7.1: Volume and Surface Area | Geometry Honors

How it works

Volume measures enclosed space in cubic units; surface area measures exposed faces in square units. For a prism use the area of its base times its perpendicular height. For a cone or pyramid include the factor one-third.

V_{\text{prism}}=Bh,\quad V_{\text{cone}}=\tfrac13\pi r^2h

Worked examples

Example 1

A rectangular prism measures $\displaystyle 3$ by $\displaystyle 4$ by $\displaystyle 5$ . Find its volume and surface area.

$\displaystyle V=3\cdot4\cdot5=60$ cubic units. $\displaystyle SA=2(12+15+20)=94$ square units.

Example 2

A cylinder has radius $\displaystyle 3$ and height $\displaystyle 4$ . Find its volume.

$\displaystyle V=\pi(3)^2(4)=36\pi$ cubic units.

Problems

Problem 7.1.1

Find the volume and surface area of a closed rectangular box of dimensions $\displaystyle 3,4,5$ .

Perspective rectangular prism labeled base dimensions 3 and 4 and perpendicular height 5. — The sketch is a perspective schematic; labels give the dimensions.

Show worked solution

Volume is $\displaystyle 60$ cubic units; surface area is $\displaystyle 2(3\cdot4+3\cdot5+4\cdot5)=94$ square units.

Perspective rectangular prism labeled base dimensions 3 and 4 and perpendicular height 5. The solution adds labeled relationships. — A horizontal section has area 12. Volume = 60 and closed-box surface area = 94.

Problem 7.1.2

A cone has radius $\displaystyle 3$ and perpendicular height $\displaystyle 8$ . Find its volume.

Show worked solution

$\displaystyle V=(1/3)\pi(9)(8)=24\pi$ cubic units.

Problem 7.1.3

A sphere has radius $\displaystyle 2$ . Find volume and surface area.

Show worked solution

$\displaystyle V=(4/3)\pi(2^3)=32\pi/3$ cubic units and $\displaystyle SA=4\pi(2^2)=16\pi$ square units.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 7.1.1

Problem 7.1.2

Problem 7.1.3

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