# Pentagon

Pronunciation: /ˈpɛntəˌgɒn/ Explain

 Click on the blue points and drag them to change the figurel. Manipulative 1 - Pentagon Created with GeoGebra.

A pentagon is a five-sided polygon. A regular pentagon is a five-sided equilateral polygon. Click on the blue points in manipulatives 1 and 2 and drag them to change the figure.

 Click on the blue points and drag them to change the figure. Why are only two points shown in the figure? Manipulative 2 - Regular Pentagon Created with GeoGebra.

### Properties of a Regular Pengagon

 Click on the blue points and drag them to change the figure. Click on the check boxes to see the various properties of a pentagon. Manipulative 3 - Properties of a Pentagon Created with GeoGebra.
• A pentagon has five sides.
• The center of a pentagon can be found at the intersection of the perpendicular bisectors of any two sides.
• There are many tilings in pentagonal patterns and tilings that use pentagons. See below.
• The central angle of a pentagon measures 72° or 2π/5 radians. The central angle is the angle between two line segments from the center of the pentagon to two adjacent vertices. See manipulative 2.
• The internal angle of a pentagon measures 108° or 3π/5 radians. The internal angle is the angle between two sides on the inside of the pentagon.
• The area of a regular pentagon in terms of one of it sides is .
• The apothem of a regular pentagon is a line segment from the midpoint of one side of the pentagon to the pentagon's center.
• There are many geometric nets that include pentagons. See below.

### Tilings of Pentagons

•  Figure 1: Tiling of a pentagon. Courtesy John Savard.
•  Figure 2: Tiling of a pentagon.

### Geometric nets that include pentagons

ExampleNamePrintable Net
Dodecahedronnet_dodecahedron.pdf
Icosidodecahedronnet_icosidodecahedron.pdf
Pentagonal Antiprismnet_pentagonal_antiprism.pdf
Pentagonal Prismnet_pentagonal_prism.pdf
Pentagonal Pyramidnet_pentagonal_pyramid.pdf
Rhombidodecahedronnet_rhombidodecahedron.pdf
Rombicosidodecahedronnet_rombicosidodecahedron.pdf
Snub Dodecahedronnet_snub_dodecahedron.pdf
Truncated Icosidodecahedronnet_trunc_icosidodecahedron.pdf
Truncated Cuboctahedronnet_truncated_cuboctahedron.pdf
Truncated Icosahedronnet_truncated_icosahedron.pdf
Truncated Octahedronnet_truncated_octahedron.pdf
Truncated Tetrahedronnet_truncated_tetrahedron.pdf

### References

1. McAdams, David E.. All Math Words Dictionary, pentagon. 2nd Classroom edition 20150108-4799968. pg 136. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Stöcker, K.H.. The Elements of Constructive Geometry, Inductively Presented. pg 27. Translated by Noetling, William A.M, C.E.. www.archive.org. Silver, Burdett & Company. 1897. Last Accessed 12/3/2018. http://www.archive.org/stream/elementsofconstr00noetrich#page/27/mode/1up. Buy the book
3. Casey, John, LL.D., F.R.S.. The First Six Books of the Elements of Euclid. pg 9. Translated by Casey, John, LL.D. F.R.S.. www.archive.org. Hodges, Figgis & Co.. 1890. Last Accessed 12/3/2018. http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/9/mode/1up/search/pentagon. Buy the book
4. Keller, Samuel Smith. Mathematics for Engineering Students, Plane and Solid Geometry. pg 9. www.archive.org. D. Van Nostrand Company. 1908. Last Accessed 12/3/2018. http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/9/mode/1up/search/pentagon. Buy the book

McAdams, David E. Pentagon. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/p/pentagon.html.

### Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/28/2018: Corrected spelling. (McAdams, David E.)
12/31/2009: Initial version. (McAdams, David E.)

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