Truncated pyramid

Truncated pyramid

A truncated pyramid is a geometry term that describes a special type of polyhedron ( polyhedron ). A truncated pyramid is created by cutting a smaller, similar pyramid (supplementary pyramid) from a pyramid (starting pyramid ) parallel to the base area on the lateral surfaces .

The two parallel surfaces of a truncated pyramid are similar to each other. The larger of these two areas is called the base area , the smaller the top area . The distance between the base and top surface is called the height of the truncated pyramid.

The volume of a truncated pyramid can be calculated using the following formula:

${\displaystyle V={\frac {h}{3}}\left(A_{\text{1}}+{\sqrt {A_{\text{1}}\cdot A_{\text{2}}}}+A_{\text{2}}\right)}$

The focus A ₁ for the base area, A ₂ on the top surface, and h for the height of the truncated pyramid.

There is no simple formula for the surface area . In the case of very crooked pyramids and truncated pyramids, it can be any size.

proofs

volume

To calculate the volume of the truncated pyramid, the height of the starting pyramid and the height of the supplementary pyramid are defined. From the centric extension it follows that ${\displaystyle h_{1}}$ ${\displaystyle h_{2}}$

${\displaystyle {\frac {h_{1}}{h_{2}}}=k}$and therefore also . ${\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}$

Here, the yield factor of the central extension. ${\displaystyle k}$

The volume of the truncated pyramid results from the difference between the volume of the initial pyramid and the volume of the supplementary pyramid:

${\displaystyle V=V_{1}-V_{2}={\frac {1}{3}}h_{1}\cdot A_{1}-{\frac {1}{3}}h_{2}\cdot A_{2}}$.

Out and follows . ${\displaystyle {\frac {h_{1}}{h_{2}}}=k}$ ${\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}$ ${\displaystyle {\frac {h_{1}}{h_{2}}}={\frac {\sqrt {A_{1}}}{\sqrt {A_{2}}}}}$

The substitution gives and . ${\displaystyle \lambda ={\frac {h_{2}}{\sqrt {A_{2}}}}}$ ${\displaystyle h_{2}=\lambda \cdot {\sqrt {A_{2}}}}$ ${\displaystyle h_{1}=\lambda \cdot {\sqrt {A_{1}}}}$

This can be used to rewrite the volume:

${\displaystyle V={\frac {\lambda \cdot {\sqrt {A_{1}}}\cdot A_{1}}{3}}-{\frac {\lambda \cdot {\sqrt {A_{2}}}\cdot A_{2}}{3}}={\frac {\lambda \cdot ({A_{1}}^{3/2}-{A_{2}}^{3/2})}{3}}}$.

Using the formula applied to and is the volume ${\displaystyle a^{3}-b^{3}=(a-b)\cdot (a^{2}+ab+b^{2})}$ ${\displaystyle a={\sqrt {A_{1}}}}$ ${\displaystyle b={\sqrt {A_{2}}}}$

${\displaystyle V={\frac {\lambda }{3}}\left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)\left({\sqrt {A_{1}}}^{2}+{\sqrt {A_{1}}}{\sqrt {A_{2}}}+{\sqrt {A_{2}}}^{2}\right)}$

or easier

${\displaystyle V={\frac {\lambda }{3}}\left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)\left(A_{1}+{\sqrt {A_{1}}}{\sqrt {A_{2}}}+A_{2}\right)}$.

The factor is the height : ${\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)}$ ${\displaystyle h}$

${\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)=\lambda \cdot {\sqrt {A_{1}}}-\lambda \cdot {\sqrt {A_{2}}}=h_{1}-{\frac {h_{2}}{\sqrt {A_{2}}}}\cdot {\sqrt {A_{2}}}=h}$.

This results in

${\displaystyle V={\frac {h}{3}}\left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)}$.

Degenerations

• If the base and top face a circle, a truncated cone is obtained , for which the same volume formula applies.
• If A2 strives towards A1, a prism is obtained whose volume formula is correspondingly simplified by A1 = A2 = A.
• If A2 tends towards 0, you get a pyramid .

literature

• Rolf Baumann: Geometry for the 9./10. Class . Centric stretching, Pythagorean theorem, circle and body calculations. 4th edition. Mentor-Verlag, Munich 2003, ISBN 3-580-63635-9 , pp. 95 ff .

See also

• Frustration

Web links

Commons : Truncated Pyramid  - Collection of images, videos, and audio files

Wiktionary: truncated pyramid  - explanations of meanings, word origins, synonyms, translations