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 Plane
For many applications, the plane will be perpendicular to the longest axis or the longitudinal axis.
Identify the Shape Projected Onto the Plane
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.
Choose the Appropriate Area Formulas
Select the appropriate area formulas to use for the calculation. Some common ones are the area of a triangle, which is 1/2 × b × h, where b is the triangle's base and h is its height; the area of a rectangle, which is b × h, where b is the rectangle's base and h is its height; and the area of a circle, which is π_r_2, where r is the circle's radius. In our example, you'd need the rectangle formula to calculate the I-beam shape.
Measure Values as Needed
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
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 first calculate the area of the two horizontal rectangles.
Each horizontal rectangle measures 4 inches × 6 inches or 24 in2, but there are two of them, so we have 24 in2 × 2 = 48 in2.
The vertical rectangle measures 2 inches × 12 inches = 24 in2.
Add these measurements together for the total area of the I-beam: 48 in2 + 24 in2 = 72 in2.
For an example of finding the cross-sectional area of a cylinder given the diameter, view the video below:
Tip: Be sure to use the correct units when you calculate the area of a cross-section: this will be "square" units, such as square inches, square meters, and so on.