Cross-Sections and Solids of Revolution

Solid Geometry — Slicing Solids and Spinning Shapes

In this lesson:

• Identify cross-sections of common 3D solids
• Predict how cut orientation changes the shape
• Determine solids formed by rotating 2D shapes

What You Will Learn Today

After this lesson, you will:

1. Identify cross-sections of prisms, cylinders, cones, spheres, and pyramids
2. Predict how cross-section shape changes with cut orientation
3. Determine the 3D solid from rotating a 2D shape around an axis
4. Solve ACT problems on cross-sections and solids of revolution

Slicing a Solid Reveals Hidden Shapes

Imagine slicing a loaf of bread:

• Cut straight across — you see a circle
• Cut at an angle — you see an ellipse
• Same bread, different cut, different shape

A cross-section is the 2D shape revealed when a plane cuts through a 3D solid.

Defining Cross-Sections of Three-Dimensional Solids

A cross-section is the new face you see when you pull the two halves apart.

Cylinder Cross-Sections Change with Cut Direction

The same cylinder produces circles, rectangles, or ellipses.

Rectangular Prism Has Many Cross-Sections

A rectangular prism reveals different polygons:

• Horizontal cut (parallel to base) — rectangle
• Vertical cut (parallel to a face) — rectangle
• Diagonal cut (corner to corner) — triangle, pentagon, or hexagon

Flat-faced solids always produce straight-edged cross-sections.

Quick Check on Cylinder Cross-Sections

A plane cuts through a cylinder parallel to its axis.

What is the shape of the cross-section?

Think about it before advancing...

Cone Cross-Sections and Conic Section Family

A cone produces the famous conic sections:

• Perpendicular to axis — circle (smaller near apex)
• Through the apex — isosceles triangle
• Angled cut — ellipse, parabola, or hyperbola

Sphere and Pyramid Cross-Section Shapes

Sphere — every cut produces a circle:

• Through center → great circle (maximum size)
• Off-center → smaller circle

Square pyramid:

• Parallel to base → smaller square
• Through apex, perpendicular to base → triangle
• Parallel to base edge, not through apex → trapezoid

Cross-Section Summary for Key Solids

Pyramid: square, triangle, or trapezoid

ACT Practice: Name the Cross-Section Shape

Problem 1: A plane cuts a cone perpendicular to the axis, halfway up. What shape appears?

Problem 2: A plane passes diagonally through a cube. What shape could result?

Problem 3: A plane cuts a sphere off-center. What shape appears?

Solutions to Cross-Section Identification Problems

Problem 1: Circle — any cut perpendicular to a cone's axis produces a circle

Problem 2: Rectangle, triangle, or hexagon — depending on the exact diagonal path

Problem 3: Circle — every cross-section of a sphere is a circle, just smaller when off-center

From Cutting Solids to Spinning Shapes

We've been cutting 3D solids to find 2D shapes.

Now let's reverse the process:

• Start with a 2D shape
• Rotate it 360 degrees around an axis
• A 3D solid appears

This is called a solid of revolution.

Rectangle Rotated Around One Edge Makes Cylinder

The side on the axis becomes the height; the perpendicular side becomes the radius.

Triangle and Semicircle Create Cone and Sphere

Right triangle rotated around a leg → Cone

• Leg on axis = height
• Other leg = radius

Semicircle rotated around its diameter → Sphere

• Semicircle radius = sphere radius

The axis determines which dimension becomes the radius.

Hollow Cylinder From Off-Axis Rectangle Rotation

When a rectangle rotates around an axis outside the shape:

• The near edge traces the inner radius
• The far edge traces the outer radius
• The result is a hollow cylinder (cylindrical shell)

This case appears occasionally on the ACT.

Quick Check on Solids of Revolution

A rectangle with sides 4 cm and 6 cm is rotated around the side of length 6 cm.

What solid is formed, and what are its dimensions?

Think about it before advancing...

ACT Problem: Identify the Resulting Solid

Triangle with legs 5 and 12 rotates around the 12 cm leg.

Step 1: Axis = the 12 cm leg

Step 2: The 5 cm leg sweeps the radius

Step 3: Result — cone with ,

Mixed Practice: Cross-Sections and Revolution

Problem 1: A plane cuts a cylinder at 45 degrees. What cross-section shape results?

Problem 2: A semicircle with radius 8 rotates around its diameter. What solid forms?

Problem 3: A vertical plane passes through a pyramid's apex. What cross-section appears?

Key Takeaways and Common Mistakes

• Same solid + different cuts = different cross-sections
• Cross-sections are new interior faces, not existing surfaces
• Always identify the axis first for solids of revolution

Watch out:

• Not all cylinder cross-sections are circles
• Angled cuts through prisms give polygons, not ellipses

Coming Up Next in Solid Geometry

Review: You've now covered all four solid geometry topics:

• Volume formulas for all five solid types
• Surface area calculations
• Composite solid problems
• Cross-sections and solids of revolution

Practice mixing all four topics in timed ACT sets!