You may encounter situations in which you have a three-dimensional 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 cross-sectional 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 cross-section depends on the shape of the solid determining the cross-section's boundaries and the angle between the solid's axis of symmetry (if any) and the plane that creates the cross section.

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 cross-section is l × w. If the cutting plane is parallel to one of the two sets the sides, the cross-sectional area is instead given by l × h or w × h.

If the cross-section 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 cross-sectional area of a plane perpendicular to the base of a cube with a volume of 27 m³.

• Since l = w = h for a cube, any one edge of the cube must be 3 m long (since 3

  × 3 

  × 3 = 27). A cross-section of the type described would therefore be a square 3 m on a side, giving an area of 9 m².

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 πr², 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 cross-section is parallel to the axis of symmetry, then the area of the cross-section is simply a circle with an area of πr². 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 cross-sectional area of the pipe under your home described in the introduction?

• This is just πr² = π(0.15 m)²=

  π(0.0225) m² = 0.071 m².  Note that the length of the pipe is irrelevant to this calculation.

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 cross-section forms, you can use the relationships C = 2πr and A = πr² 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 cross-sectional area of this chilly slice of Earth?

• Since C = 2πr = 10 m, r = 10/2π = 1.59 m;  A = πr²= π(1.59)²= 7.96 m².