way of understanding hidden structure. A loaf of bread, a tree trunk, a pipe, a bone, a building, and a machine part can all be understood by slicing. The slice reveals internal shape. In medicine, cross-sectional imaging literally saves lives by allowing doctors to see inside the body without cutting it open.

Solids of revolution are equally practical. Many manufactured objects are made by rotating profiles: bowls, bottles, vases, pipes, wheels, cups, and machine parts. If the profile is known, the solid can be understood. This connects directly to design and fabrication.

This objective also prepares students for advanced mathematics. In calculus, volumes of solids of revolution are computed by rotating regions around axes. Cross-sections are used to calculate volumes by slicing. Students do not need calculus here, but they are building the spatial foundation.

The “why” is that 3D geometry is not just about formulas. It is about seeing how solids are built, sliced, and generated.

The historical machinery: slicing and rotating in geometry

Classical geometry studied solids such as spheres, cones, cylinders, pyramids, and prisms. Conic sections — circles, ellipses, parabolas, and hyperbolas — were historically understood as slices of cones. This is one of the oldest and richest examples of cross-section thinking.

Solids of revolution also have deep mathematical history. A circle rotated in space, a region swept around an axis, or a curve generating a surface are all foundational ideas in geometry and later calculus. Many practical objects have rotational symmetry because rotating shapes are easy to manufacture and structurally useful.

Modern imaging and design technologies have made slicing and rotating even more important. CT scans use slices. CAD software builds solids from sketches, extrusions, and revolutions. 3D printing often constructs objects layer by layer, essentially using cross-sections.

The historical lesson is that visualizing slices and generated solids is a core geometric skill, not a side topic.

Where this fits in the big map of mathematics

This objective follows the trigonometric modeling sequence and begins a geometry transition. It connects Math III functions and spatial geometry.

It connects backward to volume and surface-area work in Math II. Students already studied cylinders, cones, spheres, and prisms. Now they analyze how those solids can be sliced or generated.

It connects to conic sections in Objective 170. Conics can be understood as cross-sections of cones and as equations in the coordinate plane.

It connects to calculus. Volumes by slicing and solids of revolution are major calculus applications.

It connects to engineering, medicine, architecture, manufacturing, and computer graphics.

The big-map role is spatial visualization. Students learn to move between 2D and 3D thinking.

How to execute the skill technically

For cross-sections, ask:

What solid is being sliced?
What is the plane's orientation?
Is the slice parallel to a face or base?
Does the slice pass through the center?
What 2D shape is formed by the intersection?

Examples:

Horizontal slice of a cylinder: circle.
Vertical slice through center of a cylinder: rectangle.
Slice of a sphere by any plane: circle, if it intersects the sphere.
Slice of a cube parallel to a face: square.
Diagonal slice of a cube: rectangle, triangle, or hexagon depending on the plane.
Slice of a cone parallel to base: circle.

For solids of revolution, ask:

What 2D figure is rotating?
What axis is it rotating around?
What path do points trace?
What solid is swept out?

Examples:

Rectangle rotated around one side: cylinder.
Right triangle rotated around one leg: cone.
Semicircle rotated around diameter: sphere.
Circle rotated around a line outside the circle: torus-like solid.

Students should sketch the starting figure, the axis, and the swept path.

Worked example: rotating a rectangle

Take a rectangle with height 5 and width 3. Rotate it around one of its vertical sides.

Each horizontal segment of the rectangle sweeps out a circle. The full rectangle sweeps out a cylinder. The cylinder has height 5 and radius 3, because the width of the rectangle becomes the radius of rotation.

If the same rectangle is rotated around a horizontal side, the resulting cylinder has radius 5 and length 3. The axis matters. The same 2D shape can generate different solids depending on the axis.

Worked example: slicing a cone

A cone sliced parallel to its base produces a circle. The farther the slice is from the tip and closer to the base, the larger the circle. A vertical slice through the cone's axis produces an isosceles triangle. An angled slice can produce an ellipse. These slice types connect directly to conic sections.

This example prepares students for Objective 170, where conics are studied through equations and graphs.

Problem Library

Problems in the App From This Objective

144 problems across 12 archetypes in the app.

match slice shape to base.

12 problems Warmup Practice Mixed Review Assessment

Problem 1

Identify cross-section of prism rectangular prism cut parallel to its base.

Problem 2

Identify cross-section of prism triangular prism cut parallel to its base.

Problem 3

Identify cross-section of prism hexagonal prism cut parallel to its base.

Problem 4

Identify cross-section of prism pentagonal prism cut parallel to its base.

Problem 5

Identify cross-section of prism octagonal prism cut parallel to its base.

Problem 6

Identify cross-section of prism square prism cut parallel to its base.

Problem 7

Identify cross-section of prism cube cut parallel to its base.

Problem 8

Identify cross-section of prism rhombic prism cut parallel to its base.

Problem 9

Identify cross-section of prism trapezoidal prism cut parallel to its base.

Problem 10

Identify cross-section of prism parallelogram prism cut parallel to its base.

Problem 11

Identify cross-section of prism heptagonal prism cut parallel to its base.

Problem 12

Identify cross-section of prism decagonal prism cut parallel to its base.

Open in simulator

visualize vertical slice.

12 problems Warmup Practice Mixed Review Assessment

Problem 13

Identify cross-section of prism rectangular prism cut perpendicular to its base.

Problem 14

Identify cross-section of prism triangular prism through height cut perpendicular to its base.

Problem 15

Identify cross-section of prism right prism with vertical cut across base cut perpendicular to its base.

Problem 16

Identify cross-section of prism triangular prism cut perpendicular to its base.

Problem 17

Identify cross-section of prism square prism cut perpendicular to its base.

Problem 18

Identify cross-section of prism pentagonal prism cut perpendicular to its base.

Problem 19

Identify cross-section of prism hexagonal prism cut perpendicular to its base.

Problem 20

Identify cross-section of prism octagonal prism cut perpendicular to its base.

Problem 21

Identify cross-section of prism trapezoidal prism cut perpendicular to its base.

Problem 22

Identify cross-section of prism rhombic prism cut perpendicular to its base.

Open in simulator

Problem 23

Identify cross-section of prism parallelogram prism cut perpendicular to its base.

Problem 24

Identify cross-section of prism cylinder cut perpendicular to its base.

distinguish circular, rectangular, and elliptical slices.

12 problems Warmup Practice Mixed Review Assessment

Problem 25

Identify cross-section of cylinder cut parallel to base.

Problem 26

Identify cross-section of cylinder cut perpendicular to base through diameter.

Problem 27

Identify cross-section of cylinder cut angled not parallel to base.

Problem 28

Identify cross-section of cylinder cut horizontally.

Problem 29

Identify cross-section of cylinder cut vertically.

Problem 30

Identify cross-section of cylinder cut diagonally.

Problem 31

Identify cross-section of cylinder cut parallel to the top surface.

Problem 32

Identify cross-section of cylinder cut perpendicular to the circular bases.

Problem 33

Identify cross-section of cylinder cut at an oblique angle to the base.

Problem 34

Identify cross-section of cylinder cut through the cylinder, maintaining constant radius.

Problem 35

Identify cross-section of cylinder cut along its height.

Problem 36

Identify cross-section of cylinder cut at an angle of 45 degrees to the base.

Open in simulator

recognize circle, ellipse, parabola-like, triangle slices.

12 problems Warmup Practice Mixed Review Assessment

Problem 37

Identify cross-section of cone cut parallel to base.

Open in simulator

Problem 38

Identify cross-section of cone cut through vertex and diameter of base.

Problem 39

Identify cross-section of cone cut angled across one side.

Problem 40

Identify cross-section of cone cut oblique, cutting all generators of one nappe.

Problem 41

Identify cross-section of cone cut parallel to one generator, not passing through the vertex.

Problem 42

Identify cross-section of cone cut vertical, not passing through the vertex.

Problem 43

Identify cross-section of cone cut containing the cone's axis of symmetry.

Problem 44

Identify cross-section of cone cut perpendicular to the axis of symmetry, not through the vertex.

Problem 45

Identify cross-section of cone cut tilted, cutting through the cone's body without passing through the vertex or being parallel to a generator.

Problem 46

Identify cross-section of cone cut angled such that the plane is parallel to a slant edge of the cone.

Problem 47

Identify cross-section of cone cut cutting both nappes of a double cone.

Problem 48

Identify cross-section of cone cut passing through the apex and intersecting the base.

know plane slices are circles.

12 problems Warmup Practice Mixed Review Assessment

Problem 49

Identify cross-section of sphere cut through center.

Problem 50

Identify cross-section of sphere cut off-center plane.

Problem 51

Identify cross-section of sphere cut tangent plane.

Problem 52

Identify cross-section of sphere cut by a plane passing through its north pole and south pole.

Problem 53

Identify cross-section of sphere cut by a plane that is parallel to the equatorial plane but not through the equator.

Problem 54

Identify cross-section of sphere cut by a plane that just touches its surface at one point.

Problem 55

Identify cross-section of sphere cut by a plane that does not intersect the sphere.

Open in simulator

Problem 56

Identify cross-section of sphere cut by a plane passing through its diameter.

Problem 57

Identify cross-section of sphere cut by a plane that is equidistant from the center and the surface.

Problem 58

Identify cross-section of sphere cut by a plane slicing off a cap.

Problem 59

Identify cross-section of sphere cut by a plane containing its center.

Problem 60

Identify cross-section of sphere cut by a plane perpendicular to a radius at its midpoint.

visualize cylinder generation.

12 problems Warmup Practice Mixed Review Assessment

Problem 61

Identify solid generated by rotating rectangle rectangle rotated about one side.

Problem 62

Identify solid generated by rotating rectangle rectangle rotated about a parallel external line.

Problem 63

Identify solid generated by rotating rectangle rectangle rotated about its centerline parallel to a side.

Problem 64

Identify solid generated by rotating rectangle rectangle rotated about its longer side.

Problem 65

Identify solid generated by rotating rectangle rectangle rotated about its shorter side.

Problem 66

Identify solid generated by rotating rectangle rectangle rotated about the line containing its length.

Problem 67

Identify solid generated by rotating rectangle rectangle rotated about the line containing its width.

Problem 68

Identify solid generated by rotating rectangle rectangle rotated about an external line parallel to its length.

Problem 69

Identify solid generated by rotating rectangle rectangle rotated about an external line parallel to its width.

Problem 70

Identify solid generated by rotating rectangle rectangle rotated about an internal line parallel to a side, not passing through its center.

Problem 71

Identify solid generated by rotating rectangle rectangle rotated about its centerline parallel to its length.

Open in simulator

Problem 72

Identify solid generated by rotating rectangle rectangle rotated about its centerline parallel to its width.

visualize cone generation.

12 problems Warmup Practice Mixed Review Assessment

Problem 73

Identify solid generated by rotating right triangle about one leg.

Open in simulator

Problem 74

Identify solid generated by rotating right triangle about the other leg.

Problem 75

Identify solid generated by rotating right triangle about hypotenuse.

Problem 76

Identify solid generated by rotating right triangle around one of its legs.

Problem 77

Identify solid generated by rotating right triangle by revolving it about a leg.

Problem 78

Identify solid generated by rotating right triangle about one of its perpendicular sides.

Problem 79

Identify solid generated by rotating right triangle around its remaining leg.

Problem 80

Identify solid generated by rotating right triangle by revolving it about the second leg.

Problem 81

Identify solid generated by rotating right triangle about the leg not used as the first axis.

Problem 82

Identify solid generated by rotating right triangle around its hypotenuse.

Problem 83

Identify solid generated by rotating right triangle by revolving it about its longest side.

Problem 84

Identify solid generated by rotating right triangle about the side opposite the right angle.

visualize sphere/torus-style generation where appropriate.

12 problems Warmup Practice Mixed Review Assessment

Problem 85

Identify solid generated by rotating circle or semicircle semicircle about its diameter.

Problem 86

Identify solid generated by rotating circle or semicircle circle about an external line in its plane.

Problem 87

Identify solid generated by rotating circle or semicircle circle about one of its diameters.

Problem 88

Identify solid generated by rotating circle or semicircle circle about an axis passing through its center.

Open in simulator

Problem 89

Identify solid generated by rotating circle or semicircle semicircle about its straight edge.

Problem 90

Identify solid generated by rotating circle or semicircle circle about an axis that does not intersect it.

Problem 91

Identify solid generated by rotating circle or semicircle circle about a line completely outside its boundary.

Problem 92

Identify solid generated by rotating circle or semicircle circle about a line tangent to its perimeter.

Problem 93

Identify solid generated by rotating circle or semicircle circle about a line that touches its circumference at exactly one point.

Problem 94

Identify solid generated by rotating circle or semicircle circle about a line segment that connects two points on its circumference.

Problem 95

Identify solid generated by rotating circle or semicircle circle about a chord that is not a diameter.

Problem 96

Identify solid generated by rotating circle or semicircle semicircle about a line in its plane that does not touch its arc or diameter.

infer generated shape.

12 problems Warmup Practice Mixed Review Assessment

Problem 97

Match 2D region and axis rectangle about one side to generated 3D solid.

Open in simulator

Problem 98

Match 2D region and axis right triangle about a leg to generated 3D solid.

Problem 99

Match 2D region and axis semicircle about diameter to generated 3D solid.

Problem 100

Match 2D region and axis composite rectangle plus triangle about shared axis to generated 3D solid.

Problem 101

Match 2D region and axis circle about an external axis to generated 3D solid.

Problem 102

Match 2D region and axis right trapezoid about its non-parallel side perpendicular to bases to generated 3D solid.

Problem 103

Match 2D region and axis rectangle about an external parallel axis to generated 3D solid.

Problem 104

Match 2D region and axis isosceles triangle about its altitude to generated 3D solid.

Problem 105

Match 2D region and axis right triangle about its hypotenuse to generated 3D solid.

Problem 106

Match 2D region and axis quarter circle about a straight edge to generated 3D solid.

Problem 107

Match 2D region and axis rectangle with a semicircle on top about the rectangle's base to generated 3D solid.

Problem 108

Match 2D region and axis right trapezoid about its longer parallel side to generated 3D solid.

represent slice shape and relative size.

12 problems Warmup Practice Mixed Review Assessment

Problem 109

Sketch cross-section from described cut plane parallel to base of cone near top.

Problem 110

Sketch cross-section from described cut vertical plane through center of cylinder.

Problem 111

Sketch cross-section from described cut off-center slice of sphere.

Problem 112

Sketch cross-section from described cut plane parallel to base of cone near base.

Open in simulator

Problem 113

Sketch cross-section from described cut plane parallel to base of cylinder.

Problem 114

Sketch cross-section from described cut plane through center of sphere.

Problem 115

Sketch cross-section from described cut plane parallel to the largest face of a rectangular prism.

Problem 116

Sketch cross-section from described cut plane parallel to a face of a cube.

Problem 117

Sketch cross-section from described cut plane parallel to base of square pyramid near apex.

Problem 118

Sketch cross-section from described cut plane perpendicular to base of cone and through its apex.

Problem 119

Sketch cross-section from described cut plane slanted through a cylinder, not parallel to base and not vertical.

Problem 120

Sketch cross-section from described cut plane cutting a cube through the midpoints of six edges.

transform 2D profile around an axis.

12 problems Warmup Practice Mixed Review Assessment

Problem 121

Sketch or describe solid of revolution from profile line segment from axis to radius over fixed height.

Problem 122

Sketch or describe solid of revolution from profile slanted segment from axis to base radius.

Problem 123

Sketch or describe solid of revolution from profile semicircular arc around diameter.

Problem 124

Sketch or describe solid of revolution from profile piecewise profile with rectangle then triangle.

Open in simulator

Problem 125

Sketch or describe solid of revolution from profile quarter circle with one straight edge on the axis.

Problem 126

Sketch or describe solid of revolution from profile right trapezoid with its vertical side on the axis.

Problem 127

Sketch or describe solid of revolution from profile rectangle offset from the axis, rotated around an axis parallel to its length.

Problem 128

Sketch or describe solid of revolution from profile ellipse rotated around its major axis.

Problem 129

Sketch or describe solid of revolution from profile parabolic segment rotated around its axis of symmetry.

Problem 130

Sketch or describe solid of revolution from profile rectangle with a semicircle on top, rotated around the rectangle's central vertical axis.

Problem 131

Sketch or describe solid of revolution from profile isosceles triangle rotated around its base.

Problem 132

Sketch or describe solid of revolution from profile circle rotated around an axis external to it.

catch impossible slice shapes and wrong axis rotations.

12 problems Warmup Practice Mixed Review Assessment

Problem 133

Correct the cross-section or rotation visualization error in sphere cut by plane gives ellipse.

Problem 134

Correct the cross-section or rotation visualization error in rectangle rotated about side gives cone.

Problem 135

Correct the cross-section or rotation visualization error in cylinder cut parallel to base gives rectangle.

Problem 136

Correct the cross-section or rotation visualization error in triangle rotated about leg gives sphere.

Problem 137

Correct the cross-section or rotation visualization error in cone cut parallel to base gives triangle.

Problem 138

Correct the cross-section or rotation visualization error in cylinder cut diagonally gives circle.

Problem 139

Correct the cross-section or rotation visualization error in semicircle rotated about its diameter gives cylinder.

Problem 140

Correct the cross-section or rotation visualization error in right triangle rotated about its hypotenuse gives a cone.

Problem 141

Correct the cross-section or rotation visualization error in cube cut by plane gives circle.

Open in simulator

Problem 142

Correct the cross-section or rotation visualization error in circle rotated about a tangent line gives a sphere.

Problem 143

Correct the cross-section or rotation visualization error in rectangular prism cut parallel to a face gives a triangle.

Problem 144

Correct the cross-section or rotation visualization error in rectangle rotated about a line parallel to one side but outside the rectangle gives a cylinder.