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Cross-Sections &
Rotations of 3D Objects

Learn to identify 2D shapes formed by slicing 3D solids, and discover how rotating 2D figures generates 3D objects.

Geometry

What Is a Cross-Section?

A cross-section is the 2D shape you get when a plane intersects (slices through) a 3D object. Imagine cutting a loaf of bread: each slice reveals a flat shape.

Cross-Section
Circle
Cylinder
Result: 2D shape

What Is a Solid of Revolution?

A solid of revolution is the 3D object formed when a 2D shape is rotated around an axis through a full \(360^\circ\). The 2D shape sweeps out a solid in space.

Axis
Rectangle
\(360^\circ\)
Cylinder
Solid of Revolution

Why Cross-Sections & Rotations Matter

Engineering
Engineers analyze cross-sections to understand structural strength of beams, pipes, and bridges.
Medicine
CT and MRI scans produce cross-sectional images that doctors use to diagnose conditions.
Manufacturing
Lathes and pottery wheels create solids of revolution: bowls, vases, and axles.

Cross-Section Principle

Cone
Circle
Rectangular Prism
Triangle
Sphere
Always a Circle

Solids of Revolution Principle

Rectangle → Cylinder
A rectangle rotated about one of its sides produces a cylinder.
Right Triangle → Cone
A right triangle rotated about one of its legs produces a cone.
Semicircle → Sphere
A semicircle rotated about its diameter produces a sphere.

Cross-Sections vs Solids of Revolution

Cross-Sections Solids of Revolution
Direction 3D → 2D 2D → 3D
Action Slice a solid with a plane Rotate a shape around an axis
Result A flat 2D shape A 3D solid
Key Variable Angle/position of the cut Axis of rotation
Real Example MRI scan of the brain Pottery on a wheel

How to Identify Cross-Sections

1
Identify the 3D solid. Name the shape: sphere, cylinder, cone, prism, pyramid, etc.
2
Determine the cutting plane's orientation. Is it horizontal, vertical, diagonal, or at some other angle?
3
Trace where the plane meets the solid's surface. Follow the boundary of intersection to reveal the 2D shape.
4
Name the resulting shape. Common results: circle, ellipse, triangle, rectangle, trapezoid, or polygon.

Cross-Sections of a Cylinder

Circle
Horizontal cut
Rectangle
Vertical cut
Ellipse
Diagonal cut

Cross-Sections of a Cone

Circle
Horizontal
Triangle
Vertical (through apex)
Ellipse
Angled cut

Example: Cross-Section of a Rectangular Prism

A rectangular prism (box) is cut by a diagonal plane that passes through two opposite edges. What shape is the cross-section?

Rectangle

Example: Rotating a Right Triangle

A right triangle with legs of length \(r\) and \(h\) is rotated \(360^\circ\) about the leg of length \(h\). What 3D solid is generated?

Axis
\(h\)
\(r\)
\(360^\circ\)
Cone
\(h\)
\(r\)

Example: Rotating a Shape Off-Axis

A rectangle that does not touch the axis of rotation produces a hollow cylinder.

Axis
gap
Rect
Hollow Cylinder
Hollow center from the gap

2D to 3D Rotation Reference

2D Shape Axis Position 3D Solid
Rectangle Along one side Cylinder
Right triangle Along one leg Cone
Semicircle Along the diameter Sphere
Rectangle Offset (gap from axis) Hollow Cylinder
Circle Offset (gap from axis) Torus
Trapezoid Along its height (perpendicular leg) Frustum (truncated cone)

Common Mistakes

Ignoring the Cut Angle
Assuming every cross-section of a cylinder is a circle. A vertical cut through the axis gives a rectangle, not a circle.
Wrong Axis of Rotation
Rotating a right triangle about the hypotenuse does not make a simple cone: it produces a double cone (bicone).
Forgetting the Gap
If a shape is offset from the axis, the rotation produces a hollow solid (a hollow cylinder), not a filled one like a cylinder.

Question 1 of 5

A plane cuts horizontally through a cone, parallel to its base. What is the shape of the cross-section?

Question 2 of 5

A semicircle is rotated \(360^\circ\) about its diameter. What 3D solid is formed?

Question 3 of 5

A plane cuts vertically through the center of a sphere. What shape is the cross-section?

Question 4 of 5

A rectangle is placed 3 units away from an axis (not touching it) and rotated \(360^\circ\). What solid is formed?

Question 5 of 5

A plane cuts diagonally through a rectangular prism, passing through three faces. What is a possible cross-section?