A cross section is the flat shape you see when a plane slices through a solid. Slicing a rectangular prism parallel to its base gives a rectangle the same size as the base. Slicing a rectangular pyramid parallel to its base gives a smaller rectangle, and a vertical slice through its apex gives a triangle.
Key facts:
- A cross section is the two-dimensional shape where a plane meets a solid.
- Any slice of a right rectangular prism that is parallel or perpendicular to the base is a rectangle.
- A slice of a right rectangular pyramid parallel to the base is a smaller rectangle.
- A slice of a right rectangular pyramid perpendicular to the base is a triangle if it passes through the apex, and a trapezoid if it does not.
- The area of a cross section is found with the usual area formula for its shape.
What is a cross section of a 3D figure?
Imagine slicing a loaf of bread with a straight knife. The cut surface is flat, and its outline is the cross section. In geometry the knife is a plane, and the cross section is the 2D figure where the plane meets the solid.
The two-dimensional shape formed when a plane slices through a three-dimensional figure. In Grade 7 the plane is either parallel to the base or perpendicular to the base.
What are the cross sections of a rectangular prism?
A right rectangular prism has rectangular faces, so every slice parallel or perpendicular to the base is a rectangle. A slice parallel to the base is a copy of the base. A vertical slice has the height of the prism as one side.
A vertical slice through two opposite corners of the base is still a rectangle, but it is longer than the sides of the base.
What are the cross sections of a rectangular pyramid?
A right rectangular pyramid narrows to a point called the apex. The shape of the slice depends on where the plane cuts.
| Plane | Cross section |
|---|---|
| Parallel to the base | Rectangle smaller than the base |
| Perpendicular to the base, through the apex | Triangle |
| Perpendicular to the base, parallel to a base edge, not through the apex | Trapezoid |
Higher slices parallel to the base are smaller. Each side shrinks by the same fraction, so the slice keeps the shape of the base.
A vertical slice through the apex cuts the pyramid down the middle and leaves a triangle.
A vertical slice that misses the apex has parallel top and bottom edges but slanted sides, so it is a trapezoid.
How do you find the size and area of a cross section?
A pyramid has a 9 in. by 6 in. base and a height of 6 in. A plane parallel to the base cuts it 2 in. above the base. Find the area of the cross section.
- Name the shape. A slice parallel to the base is a rectangle.
- Find the fraction of the height left above the cut: \(\frac{6-2}{6}=\frac{2}{3}\).
- Multiply each base side by that fraction: \(9\times \frac{2}{3}=6\) and \(6\times \frac{2}{3}=4\).
- Find the area: \(6\times 4=24\text{ in.}^{2}\).
Cross Sections of 3D Figures: frequently asked questions
What is a cross section in geometry?
A cross section is the 2D shape that appears where a plane slices through a 3D solid. For a rectangular prism cut parallel to its base, the cross section is a rectangle that matches the base.
What shape is a slice of a rectangular prism perpendicular to the base?
It is always a rectangle. One side of the rectangle is the height of the prism, and the other side is the length of the cut across the base.
Which cross sections can a rectangular pyramid have?
With a plane parallel to the base, the cross section is a rectangle. With a plane perpendicular to the base, it is a triangle when the plane passes through the apex and a trapezoid when it does not.
Why is a slice parallel to the base of a pyramid smaller than the base?
The pyramid gets narrower as it rises toward the apex. The higher the cut, the smaller the rectangle, and at the apex it shrinks to a single point.
How do you find the area of a cross section?
First name the shape, then find its side lengths, then use the area formula. A rectangle uses length times width, a triangle uses \(\frac{1}{2}\times \text{base}\times \text{height}\), and a trapezoid uses \(\frac{1}{2}(b_1+b_2)h\).
See every chapter on the Grade 7 math page.