- Definition
- characteristics
- Concave or convex
- Edges
- Apothem
- Denotations
- How to calculate the area? Formulas
- Calculation in irregular hexagonal pyramids
- How to calculate the volume? Formulas
- Calculation in irregular hexagonal pyramids
- Example
- Solution
- References
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 meet at a point outside the plane that contains the base. This point of concurrency is known as the vertex or apex of the pyramid.
A polyhedron is a closed three-dimensional geometric body whose faces are plane figures. A hexagon is a closed plane figure (polygon) made up of six sides. If all six sides are the same length and form equal angles, it is said to be regular; otherwise it is irregular.
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.
The pyramid is said to be 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. The pyramid is said to be 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 the polygon is said to be concave.
If the hexagon is convex, the pyramid is said to be a convex hexagonal pyramid. Otherwise, it will be said to be 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.
On the other hand, in regular pyramids the apothem will correspond to the height of each triangle (since each one is isosceles) and it 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. From the way it is defined, the apothem of the base also makes sense only in regular pyramids.
Denotations
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, and AP form a right triangle with hypotenuse AP and legs h and APb. By the Pythagorean theorem we have that AP = √ (h ^ 2 + APb ^ 2).
The image above represents a regular pyramid.
How to calculate the area? Formulas
Consider a regular hexagonal pyramid. Let A be the measure of 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) and 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 in 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 to calculate the area as in the previous case. This is because each triangle in the pyramid is going to 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 areas previously calculated.
How to calculate the volume? Formulas
The volume of a pyramid of regular hexagonal shape is the product of the height of the pyramid and the area of the base divided by 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 to calculate the volume since the edges of the base do not have the same measurement because it is an irregular polygon.
In this case, the area of the base must be calculated separately and the volume will be (h * Area of the base) / 3.
Example
Find the area and volume of a regular hexagonal pyramid with a height of 3 cm, the base of which is a regular hexagon of 2 cm on each side and the apothem of the base is 4 cm.
Solution
First, the apothem of the pyramid (AP) must be calculated, which is the only missing data. Looking at the image above, it can be seen 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, the Pythagorean theorem is used:
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 it is obtained that the volume of the given pyramid is 2 * 4 * 3 = 24cm ^ 3.
References
- Billstein, R., Libeskind, S., & Lott, JW (2013). Mathematics: A Problem Solving Approach for Elementary Education Teachers. López Mateos Editors.
- Fregoso, RS, & Carrera, SA (2005). Mathematics 3. Editorial Progreso.
- Gallardo, G., & Pilar, PM (2005). Mathematics 6. Editorial Progreso.
- Gutiérrez, CT, & Cisneros, MP (2005). 3rd Mathematics Course. Editorial Progreso.
- Kinsey, L., & Moore, TE (2006). Symmetry, Shape and Space: An Introduction to Mathematics Through Geometry (illustrated, reprint ed.). Springer Science & Business Media.
- Mitchell, C. (1999). Dazzling Math Line Designs (Illustrated ed.). Scholastic Inc.
- R., MP (2005). I draw 6th. Editorial Progreso.