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The Frontiers of Society, Science and Technology, 2021, 3(1); doi: 10.25236/FSST.2021.030125.

A Study of Crofton Type Formulas for Surface Area in Three-Dimension


Zhaoxuan Zhu1, Tingxuan Xu2, Yueying Pan3, Chenrui Hu4, Ai Li4, Kaixin Ma5

Corresponding Author:
Zhaoxuan Zhu

1Singapore International School (Hong Kong), Hong Kong SAR 999077, China

2International Department, The Affiliated High School of SCNU, Guangzhou, Guangdong 510630, China

3Alcanta International College, Guangzhou, Guangdong 511458, China

4WLSA Shanghai Academy, Shanghai 200433, China

5Hangzhou foreign languages school CAL centre, Hangzhou, Zhejiang 310023, China


Every three-dimensional object can be computed by a two-dimensional plane with the help of integration. Inspired by Crofton formulas and the Cavalieri's Principle, this work derives a general method for the surface area of a polygonal in three-dimensional space. In fact, the surface area of a three-dimensional object can be subdivided into a finite number of small rectangle. This research represents the area of a rectangle by the number of the intersection point between the rectangle and the line passing through the rectangle in all directions. Next, this research computes the proportionality constant D of integration. Eventually, this research extends the result to a boarder discussion on the application of the obtained result to a smooth surface in R^3. Within the process of integration , the volume of a four-dimensional object in TS2 is calculated. This research jumps out from the conventional representation of the surface area using the one-dimension integral geometry. The reader will realize another technique of representing the surface area with the integration of the number of intersection points in a sub-divided parallelograms. Moreover, not only does the research extent the concept of Cavalieri's principle to a three-dimensional application, but also the solution incites a possible way using the intersection point to explore the volume or the surface area of an object in a higher dimension world.


Crofton Theorem, Cavalieri's Principle, Scissors Congruence, Curvilinear Parallelograms

Cite This Paper

Zhaoxuan Zhu, Tingxuan Xu, Yueying Pan, Chenrui Hu, Ai Li, Kaixin Ma. A Study of Crofton Type Formulas for Surface Area in Three-Dimension. The Frontiers of Society, Science and Technology (2021) Vol. 3, Issue 1: 155-163. https://doi.org/10.25236/FSST.2021.030125.


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