Hexagonal Pyramid: Definition, Characteristics and Examples of Calculation

A hexagonal pyramid is a polyhedron formed by a hexagon, which is the base, and six triangles that start from the vertices of the hexagon and concur in a point outside the plane containing the base. At this point of concurrence it is known as the vertex or the apex of the pyramid.

A polyhedron is a closed three-dimensional geometric body whose faces are flat figures. A hexagon is a closed flat figure (polygon) formed by six sides. If the six sides have the same length and form equal angles, it is said to be regular; otherwise it is irregular.

Hexagonal pyramid

Index

  • 1 Definition
  • 2 characteristics
    • 2.1 Concave or convex
    • 2.2 Edges
    • 2.3 Apothem
    • 2.4 Denotes
  • 3 How to calculate the area? Formulas
    • 3.1 Calculation in irregular hexagonal pyramids
  • 4 How to calculate the volume? Formulas
    • 4.1 Calculation in irregular hexagonal pyramids
  • 5 Example
    • 5.1 Solution
  • 6 References

Definition

A hexagonal pyramid contains seven faces, the base and the six lateral triangles, of which the base is the only one that does not touch the vertex.

It is said that the pyramid is straight if all the lateral triangles are isosceles. In this case the height of the pyramid is the segment that goes from the vertex to the center of the hexagon.

In general, the height of a pyramid is the distance between the vertex and the plane of the base. It is said that the pyramid is oblique if not all the lateral triangles are isosceles.

If the hexagon is regular and the pyramid is also straight, it is said to be a regular hexagonal pyramid. Similarly, if the hexagon is irregular or the pyramid is oblique, it is said to be an irregular hexagonal pyramid.

characteristics

Concave or convex

A polygon is convex if the measure of all interior angles is less than 180 degrees. Geometrically, this is equivalent to saying that, given a pair of points within the polygon, the line segment that joins them is contained in the polygon. Otherwise, it is said that the polygon is concave.

Hexagonal pyramid 1

If the hexagon is convex, it is said that the pyramid is a hexagonal convex pyramid. Otherwise, it will be said that it is a concave hexagonal pyramid.

Edges

The edges of a pyramid are the sides of the six triangles that make it up.

Apothem

The apothem of the pyramid is the distance between the vertex and the sides of the base of the pyramid. This definition only makes sense when the pyramid is regular, because if it is irregular this distance varies depending on the triangle that is considered.

In contrast, in the regular pyramids the apothem corresponds to the height of each triangle (since each is isosceles) and will be the same in all triangles.

The apothem of the base is the distance between one of the sides of the base and the center of it. By the way it is defined, the apothem of the base also makes sense only in regular pyramids.

Denotes

The height of a hexagonal pyramid will be denoted by h , the apothem of the base (in the regular case) by APb and the apothem of the pyramid (also in the regular case) by AP .

A characteristic of regular hexagonal pyramids is that h , APb Y AP form a right triangle of hypotenuse AP and legs h Y APb . By the Pythagorean theorem you have to AP = √ (h ^ 2 + APb ^ 2).

Hexagonal pyramid 2

The previous image represents a regular pyramid.

How to calculate the area? Formulas

Consider a regular hexagonal pyramid. Be tailored to each side of the hexagon. Then A corresponds to the measure of the base of each triangle of the pyramid and, therefore, to the edges of the base.

The area of ​​a polygon is the product of the perimeter (the sum of the sides) by the apothem of the base, divided by two. In the case of a hexagon it would be 3 * A * APb.

It can be seen that the area of ​​a regular hexagonal pyramid is equal to six times the area of ​​each triangle of the pyramid plus the area of ​​the base. As previously mentioned, the height of each triangle corresponds to the apothem of the pyramid, AP.

Therefore, the area of ​​each triangle of the pyramid is given by A * AP / 2. Thus, the area of ​​a regular hexagonal pyramid is 3 * A * (APb + AP), where A is an edge of the base, APb is the apothem of the base and AP the apothem of the pyramid.

Calculation in irregular hexagonal pyramids

In the case of an irregular hexagonal pyramid there is no direct formula for calculating the area as in the previous case. This is because each triangle of the pyramid will have a different area.

In this case, the area of ​​each triangle must be calculated separately and the area of ​​the base. Then, the area of ​​the pyramid will be the sum of all the previously calculated areas.

How to calculate the volume? Formulas

The volume of a pyramid of regular hexagonal shape is the product of the height of the pyramid by the area of ​​the base between three. Thus, the volume of a regular hexagonal pyramid is given by A * APb * h, where A is an edge of the base, APb is the apothem of the base, and h is the height of the pyramid.

Calculation in irregular hexagonal pyramids

Analogously to the area, in the case of an irregular hexagonal pyramid there is no direct formula for calculating the volume since the edges of the base do not have the same measure because it is an irregular polygon.

In this case the base area must be calculated separately and the volume will be (h * Base area) / 3.

Example

Calculate the area and volume of a regular hexagonal pyramid of height 3 cm, whose base is a regular hexagon of 2 cm each side and the apothem of the base is 4 cm.

Solution

First you must calculate the apothem of the pyramid (AP), which is the only missing data. Looking at the image above, you can see that the height of the pyramid (3 cm) and the apothem of the base (4 cm) form a right triangle; therefore, to calculate the apothem of the pyramid we use the Pythagorean theorem:

AP = √ (3 ^ 2 + 9 ^ 2) = √ (25) = 5.

Thus, using the formula written above, it follows that the area is equal to 3 * 2 * (4 + 5) = 54cm ^ 2.

On the other hand, using the volume formula we obtain that the volume of the given pyramid is 2 * 4 * 3 = 24cm ^ 3.

References

  1. Billstein, R., Libeskind, S., & Lott, J. W. (2013). Mathematics: a problem-solving approach for basic education teachers. López Mateos Editores.
  2. Fregoso, R. S., & Carrera, S. A. (2005). Mathematics 3. Progress Editorial.
  3. Gallardo, G., & Pilar, P. M. (2005). Mathematics 6. Progress Editorial.
  4. Gutiérrez, C. T., & Cisneros, M. P. (2005). Mathematics Course 3rd. Progress Editorial.
  5. Kinsey, L., & Moore, T. E. (2006). Symmetry, Shape and Space: An Introduction to Mathematics Through Geometry (illustrated, reprint ed.). Springer Science & Business Media.
  6. Mitchell, C. (1999). Dazzling Math Line Designs (Illustrated ed.). Scholastic Inc.
  7. R., M. P. (2005). I draw 6th. Progress Editorial.

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