Cross-Sections and Solids of Revolution

Lesson 4 of 4: Solid Geometry

In this lesson:

• Name the shape a flat cut makes in a solid
• Predict how that shape changes with the cut
• Name the solid a spinning flat shape makes

What You Will Learn Today

1. Describe a cross-section: the face a flat cut makes
2. Name the cross-sections of five common solids
3. Predict how tilting or moving the cut changes the shape
4. Name the solid a flat shape makes when spun around an axis
5. Solve ACT-style cut and spin questions

Warm Up: Sketch These Five Shapes

• Sketch and name: rectangle, triangle, trapezoid
• Then the solids: rectangular prism (a box), square pyramid
• Predict: can one straight cut through a box make a hexagon?

Slicing a Block of Cheese

A cross-section is the new face a flat cut exposes — not a face you could see before.

Three Cuts Through One Box

Compare each count of faces crossed with the sides of its cut face.

Predict the Shape of This Cut

• A vertical plane cuts the pyramid, parallel to its front base edge, missing the apex
• (a) triangle (b) square (c) trapezoid (d) pentagon
• Commit to a letter before you advance.

One Pyramid, Three Cuts, Three Shapes

Halfway up, parallel to the base: a square with half the base edge.

Count the Faces, Get the Sides

• Flat faces always give a polygon, never a curve
• Sides = faces crossed: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon
• A box has 6 faces: crossing all 6 gives a hexagon, never more
• A cube cut through three corners sharing no edge: equilateral triangle

Warm Up: Curved Shapes and a Baguette

• Sketch and name: circle, ellipse, cylinder, cone
• Baguette: cut straight across → circle; cut on a slant → ellipse
• Commit: can a flat cut through a cylinder ever make a rectangle? Yes or no?

One Cylinder, Three Cuts, Three Shapes

Yes — a cut parallel to the axis gives a rectangle; through the axis, its width is the diameter.

Same Slanted Cut, Two Different Solids

• Across a curved side, a slanted cut makes a curved edge: ellipse
• Across flat faces, every edge is straight: a polygon, never an ellipse

Five Cuts Through a Cone

Missing the apex, tilt decides the curve: flatter than the side → ellipse; parallel → parabola; steeper → hyperbola.

Every Cut Through a Sphere Is a Circle

• Through the center → the largest circle, a great circle
• Off center → a smaller circle
• On Earth, the equator is a great circle; lines of latitude are smaller ones

Five Solids, Three Cuts, and Position

Quick Check: Name the Cut Face

1. A plane cuts a cone parallel to its slant side. What shape?
2. A plane cuts a sphere of radius 5, three units from its center. What shape?
3. Jae says every vertical cut through a cone makes a triangle. Right or wrong — and why?

Spin a Flag Around Its Pole

• Spin a rectangular flag on its pole: the pole stays put and the free edge sweeps around it
• The spinning flag sweeps out a solid — a solid of revolution
• Predict: a 3 × 5 rectangle spun around its 5-unit side makes what solid, and what size?

Rectangle Spun Around Its Side Makes Cylinder

Axis on an edge: side on the axis (5) → height; side perpendicular to it (3) → radius.

Predict: Spin This Triangle Around Its 3 Leg

A right triangle has legs 3 and 5.

• (a) cone: ,
• (b) cone: ,
• (c) cylinder: ,
• (d) cone: ,

Commit to a letter first.

Same Triangle, Two Axes, Two Cones

The leg on the axis is the height; the other leg is the radius.

A Semicircle Spun Around Its Diameter

• The axis must be the diameter — the semicircle's straight side
• The curved edge sweeps out a sphere
• Semicircle radius 3 → sphere radius 3
• Every cut of this solid is a circle — it is the sphere from earlier

Same Rectangle, Three Axes, Three Solids

Outside axis → hollow: inner radius = distance to the near side; outer = distance to the far side.

Slice the Cylinder You Just Spun

• Perpendicular to the axis: a circle of radius 3, the rectangle's width swept around
• Through the axis: the 3 × 5 rectangle + its mirror image = 6 × 5

Your Turn: Four Questions, No Choices

1. Slant-cut a square pyramid between base and apex. Shape?
2. Cut a cone of radius 3, height 5 through its axis. Shape, size?
3. A right triangle (legs 4, 7) spins on its 4 leg. Solid, size?
4. What shape and axis make a cylinder, radius 2, height 9?

Check Your Answers to All Four

1. Quadrilateral: four side faces crossed
2. Isosceles triangle, base 6, height 5
3. Cone, , (4 leg on the axis)
4. 2 × 9 rectangle on a 9 side, or 4 × 9 on the central line parallel to its 9 sides

What to Take Away From Today

✓ A cross-section is a new face, shaped by what the cut crosses and where
✓ Slice a spun solid through its axis: the shape returns, mirrored
Same solid, new cut, new shape
Slanted ≠ ellipse: flat faces give polygons
Axis first: it sets radius and height

End of the Solid Geometry Unit

You've now covered all four solid geometry lessons: volume, surface area, composite solids, and cross-sections with solids of revolution.

Next: mixed, timed practice — first decide which of the four a question is asking about.