How To Calculate A Cross-Sectional Area

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.

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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.

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.

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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.

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 be the area of one of the circles forming its base.

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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.

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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.

Cite This Article

MLA

Beck, Kevin. "How To Calculate A Cross-Sectional Area" sciencing.com, https://www.sciencing.com/calculate-crosssection-area-4913182/. 7 February 2020.

APA

Beck, Kevin. (2020, February 7). How To Calculate A Cross-Sectional Area. sciencing.com. Retrieved from https://www.sciencing.com/calculate-crosssection-area-4913182/

Chicago

Beck, Kevin. How To Calculate A Cross-Sectional Area last modified March 24, 2022. https://www.sciencing.com/calculate-crosssection-area-4913182/

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