When you slice through a three-dimensional figure, you'll see a new face. For example, if you slice a rectangular pyramid parallel to the base, the cross section is a smaller rectangle. One way to think about it is that the bottom half of the original pyramid has been cut off and replaced with a flat piece.

If you imagine the original pyramid as a clock with its hands at 12 o'clock, then the sliced-off portion will be like a pie slice taken out of the middle of the clock. The remaining part still keeps time but it's now half as long. In this case, the slice removes a third of the volume while leaving the original length of the pyramid intact. This means that the total number of pyramids remains the same but each one contains **only half as much material**.

As **another example**, suppose that we start with a regular octahedron and slice it along an edge. The cross section will be a triangle with two equal sides. It's like cutting a corner off of the octahedron. Now there are seven such slices or "facets". Each one is a triangular prism with **two opposite edges** equal in length. The remaining four edges can't be measured directly because they're inside the cube itself. They must be calculated by using Pythagoras' theorem and the fact that all right angles are equal.

- What would result from slicing a rectangular pyramid parallel to the base?
- What is the cross section of a cube?
- What is the cross section of a rectangular pyramid?
- What shape is made by the vertical cross section of this square pyramid?
- Which shapes are possible cross sections of a triangular pyramid?
- What cross section do you get when you give a vertical cut to a cube?

A cross section is the face formed when a single slice is made across an item. A sample slice through a cube is shown below, along with one of **the cross sections** that can be obtained. The cross section cannot include any of **the original face** because it all originates from "inside" the solid.

The word "cross-section" comes from the fact that this type of picture looks like a cross when viewed sideways.

You should know that the term "cross section" is also used to describe the face of **some solids** such as cylinders and cones. However, while these other types of solids can have **their own unique features**, cubes only have faces that are parallel to the axes of the cube. Thus, they can only be sliced in **half perpendicular directions**.

Cubes are used in science labs as well as in industry. They are also useful for demonstrating how different materials affect sound waves. Scientists use wave guides to study how particles interact with light and other forms of radiation. In industry, cubes are used to produce small quantities of products that are very similar one to another.

There are several methods for slicing cubes. You can use a sharp knife and cut straight down into the middle of the cube. This will divide the cube into two parts that are equal in size and shape. Each piece can then be studied in isolation from the others.

The cross section is shaped like a rectangle. Figure 5 depicts a cross section of a right rectangular pyramid taken perpendicular to its base and through **its peak**. The cross section is shaped like a triangle. However, it should be noted that even though there are three sides, it has **four corners**.

The area of **a square pyramid** is $ht$, where $h$ is the height and $t$ is the capstone (or tipping) ratio. The formula for calculating the area of **a rectangular pyramid** is similar to that of a square pyramid: $\frac{1}{4}ht$. You can calculate the area of a rectangular pyramid by using this formula and substituting values for $h$ and $t$.

It is important to understand that the cross section of a pyramid is a rectangle. Therefore, you should know that the area of a rectangle is length times width. There are several ways to find the area of a rectangle. One way is to use the formula $A = lw$, where $l$ is the length of the rectangle and $w$ is the width. In order to use this method, you need to know both the length and the width of the rectangle's cross section.

Another way to find the area of a rectangle is to think about how much space it takes up.

This object's cross section is a triangle. It's like getting a peek inside something by chopping through it. The edges of the base are the same length as one of the sides of the pyramid, which means that the base must be equilateral.

The area of a triangle can be calculated by using the Pythagorean theorem. In this case, the area is:

$$\frac{1}{2}bh = \frac{1}{2}\times 12\times 7.5 = 37.5$$

There are several ways to find the height of the pyramid. You could use the angle addition formula for two right triangles with equal legs and identical angles (in this case, three right angles). Or you could use the law of cosines, which says the distance from **any point** on a circle to another point is equal to the radius of the circle multiplied by the cosine of the angle between them. In this case, the radius is 7.5, so the distance between **any two points** on the pyramid is 9. That means its height must be $9/7.5 = 1.2$.

Because it is a triangular pyramid, the cross-section view is also a triangle. The sides of the cross section will be equal if the base of the original pyramid was also equal.

The only two regular polyhedra that are soluble (that is, can be cut into **non-overlapping pieces** that will completely dissolve in water) are the cube and the tetrahedron. There are also some irregular polyhedra that are soluble, such as the dodecahedron and icosahedron. All the other regular polyhedra are not soluble.

A convex polyhedron cannot have any concave faces, so all its faces are either flat or spherical caps. A polyhedron is called "regular" if there is exactly one way to divide it up into **non-overlapping pieces** that will completely dissolve in water. Some polyhedrons are regular, such as the cube and tetrahedron. Most aren't, including the dodecahedron and icosahedron.

A solid is called "soluble" if you can break it up into **non-overlapping pieces** that will completely dissolve in water. Solubility is a property of **the whole object**, not each part.

Similarly, a vertical die cut looks like this: Observing the pictures above, we can see that **both the horizontal and vertical cross sections** of a die result in a rectangle. This is because any plane cutting through **a solid object** will divide it into two parts, one on each side of the blade. For a square block, these two parts are identical.

A third option is a diagonal cut. We can see from the picture below that if we were to slice this block diagonally down the middle, we would end up with two triangles. Each triangle is simply one half of the original block.

This shows that if you want to divide a block into **different shapes**, you can use a die with different-shaped openings.

However, not all cuts are equal. If you look at the diagram below, you can see that a circle has been divided into eight pieces by using four separate dies with rectangular holes. A square has been divided into nine pieces by using **five separate dies** with rectangles holes. A regular octagon has been divided into 24 pieces by using **twelve separate dies** with triangular holes.

This proves that some cuts are better than others. A die with rectangular holes will always yield a rectangle for its cross section.