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# Chapter 11: Surface Area and Volume

Difficulty Level: At Grade Created by: CK-12

## Introduction

In this chapter we extend what we know about two-dimensional figures to three-dimensional shapes. First, we will determine the parts and different types of 3D shapes. Then, we will find the surface area and volume of prisms, cylinders, pyramids, cones, and spheres. Lastly, we will expand what we know about similar shapes and their areas to similar solids and their volumes.

Chapter Outline

## Summary

This chapter presents three-dimensional geometric figures beginning with polyhedrons, regular polyhedrons, and an explanation of Euler's Theorem. Three-dimensional figures represented as cross sections and nets are discussed. Then the chapter branches out to the formulas for surface area and volume of prisms, cylinders, pyramids, cones, spheres and composite solids. The relationship between similar solids and their surface areas and volumes are explored.

### Chapter Keywords

• Polyhedron
• Face
• Edge
• Vertex
• Prism
• Pyramid
• Euler’s Theorem
• Regular Polyhedron
• Regular Tetrahedron
• Cube
• Regular Octahedron
• Regular Dodecahedron
• Regular Icosahedron
• Cross-Section
• Net
• Lateral Face
• Lateral Edge
• Base Edge
• Right Prism
• Oblique Prism
• Surface Area
• Lateral Area
• Surface Area of a Right Prism
• Cylinder
• Surface Area of a Right Cylinder:
• Surface Area of a Regular Pyramid
• Cone
• Slant Height
• Surface Area of a Right Cone
• Volume
• Volume of a Cube Postulate
• Volume Congruence Postulate
• Volume of a Rectangular Prism
• Volume of a Prism
• Cavalieri’s Principle
• Volume of a Cylinder
• Volume of a Pyramid
• Volume of a Cone
• Sphere
• Great Circle
• Surface Area of a Sphere
• Volume of a Sphere
• Similar Solids
• Surface Area Ratio
• Volume Ratio

### Chapter Review

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. 13πr2h\begin{align*}\frac{1}{3} \pi r^2 h\end{align*}
2. Volume of a Pyramid - B. πr2h\begin{align*}\pi r^2 h\end{align*}
3. Volume of a Cone - C. 4πr2\begin{align*}4 \pi r^2\end{align*}
4. Volume of a Cylinder - D. 43πr3\begin{align*}\frac{4}{3} \pi r^3\end{align*}
5. Volume of a Sphere - E. πr2+πrl\begin{align*}\pi r^2+ \pi rl\end{align*}
6. Surface Area of a Prism - F. 2πr2+2πrh\begin{align*}2 \pi r^2+2 \pi rh\end{align*}
7. Surface Area of a Pyramid - G. 13Bh\begin{align*}\frac{1}{3} Bh\end{align*}
8. Surface Area of a Cone - H. Bh\begin{align*}Bh\end{align*}
9. Surface Area of a Cylinder - I. B+12Pl\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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