You may encounter situations in which you have a threedimensional solid shape and need to figure out the area of an imaginary plane inserted through the shape and having borders defined by the boundaries of the solid.
For example, if you had a cylindrical pipe running under your home measuring 20 meters (m) in length and 0.15 m across, you might want to know the crosssectional area of the pipe.
Cross sections can be perpendicular to the orientation of the axes of the solid if any exist. In the case of a sphere, any cutting plane through the sphere regardless of orientation will result in a disk of some size.
The area of the crosssection depends on the shape of the solid determining the crosssection's boundaries and the angle between the solid's axis of symmetry (if any) and the plane that creates the cross section.
CrossSectional Area of a Rectangular Solid
The volume of any rectangular solid, including a cube, is the area of its base (length times width) multiplied by its height: V = l × w × h.
Therefore, if a cross section is parallel to the top or bottom of the solid, the area of the crosssection is l × w. If the cutting plane is parallel to one of the two sets the sides, the crosssectional area is instead given by l × h or w × h.
If the crosssection is not perpendicular to any axis of symmetry, the shape created may be a triangle (if placed through a corner of the solid) or even a hexagon.
Example: Calculate the crosssectional area of a plane perpendicular to the base of a cube with a volume of 27 m^{3}.

Since l = w = h for a cube, any one edge of the cube must be 3 m long (since 3
× 3
× 3 = 27). A crosssection of the type described would therefore be a square 3 m on a side, giving an area of 9 m^{2}.
CrossSectional Area of a Cylinder
A cylinder is a solid created by extending a circle through space perpendicular to its diameter. The area of a circle is given by the formula πr^{2}, where r is the radius. It therefore makes sense that the volume of a cylinder would be the area of one of the circles forming its base.
If the crosssection is parallel to the axis of symmetry, then the area of the crosssection is simply a circle with an area of πr^{2}. If the cutting plane is inserted at a different angle, the shape generated is an ellipse. The area uses the corresponding formula: πab (where a is the longest distance from the center of the ellipse to the edge, and b is the shortest).
Example: What is the crosssectional area of the pipe under your home described in the introduction?

This is just πr^{2} = π(0.15 m)^{2}=
π(0.0225) m^{2} = 0.071 m^{2}. Note that the length of the pipe is irrelevant to this calculation.
CrossSectional Area of a Sphere
Any theoretical plane placed through a sphere will result in a circle (think about this for a few moments). If you know either the diameter or the circumference of the circle the crosssection forms, you can use the relationships C = 2πr and A = πr^{2} to obtain a solution.
Example: A plane is rudely inserted through the Earth very close to the North Pole, removing a section of the planet 10 m around. What is the crosssectional area of this chilly slice of Earth?
 Since C = 2πr = 10 m, r = 10/2π = 1.59 m; A = πr^{2}= π(1.59)^{2}= 7.96 m^{2}.
References
Tips
 Be sure to use the correct units when you calculate the area of a crosssection: this will be "square" units, such as square inches, square meters, and so on.
About the Author
Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs.com and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at www.kemibe.com.