G.3.J Vocabulary of Three-Dimensional Figures

POLYHEDRONS Definition: A polyhedron is a solid formed by polygons where all the faces are flat. A face of a polyhedron is one of the flat polygon surfaces. An edge of a polyhedron is the line segment formed when two faces intersect. A vertex of a polyhedron is the point of intersection of three or more edges. In other words, a corner of the polyhedron. Edge Face Vertex

EXAMPLE Determine if the following figures are polyhedra. Explain your reasoning. There are no faces. There are no polygons. Yes All faces are polygons. No There are faces that are not polygons. Yes No All faces are polygons. Yes No All faces are polygons. One of the faces is not a polygon.

NAMING POLYHEDRONS When naming a polyhedron, use the same prefix as its polygon counterpart and place “hedron” at the end. # of faces + hedron

EXAMPLE Determine the number of faces of each polyhedron and then name it. 4 1. Number of Faces: _______ Name: ________________ Tetrahedron 6 2. Number of Faces: _______ Name: ________________ Hexahedron 7 3. Number of Faces: _______ Name: ________________ Heptahedron

8 Octahedron 10 Decahedron 12 Dodecahedron 20 Icosahedron 4. Number of Faces: _______ Name: ________________ Octahedron 10 5. Number of Faces: _______ Name: ________________ Decahedron 12 6. Number of Faces: _______ Name: ________________ Dodecahedron 20 7. Number of Faces: _______ Name: ________________ Icosahedron

REGULAR POLYHEDRA Definition: A regular polyhedron is a solid that uses only one type of regular polygon for its faces. There are only five regular polyhedra, called Platonic Solids. Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron There are only three shapes used as faces are: Equilateral Triangles Squares Regular Pentagons They are named after the Greek mathematician and philosopher Plato.

CONVEX vs. CONCAVE Definition: A polyhedron will be convex when any two points on its surface can be connected by a segment that lies completely in its interior. Said another way, pick any two points on different edges and make sure that the segment connecting them stays inside the polyhedron. All faces should be convex polygons.

Definition: A polyhedron will be concave when any two points on its surface can be connected by a segment that leaves the interior and returns. Said another way, pick any two points on different edges and see that the segment goes outside the polyhedron and then back in. If one face of the polyhedron is concave, then the entire polyhedron is said to be concave.

EXAMPLE Determine if the polyhedron is convex or concave. Then determine if it is regular. 1. 2. 3. Convex Non-regular Convex Regular Concave Non-regular

PRISMS Definition: A prism is a polyhedron with two congruent faces that are parallel to each other. The congruent parallel faces are called bases. The other faces are called lateral faces. Lateral faces will be some type of parallelogram. Lateral edges are the segments where the lateral faces intersect. When naming a prism, they are always named by the shape of their bases. Hexagonal Prism (Octahedron)

PYRAMIDS Definition: A pyramid is a polyhedron with one base and lateral faces that meet at one common vertex. The base must be a polygon. The lateral faces will always be triangles. When naming a pyramid, they are always named by the shape of their base. Rectangular Pyramid (Pentahedron) Hexagonal Pyramid (Heptahedron)

CYLINDERS Definition: A cylinder is a solid with congruent and parallel circles for its bases. A cylinder is NOT a polyhedron. The lateral surface is a rectangle that is wrapped around its circle bases.

CONES Definition: A cone is a solid with one circular base and a vertex that is not in the same plane as the base. A cone is NOT a polyhedron. You may think of a cone as a circle based pyramid.

RIGHT vs. OBLIQUE SOLIDS Definition: A right solid is a geometrical figure whose axis or lateral edges are perpendicular to its base. A right solid is one that stands up straight. The height of a solid is the perpendicular distance between the bases or the base and the vertex. The height of a solid is called its altitude.

Definition: An oblique solid is a geometrical figure whose axis or lateral edges are NOT perpendicular to its base. An oblique solid is one that is slanted to one side or the other. The length of the slanted lateral edge is called the slant height. The height (or altitude) must be drawn in so that it is perpendicular to the bases or the base and vertex.

EXAMPLE Name the solid and determine if it is right or oblique. 1. 2. 3. Right, Rectangular Prism (Hexahedron) Right, Pentagonal Pyramid (Hexahedron) Oblique, Triangular Prism (Pentahedron)

Oblique, Hexagonal Prism 5. 4. 6. Oblique Cone Oblique, Hexagonal Prism (Octahedron) Right Cylinder

SPHERES Definition: A sphere is a set of points in space that are equidistant from one given point. A sphere is a shell of points that are the same distance from the center. A sphere is a three-dimensional circle.

The point inside the sphere where all points are equidistant to is called the center of the sphere. A radius of the sphere is a segment drawn from the center to a point on the sphere. Any two-dimensional circle that contains the center of the sphere is called a great circle. Every great circle of a sphere splits a sphere into two congruent halves called hemispheres.