How Many Faces, Edges, And Vertices Does A Pentagonal Pyramid Have?


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Ray Hueston answered
A pentagonal based pyramid is a three dimensional geometric shape with a base that has five equal length edges, with an angle between each edge of 108o. This can be calculated using the formula (((n-2)*180)/n) where n is the number of sides of the regular shape. In our example of a pentagon, n=5, therefore (((5-2)*180)/5) which is 540/3=108o. From each of the sides of the pentagon, five equal triangles with their bases on each of the five edges of the pentagon meet with their point at the top of the shape. In a regular pentagonal based pyramid, each of the triangles that are erected from the pentagon will be an equilateral triangle which means that each one will have equal length sides, and all the internal angles will be 60o, this shape is known as a regular pentagonal pyramid, and was classed by Norman Johnson in 1966 as number two, out of the ninety two Johnson solids, giving it the symbol J2. The resulting shape caused by this regular pentagon and five regular triangles will give rise to a geometric object with six faces, one of which is the pentagonal base, whilst the other five are the triangles which form the point of the pyramid. Thus it can be visualised that this shape has ten edges. There are not twenty which may be expected due to the sum (5+(3*5))=20 because a fast majority of the edges are shared, and thus are only counted once. Secondly, there are six vertices (corners or points) on this shape, one on each of the points of the pyramidal base, and one at the top of the shape where the three triangular faces converge. To assist in visualising the shape, it is possible to find online, interactive three dimensional models of this shape, one such can be found here where it is possible to drag the shape to look at it from any view, but also possible to "explode" the shape to see its composition of five triangles and one pentagon.

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