Math Core

Slice a loaf of bread and you see a flat face that looks a bit like a rectangle with a rounded top. Cut a block of cheese straight down and the face is a rectangle. That flat face is a cross section, and its shape depends on the solid and on how you slice it.

What a cross section is

Definition

Cross section

A cross section is the flat, two-dimensional shape you get where a plane (a perfectly flat cut) slices through a three-dimensional solid.

In this lesson you'll slice two kinds of solids:

A right rectangular prism is a box. It has six rectangular faces.
A right rectangular pyramid has a rectangular base and four triangular faces that meet at one point on top, called the apex. If the base is a square, it's a square pyramid.

Slicing a rectangular prism

A 4 in by 2 in by 3 in box sliced halfway up, parallel to its base. The slice is a 4 in by 2 in rectangle.

Parallel to the base. A horizontal slice cuts across the box like a shelf. The cross section is a rectangle with the same size and shape as the base. That's true at any height.

Parallel to a side face. A vertical slice parallel to the front makes a rectangle the same size as the front face. A vertical slice parallel to the side makes a rectangle the same size as the side face.

Slanted cuts. A cut that tilts can make a rectangle of a different size. A cut that slices off a corner makes a triangle. Cutting a cube at tricky angles can even make shapes with five or six sides.

Worked example: Area of a cross section

The box in the figure is $4$ in long, $2$ in deep and $3$ in tall. Find the area of (a) the horizontal slice shown, and (b) a vertical slice parallel to the front face.

(a) The slice matches the base, which is $4$ in by $2$ in. Area: $4 \times 2 = 8$ square inches.

(b) The slice matches the front face, which is $4$ in by $3$ in. Area: $4 \times 3 = 12$ square inches.

Slicing a pyramid

A square pyramid sliced parallel to its base, halfway up. The cross section is a smaller square.

Parallel to the base. The cross section has the same shape as the base, but it's smaller. The closer the cut is to the apex, the smaller it gets. For a square pyramid, a horizontal slice is always a square.

Straight down through the apex. A vertical cut through the apex makes a triangle. Its bottom side lies on the base, and its top corner is the apex.

Straight down, but missing the apex. A vertical cut parallel to a base edge that doesn't pass through the apex makes a trapezoid: the top is chopped off the triangle.

Worked example: Naming the shape

A square pyramid is cut by a vertical plane that passes through the apex and is parallel to one edge of the base. What shape is the cross section?

The cut goes straight down from the apex to the base, so the top of the slice is a single point and the bottom is a segment across the base. The cross section is a triangle.

Quick guide to cross sections

Solid	Cut	Cross section
Rectangular prism	parallel to base or a face	rectangle congruent to that face
Rectangular prism	slicing off a corner	triangle
Rectangular pyramid	parallel to base	smaller rectangle, same shape as the base
Rectangular pyramid	vertical, through the apex	triangle
Rectangular pyramid	vertical, parallel to a base edge, not through the apex	trapezoid

Common mistake

A pyramid's horizontal slices shrink as you move up. Don't assume a slice halfway up a pyramid is the same size as the base. That's only true for prisms.

Tip

Picture a real object: a box of cereal for a prism, or a tent for a pyramid. Imagine a knife moving through it and look at the cut face.

Practice

Practice 1

A square pyramid is sliced by a plane parallel to its base. What shape is the cross section?

Practice 2

A box is $10$ cm long, $6$ cm wide and $4$ cm tall. It is sliced parallel to its $10$ cm by $6$ cm base. What is the area of the cross section, in square centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

The same $10$ cm by $6$ cm by $4$ cm box is sliced vertically, parallel to its $6$ cm by $4$ cm end. What is the area of the cross section, in square centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A square pyramid is cut by a vertical plane that passes through the apex. What shape is the cross section?

Practice 5

A cube is sliced by a plane that cuts off one corner, passing through three edges that meet at that corner. What shape is the cross section?

Practice 6

Which shape can not be a cross section of a cube?

Practice 7

A square pyramid has a base $6$ cm on each side. It is sliced parallel to the base exactly halfway up. The slice is a square whose sides are half as long as the base's sides. What is the area of the cross section, in square centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.