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# 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.
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.

1. Triangular Prism
2. Icosahedron
3. Cylinder
4. Cone
5. Tetrahedron
6. Pentagonal Prism
7. Octahedron
8. Hexagonal Pyramid
9. Octagonal Prism
10. Sphere
11. Cube
12. Dodecahedron

Match the formula with its description.

1. Volume of a Prism - A. \begin{align*}\frac{1}{3} \pi r^2 h\end{align*}
2. Volume of a Pyramid - B. \begin{align*}\pi r^2 h\end{align*}
3. Volume of a Cone - C. \begin{align*}4 \pi r^2\end{align*}
4. Volume of a Cylinder - D. \begin{align*}\frac{4}{3} \pi r^3\end{align*}
5. Volume of a Sphere - E. \begin{align*}\pi r^2+ \pi rl\end{align*}
6. Surface Area of a Prism - F. \begin{align*}2 \pi r^2+2 \pi rh\end{align*}
7. Surface Area of a Pyramid - G. \begin{align*}\frac{1}{3} Bh\end{align*}
8. Surface Area of a Cone - H. \begin{align*}Bh\end{align*}
9. Surface Area of a Cylinder - I. \begin{align*}B+\frac{1}{2} Pl\end{align*}
10. 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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