surface area and volume of a hexagonal prism

In the polyhedron above, line segment AB is inside of the polyhedron, and there is no way to draw any line segment between two points on the surface of the polyhedron such that the line segment will be outside of the polyhedron. Convex polyhedronĪ convex polyhedron is a polyhedron in which a line segment cannot be drawn between two points such that the segment goes outside the polyhedron. As long as a line segment can be drawn anywhere between two points on the surface of the polyhedron, the polyhedron is classified as concave.

In the polyhedron above, line segment AB is outside of the polyhedron, making it a concave polyhedron. For a polyhedron to be classified as concave, it must be possible to draw a line segment between two points such that the line segment goes outside the polyhedron. The base for the trapezoidal pyramid above is a trapezoid with unequal sides (so it is an irregular polygon).Ī concave polyhedron has at least one face that is a concave polygon. The base of the square pyramid above has a base that is a square (a regular polygon). A pyramid is typically named by the shape of its polygonal base. Pyramids are polyhedra that have polygons as bases and triangles as all other faces. The bases for the hexagonal prism above are irregular hexagons. The bases for the regular hexagonal prism above have bases that are regular hexagons. The lateral faces (the sides that are not bases) are parallelograms, rectangles, or squares. A prism is typically named by the shape of its polygonal bases. Prisms are polyhedra that have two congruent faces, called bases, lying in parallel planes. Likewise, a regular tetrahedron is the only regular pyramid that is also a regular polyhedron. A cube is the only regular prism that can also be classified as a regular polyhedron. Most regular prisms are generally not considered regular polyhedra. The figure below shows these shapes as well as the polyhedron net for each.Ī polyhedron net is a 2D pattern of polygons that can be modified to form each polyhedron. The 5 Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. There are only 5 Platonic solids, and their names indicate the number of faces they have. In a Platonic solid, the same number of faces meet at each vertex. Platonic solidsĪ Platonic solid, also referred to as a regular polyhedron, is a polyhedron whose faces are all congruent regular polygons. Only the Platonic solids are regular polyhedra prisms and pyramids are irregular polyhedra.

A regular polyhedron is a polyhedron whose faces are all congruent regular polygons any polyhedron that does not meet these conditions is considered irregular. They can further be categorized as regular or irregular and convex or concave. Polyhedrons typically fall into three groups: Platonic solids, prisms, and pyramids. Where F is the number of faces, V is the number of vertices, and E is the number of edges of a polyhedron.įor the hexagonal prism shown above, F = 8 (six lateral faces + two bases), V = 12, and E = 18: It states that the sum of the faces and vertices minus the number of edges always equals two: Euler's Theorem - Faces, edges, verticesĮuler's Theorem relates the number of faces, vertices, and edges of a polyhedron. Using this theorem, it is possible to determine the number of faces, edges, and vertices of a polyhedron given you know the number of at least two of them. There is a theorem that relates the faces, edges, and vertices of a polyhedron.

Vertices: points where three or more edges meet.įor example, a hexagonal prism has 8 faces (6 lateral and 2 bases), 18 edges, and 12 vertices.

Edges: the line segments created by two intersecting faces.

Polyhedrons typically have 1 or 2 faces that are also referred to as bases.

Faces: the polygons that form the polyhedron.

For example, a tetrahedron has 4 faces, a pentahedron has 5 faces, and a hexahedron has 6 faces. They are named based on the number of faces they have. Polyhedrons are made up of faces, edges, and vertices. Real world examples of polyhedra include the Great Pyramid of Giza, concrete blocks, dice, bricks, and more. A polyhedron is a three-dimensional ( 3D) solid figure that is made up of only polygons, meaning that the figure has no curves.

Surface area and volume of a hexagonal prism