11.8: Chapter 11 Review
Difficulty Level: At Grade
Created by: CK-12
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 \begin{align*}(F)\end{align*}, vertices \begin{align*}(V)\end{align*}, and edges \begin{align*}(E)\end{align*} of a polyhedron can be related such that \begin{align*}F+V=E+2\end{align*}.
- 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 \begin{align*}r\end{align*} is the radius of the base and \begin{align*}h\end{align*} is the height of the cylinder, then the surface area is \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}.
- Surface Area of a Regular Pyramid
- If \begin{align*}B\end{align*} is the area of the base and \begin{align*}P\end{align*} is the perimeter of the base and \begin{align*}l\end{align*} is the slant height, then \begin{align*}SA=B+\frac{1}{2} Pl\end{align*}.
- 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 \begin{align*}l\end{align*} and base radius \begin{align*}r\end{align*} is \begin{align*}SA= \pi r^2+ \pi rl\end{align*}.
- 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 \begin{align*}s^3\end{align*}.
- 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 \begin{align*}h\end{align*} units high, \begin{align*}w\end{align*} units wide, and \begin{align*}l\end{align*} units long, then its volume is \begin{align*}V=l \cdot w \cdot h\end{align*}.
- Volume of a Prism
- If the area of the base of a prism is \begin{align*}B\end{align*} and the height is \begin{align*}h\end{align*}, then the volume is \begin{align*}V=B \cdot h\end{align*}.
- 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 \begin{align*}h\end{align*} and the radius is \begin{align*}r\end{align*}, then the volume would be \begin{align*}V=\pi r^2 h\end{align*}.
- Volume of a Pyramid
- If \begin{align*}B\end{align*} is the area of the base and \begin{align*}h\end{align*} is the height, then the volume of a pyramid is \begin{align*}V=\frac{1}{3} Bh\end{align*}.
- Volume of a Cone
- If \begin{align*}r\end{align*} is the radius of a cone and \begin{align*}h\end{align*} is the height, then the volume is \begin{align*}V=\frac{1}{3} \pi r^2 h\end{align*}.
- 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 \begin{align*}r\end{align*} is the radius, then the surface area of a sphere is \begin{align*}SA=4 \pi r^2\end{align*}.
- Volume of a Sphere
- If a sphere has a radius \begin{align*}r\end{align*}, then the volume of a sphere is \begin{align*}V=\frac{4}{3} \pi r^3\end{align*}.
- 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 \begin{align*}\frac{a}{b}\end{align*}, then the surface areas are in a ratio of \begin{align*}\left ( \frac{a}{b} \right )^2\end{align*}.
- Volume Ratio
- If two solids are similar with a scale factor of \begin{align*}\frac{a}{b}\end{align*}, then the volumes are in a ratio of \begin{align*}\left ( \frac{a}{b} \right )^3\end{align*}.
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. \begin{align*}\frac{1}{3} \pi r^2 h\end{align*}
- Volume of a Pyramid - B. \begin{align*}\pi r^2 h\end{align*}
- Volume of a Cone - C. \begin{align*}4 \pi r^2\end{align*}
- Volume of a Cylinder - D. \begin{align*}\frac{4}{3} \pi r^3\end{align*}
- Volume of a Sphere - E. \begin{align*}\pi r^2+ \pi rl\end{align*}
- Surface Area of a Prism - F. \begin{align*}2 \pi r^2+2 \pi rh\end{align*}
- Surface Area of a Pyramid - G. \begin{align*}\frac{1}{3} Bh\end{align*}
- Surface Area of a Cone - H. \begin{align*}Bh\end{align*}
- Surface Area of a Cylinder - I. \begin{align*}B+\frac{1}{2} Pl\end{align*}
- 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, 2012
Last Modified:
Feb 03, 2016
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