☰ Course contents
📐 Mathematics Course
Mathematics › 🔄 Grade 7 › Cross-Sections of Three-Dimensional Solids
🔄 Grade 7 · Lesson 10 of 14

Cross-Sections of Three-Dimensional Solids

Slice a solid with an imaginary plane and the flat shape you reveal depends entirely on the solid — and exactly how you cut it.

Grade 7Middle school
Cross-Sections of Three-Dimensional Solids — illustration
💡
The big idea: A cross-section is the two-dimensional shape you see when a flat plane slices through a three-dimensional solid, like cutting through a loaf of bread. The same solid can reveal very different cross-sections depending on where and at what angle it's sliced — a cylinder can show a circle or a rectangle, and a cone can show a circle or a triangle, depending on the cut.
🎯 By the end, you'll be able to
  • Define a cross-section as the 2D shape formed when a plane slices a 3D solid
  • Predict the cross-section shape for common solids sliced in different ways
  • Distinguish cross-sections from slices parallel to a base versus slices through an axis or apex
  • Apply cross-section reasoning to everyday solids, like cutting a cake or a can
📎 You should already know
  • Identifying 3D solids and their faces
  • Basic 2D shapes (circle, triangle, rectangle)

Cutting into a solid

Imagine slicing straight through a solid object with an imaginary flat plane, the way a knife slices through a loaf of bread or a laser sweeps through a block of ice. The flat, 2D shape exposed at the cut is called a cross-section.

The same solid can produce very different cross-sections depending on where you slice and at what angle — that's what makes this topic surprisingly rich.

🔑 A cross-section is a 2D slice of a 3D solid
A cross-section is the flat shape formed where a plane intersects a solid. The same solid can have many different cross-sections, each one depending on the position and angle of the slicing plane.

Slicing a cube

Slice a cube with a plane parallel to one of its faces, and the cross-section is a square congruent to that face, no matter how far along you slice. Tilt the slicing plane at an angle instead, and the cross-section can stretch into a rectangle, or even a more complex polygon if the plane cuts through several faces at once.

🎮 Laser Cross-Section LIVE
Slice a 3D solid with a plane and see the 2D cross-section it reveals.

Slicing a cylinder and a cone

Slice a cylinder horizontally, parallel to its circular base, and every cut reveals a circle the same size as the base. Slice it vertically, straight down through its central axis, and the cross-section becomes a rectangle instead.

A cone behaves differently: a horizontal slice parallel to its base gives a circle (smaller the closer you get to the tip), while a vertical slice straight through the apex gives a triangle.

✨ A sphere always slices into a circle
No matter how you orient the slicing plane, a sphere always produces a circle as its cross-section. The circle is largest when the plane passes through the center (called a great circle) and shrinks the farther off-center the slice is taken.
📝 Worked example: A cylindrical can is sliced by a plane parallel to its circular base (a flat, horizontal cut). What is the cross-section?
  1. The slicing plane is parallel to both circular bases of the cylinder.
  2. Every cross-section parallel to a cylinder's base is congruent to that base.
  3. Since the base is a circle, the cross-section must match it.
✓ The cross-section is a <strong>circle</strong>, the same size as the can's circular base.
📝 Worked example: A square pyramid is sliced by a plane parallel to its square base, partway up toward the apex. What shape is the cross-section, and how does its size compare to the base?
  1. A slice parallel to the base of a pyramid is similar in shape to the base — here, a square.
  2. Because the pyramid narrows toward its apex, the cross-section shrinks the higher up the slice is taken.
  3. Right at the apex, the 'cross-section' shrinks to a single point.
✓ The cross-section is a <strong>smaller square</strong>, similar to the base, shrinking toward a point at the apex.
⚠️ One solid, many possible cross-sections
It's tempting to assume every slice of a solid gives the same shape. It doesn't — the cross-section depends on where and at what angle you cut. Always picture the specific slicing plane described before naming the shape.

Check your understanding

1. What shape is the cross-section of a cylinder cut horizontally, parallel to its circular base?
A horizontal slice of a cylinder, parallel to its base, always matches the circular base.
2. A sphere is sliced by a plane passing through its exact center. What is the cross-section?
Slicing through a sphere's center produces the largest possible circular cross-section, called a great circle.
3. What is the cross-section of a cone sliced by a plane through its apex, perpendicular to the base?
A vertical slice through a cone's apex produces a triangular cross-section.
4. A cube is sliced by a plane parallel to one of its faces. What is the cross-section?
A slice parallel to a cube's face always matches that face, giving a congruent square.
5. Is it true that every cross-section of a given 3D solid is the same shape and size, no matter where you slice?
Cross-sections generally change with the position and angle of the slicing plane, as seen with cylinders, cones, and pyramids.
✅ Key takeaways
  • A cross-section is the flat 2D shape revealed when a plane slices through a 3D solid.
  • A cube sliced parallel to a face gives a square; tilted, it can give a rectangle or other polygon.
  • A cylinder gives a circle when sliced parallel to its base, and a rectangle when sliced through its axis.
  • A cone gives a circle sliced parallel to its base, and a triangle when sliced through its apex.
  • A sphere always gives a circle, largest when the slicing plane passes through its center.