How to Calculate a Cross-Sectional Area

When a plane cuts through an object, an area is projected onto the plane. Any plane can be used to cut through the surface, but when that plane is perpendicular to an axis of symmetry, its projection is called a cross-sectional area. For a simple three-dimensional shape, such as a cylinder, the cross-sectional projection is a circle, and the area is easy to calculate. With such shapes as an I-beam, however, calculating the cross-sectional area can be complicated.

Identify the axis of symmetry. For many applications, this will be the longest axis or the longitudinal axis.

Identify the shape projected onto a plane that passes through the shape perpendicular to the axis of symmetry. If the shape is complex, divide it into simpler shapes for ease of calculation. An I-beam, for example, can be divided into a horizontal rectangle on the top, a horizontal rectangle on the bottom and a vertical rectangle connecting them in the middle.

Select the appropriate area formulas to use for the calculation. Some common ones are the area of a triangle, which is half of the base times the height; the area of a rectangle, which is the base times the height; and the area of a circle, which is pi times the radius squared. In our example, you'd need the rectangle formula to calculate the I-beam shape.

Measure the values needed to fill in the formula or formulas. For example, suppose each of the horizontal rectangles in our I-beam shape measures 4 inches by 6 inches, and the vertical rectangle measures 2 inches by 12 inches.

Solve the area equations. For complex geometries, solve the simpler equations and add them together to get the total cross-sectional area. In our example, we add the area of two of the horizontal rectangles -- 24 + 24 = 48 square inches -- to the area of the vertical rectangle -- 24 square inches -- for the total area: 72 square inches.