Learning How to Calculate the Cross Section

David Goehring/CC-BY 2.0

In math, a cross-section is the shape you would see if you were to slice an object. Knowing how to calculate it can be useful, especially for calculating the volume of a whole object.

In technical terms, the cross section is when a plane meets a solid form. Calculating it is very similar to calculating the area of a shape; the only difference is that a cross section is part of a three-dimensional form. Different shapes have different formulas, and below are some examples of how to go about your calculations.

Rectangles You may see a rectangular cross section on a rectangular building, cereal box or anything that’s a rectangular prism. This may be the easiest cross-sectional area to calculate, as it’s simply the width x length. For example, let’s say one side of the building is eight-feet long and the other side is six-feet long. Eight multiplied by six is 48, and there you have your cross section.

Squares You can see a square-shaped cross section in a dice or in a square-based pyramid, like the ones in Egypt. Since all sides of a square are of equal length, the cross section is simply that length multiplied by itself. For example, if the length of the dice is two inches, the area of the cross section is four inches squared. An easy way to remember this is that the cross-section of a square is the length squared.

Triangles A triangular cross section may not be as common as other shapes, as you would probably only see one in a triangular-based pyramid or a triangular prism. The most basic way of finding this cross-sectional area would be the base length multiplied by the height and then divided by two. However, these lengths may be tricky to find if they aren’t stated, and you’ll need to rely on trigonometry to find out any missing figures.

Circles The area of a circle cross section is pretty complicated since it usually requires a calculator to get the precise number because it involves pi. Forms that have a circular cross section include cones, spheres and cylinders. The area of the circle is two x pi x radius squared. The radius is half the diameter, which is the length cutting across the circle. If you don’t need an exact number, you can just leave the result with pi in it. For example, if the radius is five inches, the cross sectional area would be two x pi x 25, which comes out to be 50 x pi as a final result without a calculator.

Taking an Extra Step While the main focus here is on the cross-sectional area of a form, you can easily find the volume of the form with this information. With most shapes, you just multiply the area from the cross section by the length of the object. Certain shapes, such as spheres and cones, have different formulas, meaning it’s not a one-size-fits-all rule.

This is only a simple guide on how to calculate basic cross sectional areas. Most forms can be broken down into these four simpler shapes. The hardest part about cross sections is being able to visualize what the form looks like when it’s sliced. Once you know the cross-sectional shape, you can easily apply one of these formulas and do the math.