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 that shape and whose borders are defined by the boundaries of the solid.
For example, if you had a cylindrical pipe running under your home measuring 20 meters in length and 0.15 meters 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, the angle between the solid’s axis of symmetry (if any), and the plane that creates the cross section.
Cross-sectional 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 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 meters long (since 3 × 3 × 3 = 27). A cross section of the type described would therefore be a square 3 meters on a side, giving an area of 9 m2.
Cross-sectional 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 πr2, where r is the radius. It therefore makes sense that the volume of a cylinder would use 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.
Cross-sectional 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 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 meters around. What is the cross-sectional area of this chilly slice of earth? Since C = 2πr = 10 m and r = 10/2π = 1.59 m, then A = πr2= π(1.59)2= 7.96 m2.
