# 9.6 Solve geometry applications: volume and surface area  (Page 3/11)

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A rectangular box has length $9$ feet, width $4$ feet, and height $6$ feet. Find its volume and surface area.

1. 216 cu. ft
2. 228 sq. ft

A rectangular suitcase has length $22$ inches, width $14$ inches, and height $9$ inches. Find its volume and surface area.

1. 2,772 cu. in.
2. 1,264 sq. in.

## Volume and surface area of a cube

A cube    is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

$\begin{array}{ccccc}V=LWH\hfill & & & & S=2LH+2LW+2WH\hfill \\ V=s·s·s\hfill & & & & S=2s·s+2s·s+2s·s\hfill \\ V={s}^{3}\hfill & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\hfill \\ & & & & S=6{s}^{2}\hfill \end{array}$

So for a cube, the formulas for volume and surface area are $V={s}^{3}$ and $S=6{s}^{2}.$

## Volume and surface area of a cube

For any cube with sides of length $s,$

A cube is $2.5$ inches on each side. Find its volume and surface area.

## Solution

Step 1 is the same for both and , so we will show it just once.

 Step 1. Read the problem. Draw the figure and label it with the given information.
 ⓐ Step 2. Identify what you are looking for. the volume of the cube Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate. Write the appropriate formula. $V={s}^{3}$ Step 5. Solve. Substitute and solve. $V={\left(2.5\right)}^{3}$ $V=15.625$ Step 6. Check: Check your work. Step 7. Answer the question. The volume is 15.625 cubic inches.
 ⓑ Step 2. Identify what you are looking for. the surface area of the cube Step 3. Name. Choose a variable to represent it. let S = surface area Step 4. Translate. Write the appropriate formula. $S=6{s}^{2}$ Step 5. Solve. Substitute and solve. $S=6\cdot {\left(2.5\right)}^{2}$ $S=37.5$ Step 6. Check: The check is left to you. Step 7. Answer the question. The surface area is 37.5 square inches.

For a cube with side 4.5 meters, find the volume and surface area of the cube.

1. 91.125 cu. m
2. 121.5 sq. m

For a cube with side 7.3 yards, find the volume and surface area of the cube.

1. 389.017 cu. yd.
2. 319.74 sq. yd.

A notepad cube measures $2$ inches on each side. Find its volume and surface area.

## Solution

 Step 1. Read the problem. Draw the figure and label it with the given information.
 ⓐ Step 2. Identify what you are looking for. the volume of the cube Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate. Write the appropriate formula. $V={s}^{3}$ Step 5. Solve the equation. $V={2}^{3}$ $V=8$ Step 6. Check: Check that you did the calculations correctly. Step 7. Answer the question. The volume is 8 cubic inches.
 ⓑ Step 2. Identify what you are looking for. the surface area of the cube Step 3. Name. Choose a variable to represent it. let S = surface area Step 4. Translate. Write the appropriate formula. $S=6{s}^{2}$ Step 5. Solve the equation. $S=6\cdot {2}^{2}$ $S=24$ Step 6. Check: The check is left to you. Step 7. Answer the question. The surface area is 24 square inches.

A packing box is a cube measuring $4$ feet on each side. Find its volume and surface area.

1. 64 cu. ft
2. 96 sq. ft

A wall is made up of cube-shaped bricks. Each cube is $16$ inches on each side. Find the volume and surface area of each cube.

1. 4,096 cu. in.
2. 1536 sq. in.

## Find the volume and surface area of spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

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