One indicates under the name of theorem of Guldin two statements of Euclidean Géométrie establish by the Swiss mathematician Paul Guldin. It is probable that these results were already known of Pappus of Alexandria and this is why one meets also the name of theorem of Pappus-Guldin. It expresses under certain conditions:

• the surface of the surface generated by an arc of curve;
• the measurement of the volume generated by a surface.

Another current application of this theorem is the calculation of the position of the Center of gravity of an arc of curve or a surface.

First statement

Examples:

• the surface of the open Torus of rays $r$ and $R$ is worth $A\; =\; (2\; \backslash \; pi\; R)\; (2\; \backslash \; pi\; R)\; =\; 4\; \backslash \; pi^\; 2\; r\; R$
• the surface generated by a half-circle of radius $R$ and center of gravity $G$ is the sphere of surface $A\; =\; (\backslash \; pi\; R)\; (2\; \backslash \; pi\; r\_G)\; =\; 4\; \backslash \; pi\; R^2$. It comes $r\_G=\; \backslash \; tfrac\; \{2\}\; \{\backslash \; pi\}\; R$.

Second statement

Examples:

• the interior volume of the open Torus of rays $r$ and $R$ is worth $V\; =\; (\backslash \; pi\; r^2)\; (2\; \backslash \; pi\; R)\; =\; 2\; \backslash \; pi^\; 2\; r\; ^\; 2\; R$.
• the volume generated by a half-disc of radius $R$ and center of gravity $G$ is the ball of measurement $V\; =\; (\backslash \; tfrac\; \{1\}\; \{2\}\; \backslash \; pi\; R^2)\; (2\; \backslash \; pi\; r\_G)\; =\; \backslash \; tfrac\; \{4\}\; \{3\}\; \backslash \; pi\; R^3$. It comes $r\_G=\; \backslash \; tfrac\; \{4\}\; \{3\; \backslash \; pi\}\; R$.

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