11.8: Chapter 11 Review
Difficulty Level: At Grade
Created by: CK12
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Keywords, Theorems, & Formulas
 Polyhedron
 A 3dimensional 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*}
(F) , vertices \begin{align*}(V)\end{align*}(V) , and edges \begin{align*}(E)\end{align*}(E) of a polyhedron can be related such that \begin{align*}F+V=E+2\end{align*}F+V=E+2 .
 Regular Polyhedron
 A polyhedron where all the faces are congruent regular polygons.
 Regular Tetrahedron
 A 4faced polyhedron where all the faces are equilateral triangles.
 Cube
 A 6faced polyhedron where all the faces are squares.
 Regular Octahedron
 An 8faced polyhedron where all the faces are equilateral triangles.
 Regular Dodecahedron
 A 12faced polyhedron where all the faces are regular pentagons.
 Regular Icosahedron
 A 20faced polyhedron where all the faces are equilateral triangles.
 CrossSection
 The intersection of a plane with a solid.
 Net
 An unfolded, flat representation of the sides of a threedimensional 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*}
r is the radius of the base and \begin{align*}h\end{align*}h is the height of the cylinder, then the surface area is \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}SA=2πr2+2πrh .
 Surface Area of a Regular Pyramid

If \begin{align*}B\end{align*}
B is the area of the base and \begin{align*}P\end{align*}P is the perimeter of the base and \begin{align*}l\end{align*}l is the slant height, then \begin{align*}SA=B+\frac{1}{2} Pl\end{align*}SA=B+12Pl .
 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 nonregular 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*}
l and base radius \begin{align*}r\end{align*}r is \begin{align*}SA= \pi r^2+ \pi rl\end{align*}SA=πr2+πrl .
 Volume
 The measure of how much space a threedimensional 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*}
s3 .
 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 nonoverlapping parts.
 Volume of a Rectangular Prism

If a rectangular prism is \begin{align*}h\end{align*}
h units high, \begin{align*}w\end{align*}w units wide, and \begin{align*}l\end{align*}l units long, then its volume is \begin{align*}V=l \cdot w \cdot h\end{align*}V=l⋅w⋅h .
 Volume of a Prism

If the area of the base of a prism is \begin{align*}B\end{align*}
B and the height is \begin{align*}h\end{align*}h , then the volume is \begin{align*}V=B \cdot h\end{align*}V=B⋅h .
 Cavalieri’s Principle
 If two solids have the same height and the same crosssectional 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 threedimensional 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 CK12 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:
Aug 15, 2016
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