Plane Sections of 3-D Figures

Slice a solid with a flat plane and a flat shape appears. The angle of the cut decides which shape you get.

Take a block of cheese and cut straight down. The face you expose is a rectangle.

Now cut flat across instead. Also a rectangle, but a different one.

slice a block of cheese three ways straight down flat across at a slant the cut face is flat — but not the same shape every time

Now cut at a slant, corner to corner. The face is still flat — but it is longer and thinner than either of the others.

One solid, three cuts, three different shapes. The shape depends on the angle of the knife, not only on the block.

Tilt a cut through a cube far enough and you can expose a triangle, a pentagon, even a hexagon.

A cross section is the flat face a cut leaves behind. The question is never "what shape is this solid?" but "which way did the plane go through it?"

A plane slicing through a cube to reveal a hexagon one flat cut, six faces crossed → a hexagon cut 3 faces near a corner → triangle cut parallel to a face → square one side per face the plane meets

What a cross section is

Imagine slicing a block of cheese with a single straight cut. The new flat face you expose is the cross section.

The same solid gives different cross sections depending on how you cut it, so the question always has two parts: which solid, and which direction.

Slicing a rectangular prism

Parallel to a face: you get a rectangle identical to that face, no matter how deep you cut.

Perpendicular to a face: also a rectangle.

Diagonal cuts: you can get a non-rectangular parallelogram, a triangle (cutting off one corner), a pentagon, or even a hexagon if you slice cleverly through six faces.

The rule: the cross section has one side for each face the plane passes through. A cube has six faces, so a hexagon is the most sides possible.

Count the faces you cut

The number of sides of the cross section equals the number of faces the plane passes through. Cut three faces near a corner, get a triangle. Cut all six, get a hexagon.

Slicing a pyramid

A square pyramid cut parallel to the base gives a square — smaller the higher you cut.

Cut perpendicular through the apex and you get a triangle.

Cut at a slant and you can get a trapezoid.

A square pyramid cut three ways: parallel to the base gives a square, down through the apex a triangle, at a slant a trapezoid parallel to base↓a squarethrough the apex↓a triangleat a slant↓a trapezoid

Slicing a cylinder and a cone

A cylinder cut parallel to its base gives a circle, always the same size. Cut perpendicular (straight down through the middle) and you get a rectangle. A slanted cut gives an ellipse — an oval.

A cone cut parallel to the base gives a circle; cut vertically through the tip gives a triangle.

A cylinder cut parallel to its base gives a circle, straight down the middle a rectangle, at a slant an ellipse; a cone cut parallel to its base gives a circle, down through the tip a triangle parallel to basedown the middleat a slant circle rectangle ellipse cone: parallel to basecone: down through the tip a smaller circle triangle

Nets: the other way to flatten

A net is what you get if you unfold a solid along its edges and lay it flat.

A cube has 11 different nets. A useful check when reading one: on a cube, opposite faces are never adjacent in the net and never share an edge. In a row of four squares, the 1st and 3rd are opposite, and so are the 2nd and 4th.

Nets are also the easiest route to surface area — the surface area is just the area of the net.

Worked examples

Cross sections of a cube

Name three different cross sections you can obtain from a cube.

  1. Cut parallel to a face → a square. The plane meets four faces, all at right angles.
  2. Slice off one corner → a triangle. The plane meets only the three faces at that corner.
  3. Cut on a diagonal through opposite edges → a rectangle. Longer than the square, because the diagonal is longer than an edge.
  4. A cut crossing all six faces → a regular hexagon. Six faces cut, six sides — the maximum for a cube.

Square, rectangle, triangle, and hexagon are all possible.

Reading a net

A cube net is a row of four squares labelled A, B, C, D, with E above B and F below B. Which face is opposite A?

  1. In a row of four, the 1st and 3rd are opposite. Folding a row wraps it around, so faces two apart end up facing each other.
  2. So A is opposite C, and B is opposite D. Apply the rule to both pairs.
  3. That leaves E opposite F. Six faces make three opposite pairs.

C is opposite A.

The shape depends on the cut

A cylindrical can of radius 4 cm and height 10 cm is cut straight down through its central axis. Describe and measure the cross section.

  1. A vertical cut through the axis gives a rectangle. You expose a flat slice running the full height.
  2. Its height is the can's height: 10 cm. The cut runs top to bottom.
  3. Its width is the full diameter: 2 × 4 = 8 cm. Through the centre means across the widest part.
  4. Area = 10 × 8 = 80 cm2 A rectangle, so base times height.

A 10 cm × 8 cm rectangle, area 80 cm2.

Words to know

Cross section
The 2-D face revealed when a plane cuts through a solid.
Plane
A perfectly flat surface extending forever.
Net
A 3-D shape unfolded flat.
Face, edge, vertex
A flat surface; a line where two faces meet; a corner point.

Mistakes to watch for

Assuming every slice of a cube is a square. Only cuts parallel to a face give squares. Slanted cuts give rectangles, triangles, pentagons and hexagons — the cube is the same, the plane is not.

Using the radius instead of the diameter for the width of a vertical cylinder section. A cut straight down through the middle exposes a rectangle whose width is the full diameter, since it crosses the circle from edge to edge.

Reading a net without checking which faces end up opposite. Two faces that touch on the flat net may fold to become opposite sides, and two that look far apart may meet. Fold it mentally, or mark the pairs.

Expecting the cut face to be the largest thing in the picture. A slant cut is longer than a straight one, so it can look bigger than the face it replaced — that is real, not an error.

Slice a cone the right way and you get a circle, an ellipse, a parabola or a hyperbola. The Greeks called these the conic sections and studied them for centuries as pure geometry.

Two thousand years later Kepler found that planets orbit in ellipses, and Newton showed that any object under gravity follows one of those same four curves.

A piece of geometry with no known use outside itself turned out to describe the solar system.

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