Keywords, Theorems, & Formulas
- Polyhedron
- A 3-dimensional figure that is formed by polygons that enclose a region in space.
- Face
- Each polygon in a polyhedron is called a face.
- Edge
- The line segment where two faces intersect is called an edge
- Vertex
- the point of intersection of two edges is a vertex.
- Prism
- A polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles.
- Pyramid
- A polyhedron with one base and all the lateral sides meet at a common vertex. The lateral sides are triangles.
- Euler’s Theorem
- The number of faces , vertices , and edges of a polyhedron can be related such that .
- Regular Polyhedron
- A polyhedron where all the faces are congruent regular polygons.
- Regular Tetrahedron
- A 4-faced polyhedron where all the faces are equilateral triangles.
- Cube
- A 6-faced polyhedron where all the faces are squares.
- Regular Octahedron
- An 8-faced polyhedron where all the faces are equilateral triangles.
- Regular Dodecahedron
- A 12-faced polyhedron where all the faces are regular pentagons.
- Regular Icosahedron
- A 20-faced polyhedron where all the faces are equilateral triangles.
- Cross-Section
- The intersection of a plane with a solid.
- Net
- An unfolded, flat representation of the sides of a three-dimensional shape.
- Lateral Face
- A face that is not the base.
- Lateral Edge
- The edges between the lateral faces are called lateral edges.
- Base Edge
- The edges between the base and the lateral faces are called base edges.
- Right Prism
- All prisms are named by their bases, so the prism to the right is a pentagonal prism. This particular prism is called a right prism
- Oblique Prism
- Oblique prisms lean to one side or the other and the height is outside the prism.
- Surface Area
- The sum of the areas of the faces.
- Lateral Area
- The sum of the areas of the lateral faces.
- Surface Area of a Right Prism
- The surface area of a right prism is the sum of the area of the bases and the area of each rectangular lateral face.
- Cylinder
- A solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed.
- Surface Area of a Right Cylinder
- If is the radius of the base and is the height of the cylinder, then the surface area is .
- Surface Area of a Regular Pyramid
- If is the area of the base and is the perimeter of the base and is the slant height, then .
- Cone
- A solid with a circular base and sides taper up towards a common vertex.
- Slant Height
- All regular pyramids also have a slant height that is the height of a lateral face. Because of the nature of regular pyramids, all slant heights are congruent. A non-regular pyramid does not have a slant height.
- Surface Area of a Right Cone
- The surface area of a right cone with slant height and base radius is .
- Volume
- The measure of how much space a three-dimensional figure occupies.
- Volume of a Cube Postulate
- The volume of a cube is the cube of the length of its side, or .
- Volume Congruence Postulate
- If two solids are congruent, then their volumes are congruent.
- Volume Addition Postulate
- The volume of a solid is the sum of the volumes of all of its non-overlapping parts.
- Volume of a Rectangular Prism
- If a rectangular prism is units high, units wide, and units long, then its volume is .
- Volume of a Prism
- If the area of the base of a prism is and the height is , then the volume is .
- Cavalieri’s Principle
- If two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.
- Volume of a Cylinder
- If the height of a cylinder is and the radius is , then the volume would be .
- Volume of a Pyramid
- If is the area of the base and is the height, then the volume of a pyramid is .
- Volume of a Cone
- If is the radius of a cone and is the height, then the volume is .
- Sphere
- The set of all points, in three-dimensional space, which are equidistant from a point.
- Great Circle
- The great circle is a plane that contains the diameter.
- Surface Area of a Sphere
- If is the radius, then the surface area of a sphere is .
- Volume of a Sphere
- If a sphere has a radius , then the volume of a sphere is .
- Similar Solids
- Two solids are similar if and only if they are the same type of solid and their corresponding linear measures (radii, heights, base lengths, etc.) are proportional.
- Surface Area Ratio
- If two solids are similar with a scale factor of , then the surface areas are in a ratio of .
- Volume Ratio
- If two solids are similar with a scale factor of , then the volumes are in a ratio of .
Review Questions
Match the shape with the correct name.
- Triangular Prism
- Icosahedron
- Cylinder
- Cone
- Tetrahedron
- Pentagonal Prism
- Octahedron
- Hexagonal Pyramid
- Octagonal Prism
- Sphere
- Cube
- Dodecahedron
Match the formula with its description.
- Volume of a Prism - A.
- Volume of a Pyramid - B.
- Volume of a Cone - C.
- Volume of a Cylinder - D.
- Volume of a Sphere - E.
- Surface Area of a Prism - F.
- Surface Area of a Pyramid - G.
- Surface Area of a Cone - H.
- Surface Area of a Cylinder - I.
- Surface Area of a Sphere - J. The sum of the area of the bases and the area of each rectangular lateral face.
Texas Instruments Resources
In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9696.
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Date Created:
Feb 22, 2012Last Modified:
Dec 13, 2013
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