9.6 Solve geometry applications: volume and surface area (Page 2/11)

Prealgebra

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Notice for each of the three faces you see, there is an identical opposite face that does not show.

\begin{array}{l} S = (front + back) + (left side + right side) + (top + bottom) \\ S = (2 \cdot front) + (2 \cdot left side) + (2 \cdot top) \\ S = 2 \cdot 12 + 2 \cdot 6 + 2 \cdot 8 \\ S = 24 + 12 + 16 \\ S = 52 sq. units \end{array}

The surface area $S$ of the rectangular solid shown in [link] is $52$ square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see [link] ). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

S = 2 L H + 2 L W + 2 W H

A rectangular solid is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH. — For each face of the rectangular solid facing you, there is another face on the opposite side. There are $6$ faces in all.

Volume and surface area of a rectangular solid

For a rectangular solid with length $L,$ width $W,$ and height $H :$

A rectangular solid is shown. The sides are labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.

Doing the Manipulative Mathematics activity “Painted Cube” will help you develop a better understanding of volume and surface area.

For a rectangular solid with length $14$ cm, height $17$ cm, and width $9$ cm, find the ⓐ 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 rectangular solid
Step 3. Name. Choose a variable to represent it.	Let $V$ = volume
Step 4. Translate. Write the appropriate formula. Substitute.	$V = L W H$ $V = 14 \cdot 9 \cdot 17$
Step 5. Solve the equation.	$V = 2,142$
Step 6. Check We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is $1,034$ square centimeters.

ⓑ
Step 2. Identify what you are looking for.	the surface area of the solid
Step 3. Name. Choose a variable to represent it.	Let $S$ = surface area
Step 4. Translate. Write the appropriate formula. Substitute.	$S = 2 L H + 2 L W + 2 W H$ $S = 2 (14 \cdot 17) + 2 (14 \cdot 9) + 2 (9 \cdot 17)$
Step 5. Solve the equation.	$S = 1,034$
Step 6. Check: Double-check with a calculator.
Step 7. Answer the question.	The surface area is 1,034 square centimeters.

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Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length $8$ feet, width $9$ feet, and height $11$ feet.

ⓐ 792 cu. ft
ⓑ 518 sq. ft

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Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length $15$ feet, width $12$ feet, and height $8$ feet.

ⓐ 1,440 cu. ft
ⓑ 792 sq. ft

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A rectangular crate has a length of $30$ inches, width of $25$ inches, and height of $20$ inches. 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 crate
Step 3. Name. Choose a variable to represent it.	let $V$ = volume
Step 4. Translate. Write the appropriate formula. Substitute.	$V = L W H$ $V = 30 \cdot 25 \cdot 20$
Step 5. Solve the equation.	$V = 15,000$
Step 6. Check: Double check your math.
Step 7. Answer the question.	The volume is 15,000 cubic inches.

ⓑ
Step 2. Identify what you are looking for.	the surface area of the crate
Step 3. Name. Choose a variable to represent it.	let $S$ = surface area
Step 4. Translate. Write the appropriate formula. Substitute.	$S = 2 L H + 2 L W + 2 W H$ $S = 2 (30 \cdot 20) + 2 (30 \cdot 25) + 2 (25 \cdot 20)$
Step 5. Solve the equation.	$S = 3,700$
Step 6. Check: Check it yourself!
Step 7. Answer the question.	The surface area is 3,700 square inches.

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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