Perimeter and Fencing Problems Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

A city is installing a new fence around a rectangular dog park. The park has a length of $80$ feet and a width of $60$ feet. How many feet of fencing are required to completely enclose the park?

Show answer & explanation

Answer: 280

Answer

280

The fencing must cover the entire outer edge of the rectangular park. The perimeter is found by adding the length and width together, then multiplying the sum by 2 to account for all four sides. Thus, 2 × (80 + 60) = 280 feet.

Step-by-Step Solution

1

Identify the given dimensions of the rectangular park.

Length = 80 feet, Width = 60 feet.

To determine the values needed for the perimeter formula.

2

Apply the formula for the perimeter of a rectangle.

Perimeter = 2 × (80 + 60)

The total fencing required is equal to the perimeter of the rectangle.

3

Perform the addition inside the parentheses.

80 + 60 = 140

Following the order of operations.

4

Multiply the sum by 2.

2 × 140 = 280

To account for all four sides of the rectangular boundary.

Key Concept

The perimeter of a rectangle is the total distance around the outside edge, calculated using the formula 2 × (Length + Width).

Question 2Question →

A homeowner is enclosing a rectangular garden with a new fence. The garden has a length of $5$ yards and a width of $8$ feet. The homeowner will leave a $36$ -inch wide opening along one of the sides for a wooden gate. If the fencing material costs $\$14.50$ per linear foot, what is the total cost of the fencing material required?

\$333.50

\$623.50

\$1,740.00

\$6,235.00

Show answer & explanation

Answer: $\$623.50$

Answer

\$623.50

To find the correct total cost, all dimensions must first be converted to feet (

5

yards

= 15

feet,

36

inches

= 3

feet). The perimeter is found by adding all four sides (

15 + 15 + 8 + 8 = 46

feet), then subtracting the

3

-foot gate opening to determine that exactly

43

feet of fencing is needed. Multiplying

43

feet by the

\$14.50

cost per foot yields the correct total of

\$623.50

.

Step-by-Step Solution

1

Convert all given measurements to a single common unit (feet) since the cost is given per linear foot.

Length:

5 \text{ yards} = 15 \text{ feet}

. Gate width:

36 \text{ inches} = 3 \text{ feet}

.

Measurements must be in the same units as the pricing rate to calculate accurate costs.

2

Calculate the total perimeter of the rectangular garden.

P = 2 \times (\text{Length} + \text{Width}) = 2 \times (15 + 8) = 2 \times 23 = 46 \text{ feet}

.

The perimeter represents the total outline of the garden before accounting for the gate.

3

Subtract the width of the gate opening from the total perimeter.

46 \text{ feet} - 3 \text{ feet} = 43 \text{ feet}

.

Fencing material is not needed across the opening where the wooden gate will be installed.

4

Multiply the required length of fencing by the cost per linear foot.

43 \text{ feet} \times \$14.50/\text{foot} = \$623.50

.

This provides the final total monetary cost for the required material.

Key Concept

Perimeter and Unit Conversion

Question 3Question →

A military logistics facility is upgrading the security fence around a rectangular motor pool. The motor pool is $50$ yards long and $84$ feet wide. The new design requires leaving a $24$ -foot gap for a main vehicle gate and two separate $3$ -foot gaps for pedestrian access. The security fencing is purchased in pre-assembled panels that are $72$ inches long each. If each panel costs $\$120$ , what is the total cost of the fencing panels needed to enclose the motor pool?

Show answer & explanation

Answer: 8760

Answer

8760 dollars

The correct calculation requires converting all linear measurements to a common unit (feet), calculating the perimeter of the rectangle, subtracting the required gaps for the gates, dividing the remaining length by the panel length to find the total number of panels, and finally multiplying by the cost per panel.

Step-by-Step Solution

1

Convert yards and inches to feet.

Length is 150 feet; panel length is 6 feet.

To ensure consistency in perimeter and material calculations before determining the number of panels.

2

Calculate the full perimeter of the rectangular motor pool.

468 feet.

To find the total boundary length before accounting for the gaps left for gates.

3

Subtract the total width of the gate gaps from the full perimeter.

438 feet.

Because fencing panels are not installed where the vehicle and pedestrian gates are located.

4

Divide the required fencing length by the length of a single panel.

73 panels.

To find how many discrete 6-foot sections make up the 438-foot fence line.

5

Multiply the number of panels by the cost per panel.

$8,760.

To find the final total price of purchasing the required materials.

Key Concept

Calculating perimeter with unit conversions and subtracting gaps to find total material cost.

Estimated Time:2m 0s

Question 4Question →

A contractor is installing a decorative border around a newly built rectangular patio. The patio is $15$ feet long and $10$ feet wide. If the border material costs $\$2$ per foot, what is the total cost of the material needed?

\$50

\$100

\$150

\$300

Show answer & explanation

Answer: $\$100$

Answer

The total cost of the material needed is

\$100

.

The correct answer is found by first calculating the perimeter of the rectangle, which is

2 \times (15 + 10) = 50

feet. Then, this total length is multiplied by the price per foot (

50 \times \$2

) to get a final cost of

\$100

.

Step-by-Step Solution

1

Identify that the border goes around the outside of the patio, which means you need to find the perimeter.

Perimeter =

2 \times (\text{Length} + \text{Width})

Understanding the difference between the boundary (perimeter) and the inside space (area).

2

Calculate the total perimeter of the rectangular patio.

Perimeter =

2 \times (15 + 10) = 2 \times 25 = 50

feet.

This determines the total linear feet of border material required.

3

Multiply the total perimeter by the cost per foot to find the total cost.

50 \text{ feet} \times \$2\text{/foot} = \$100

.

Applying the unit rate to the total length gives the final financial cost.

Key Concept

Calculating the perimeter of a rectangle and applying a unit rate to find total cost.

Estimated Time:45s

Question 5Question →

A volunteer group is installing a wooden fence around a rectangular community garden. The garden measures $30$ feet in length and $18$ feet in width. How many feet of fencing are needed to completely enclose the garden?

Show answer & explanation

Answer: 96

Answer

96

The amount of fencing needed to completely enclose a rectangular space is equal to the perimeter of that space. The formula for the perimeter of a rectangle is

P = 2 \times (L + W)

. By substituting the given length (

30

feet) and width (

18

feet), the perimeter is

2 \times (30 + 18) = 2 \times 48 = 96

feet.

Step-by-Step Solution

1

Identify the formula for the perimeter of a rectangle.

P = 2 \times (\text{Length} + \text{Width})

The total fencing needed to completely enclose a rectangular area is equal to its perimeter.

2

Substitute the given dimensions into the formula.

P = 2 \times (30 + 18) = 2 \times 48 = 96

This calculates the total linear feet of the boundary for a garden with a length of 30 feet and a width of 18 feet.

Key Concept

The total length of fencing required to enclose a rectangular area is equal to its perimeter, calculated as

2 \times (\text{length} + \text{width})

.

Question 6Question →

A school is installing a new fence around a square play area. Each side of the play area measures $5$ yards. If the fencing material costs $\$0.50$ per foot, what is the total cost of the fencing required?

\$10.00

\$30.00

\$112.50

\$300.00

Show answer & explanation

Answer: $\$30.00$

Answer

\$30.00

The correct answer is found by first ensuring the units match. Since the cost is provided per foot, convert the

5

-yard sides into feet (

5 \times 3 = 15

feet). Next, find the perimeter of the square play area (

15 \times 4 = 60

feet). Finally, multiply the

60

-foot perimeter by the

\$0.50

per foot cost to determine a total fencing cost of

\$30.00

.

Step-by-Step Solution

1

Convert the side length from yards to feet.

5 \text{ yards} \times 3 = 15 \text{ feet}

The cost of fencing is given per foot, so all geometric measurements must be converted to feet before calculating the total cost.

2

Calculate the perimeter of the square play area.

4 \times 15 \text{ feet} = 60 \text{ feet}

The fence goes around the outer boundary, which is the perimeter of the square.

3

Calculate the total cost by multiplying the perimeter by the cost per foot.

60 \text{ feet} \times \$0.50/\text{foot} = \$30.00

To find the total cost, multiply the total linear feet of fencing needed by the given price per foot.

Key Concept

Calculating perimeter of a square and applying unit conversions to determine total cost.

Question 7Question →

A rancher is building a fence around a rectangular grazing pasture. The pasture is $140$ feet long and $70$ feet wide. If the rancher needs to leave a $10$ -foot wide opening for a gate, how many total feet of fencing material will be required?

Show answer & explanation

Answer: 410

Answer

410 feet

To find the total amount of fencing required, you first determine the perimeter of the rectangle, which is

140 + 140 + 70 + 70 = 420

feet. Because there is a

10

-foot gate that does not require fencing, you subtract

10

from the total perimeter, leaving

410

feet.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular pasture.

2 \times (140 + 70) = 420

feet.

The fence runs along the entire outer boundary of the pasture, which requires finding the perimeter of the rectangle.

2

Subtract the gate opening from the total perimeter.

420 - 10 = 410

feet.

The

10

-foot opening for the gate does not use fencing material, so it must be subtracted from the total.

Key Concept

Calculating perimeter of a rectangle and adjusting for gaps or openings.

Question 8Question →

A commercial facility is securing a right-triangular storage lot. The two perpendicular boundaries of the lot measure $60$ yards and $80$ yards. Security fencing will be installed along the entire perimeter to create a closed loop. If a support post must be placed every $12$ feet, how many total posts are required to complete the fence?

20

35

60

200

Show answer & explanation

Answer: 60

Answer

60

The correct answer is found by first determining the hypotenuse of the right-triangular lot, which is 100 yards (

60^2 + 80^2 = 100^2

). The total perimeter is 240 yards (60 + 80 + 100). Because the post spacing is given in feet, the perimeter must be converted to 720 feet (240 × 3). Finally, dividing the 720-foot boundary by the 12-foot spacing requirement yields exactly 60 posts.

Step-by-Step Solution

1

Find the length of the third side of the lot using the Pythagorean theorem (

a^2 + b^2 = c^2

).

60^2 + 80^2 = 3600 + 6400 = 10000

. The square root of

10000

is

100

yards.

The boundary of a right-triangular lot includes the hypotenuse, which must be found to calculate the total perimeter.

2

Calculate the total perimeter of the triangular lot.

60 + 80 + 100 = 240

yards.

The fencing will enclose the entire boundary of the lot.

3

Convert the perimeter into feet to match the unit of the post spacing.

240 \text{ yards} \times 3 \text{ feet/yard} = 720 \text{ feet}

.

Measurements must be in the same unit (feet) before dividing.

4

Divide the total perimeter by the distance required between each post.

720 \div 12 = 60

posts.

For a closed loop, the total number of posts equals the total perimeter divided by the spacing distance.

Key Concept

Calculating the perimeter of a right triangle using the Pythagorean theorem, converting units of measurement, and determining spacing requirements for closed boundaries.

Alternative Method

Instead of converting the perimeter to feet, convert the post spacing to yards. Since 3 feet equals 1 yard, 12 feet equals 4 yards. Divide the 240-yard perimeter by 4 yards to get 60 posts.

Estimated Time:1m 30s

Question 9Question →

A renewable energy company is installing a chain-link fence around a newly constructed rectangular solar farm. The solar farm measures $120$ yards in length and $85$ yards in width. The fence will enclose the entire perimeter, with the exception of two access gates that each measure $5$ yards wide. If the fencing material costs $\$22$ per linear yard, what is the total cost of the fencing material in dollars?

Show answer & explanation

Answer: 8800

Answer

The total cost of the fencing material is 8800 dollars.

The total perimeter of the solar farm is

410

yards (

120 + 120 + 85 + 85

). Subtracting the

10

yards for the two

5

-yard gates leaves

400

yards of fencing to be installed. Multiplying

400

yards by the

\$22

per yard cost gives a total of

\$8800

.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular solar farm.

P = 2 \times (120 + 85) = 2 \times 205 = 410

yards.

The perimeter formula for a rectangle is

P = 2(L + W)

, which represents the total boundary length before any openings are considered.

2

Determine the total width of the two access gates.

2 \times 5 = 10

yards.

Since there are two gates and each is

5

yards wide, their combined width must be subtracted from the total perimeter.

3

Calculate the actual length of fencing needed.

410 - 10 = 400

yards.

The fencing covers the perimeter minus the space left open for the access gates.

4

Calculate the total cost of the fencing material.

400 \times 22 = 8800

.

Multiplying the total yards of fencing by the cost per yard gives the total cost in dollars.

Key Concept

Calculating the perimeter of a rectangle, subtracting openings, and finding the total cost based on a unit price.

Question 10Question →

A homeowner is planning to enclose a rectangular residential lot with a new privacy fence. The lot has a length of $20$ yards and a width of $45$ feet. The fencing material costs $\$12.50$ per linear foot. If the homeowner needs to leave a $6$ -foot opening for a driveway gate, what will be the total cost of the fencing materials?

\$255

\$1,550

\$2,550

\$33,675

Show answer & explanation

Answer: $\$2,550$

Answer

The total cost of the fencing materials is

\$2,550

.

The correct answer is found by first converting all units to feet. The length of

20

yards is equal to

60

feet. The total perimeter of the rectangular lot is

2 \times (60 + 45) = 210

feet. Subtracting the

6

-foot gap for the gate leaves

204

linear feet of fencing required. Multiplying this by the cost of

\$12.50

per foot yields a total cost of

\$2,550

.

Step-by-Step Solution

1

Convert the length of the lot from yards to feet.

20 \text{ yards} \times 3 = 60 \text{ feet}

All measurements must be in the same unit (feet) because the fencing cost is given per linear foot.

2

Calculate the total perimeter of the rectangular lot.

P = 2 \times (60 + 45) = 2 \times 105 = 210 \text{ feet}

The fencing will go around the entire boundary of the rectangular lot.

3

Subtract the length of the driveway gate from the total perimeter.

210 - 6 = 204 \text{ feet}

Fencing material is not needed for the space where the gate will be installed.

4

Calculate the total cost by multiplying the fencing length by the cost per foot.

204 \times \$12.50 = \$2,550

The total cost is the required length in feet multiplied by the price per linear foot.

Key Concept

Calculating the perimeter of a rectangle with mixed units and adjusting for an opening.

Question 11Question →

A military engineering unit is installing a security fence around a rectangular vehicle depot. The depot is $150$ yards long and $80$ yards wide. The fencing plan includes leaving a single $30$ -foot wide opening for a main access gate. If the fencing material costs $\$15$ per linear foot, what is the total cost of the fencing material needed? (Do not include commas or dollar signs in your answer)

Show answer & explanation

Answer: 20250

Answer

20250

To find the total cost, first determine the total perimeter of the depot in yards:

2 \times (150 + 80) = 460

yards. Since the fencing cost and gate width are in feet, convert the perimeter to feet:

460 \times 3 = 1380

feet. Next, subtract the

30

-foot gate opening where no fencing is needed:

1380 - 30 = 1350

feet of fencing. Finally, multiply the required fencing by the cost per foot:

1350 \times \$15 = \$20,250

.

Step-by-Step Solution

1

Calculate the perimeter of the rectangular depot in yards.

2(150 + 80) = 2(230) = 460

yards.

The fencing will enclose the boundary of the rectangular area.

2

Convert the perimeter from yards to feet.

460 \times 3 = 1380

feet.

The cost of the fencing is given per linear foot and the gate is measured in feet, so all measurements must be in feet.

3

Subtract the gate opening from the total perimeter.

1380 - 30 = 1350

feet.

Fencing material is not needed for the gate opening.

4

Calculate the total cost of the fencing material.

1350 \times 15 = 20250

dollars.

The total length of fencing required is multiplied by the cost per linear foot.

Key Concept

Calculating perimeter, converting units (yards to feet), and determining total cost based on a linear rate.

Question 12Question →

An event coordinator is setting up a temporary security perimeter for an outdoor festival. The secured VIP zone is rectangular, measuring $30$ yards in length and $20$ yards in width. An un-fenced opening of $5$ yards must be left on one side for the main entrance gate. If the temporary fencing costs $\$4.50$ per linear foot, what is the total cost, in dollars, of the fencing required?

Show answer & explanation

Answer: 1282.5

Answer

1282.5

The correct total cost is

1282.50. To find this, the full perimeter of the 30-by-20 yard rectangle is calculated as 100 yards. After subtracting the 5-yard opening, 95 yards of fencing are needed. Because the fencing is priced per foot, the 95 yards must be converted to 285 feet by multiplying by 3. Finally, multiplying 285 feet by the

4.50 per foot cost yields 1282.50.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular zone in yards.

100 yards

The perimeter represents the total distance around the rectangle before accounting for any openings.

2

Subtract the gate opening from the perimeter.

95 yards

Fencing is not required for the 5-yard entrance gate.

3

Convert the required fencing length from yards to feet.

285 feet

The cost of the fencing is provided per linear foot, so the length must be in feet.

4

Calculate the total cost by multiplying the length in feet by the price per foot.

1282.5

Total cost equals the total amount of material multiplied by the unit price.

Key Concept

Calculating perimeter of a rectangle, adjusting for openings, and converting units for cost calculation.

Question 13Question →

A logistics company is securing an L-shaped equipment depot with a chain-link fence. The facility has a 120-foot southern border, a 90-foot western border, a 75-foot northern border, and a 45-foot eastern border. The remaining two inner boundary walls connect the northern and eastern borders to complete the L-shape. The entire perimeter will be fenced except for a 15-foot wide entrance gate on the southern border. If the fencing material costs $25$ per yard, what is the total cost of the fencing?

$2,625

$3,375

$10,125

$24,375

Show answer & explanation

Answer: $3,375

Answer

The total cost of the fencing is $3,375.

The correct answer is found by first determining the missing inner dimensions of the L-shape (120 - 75 = 45 feet, and 90 - 45 = 45 feet) to find the total 6-sided perimeter (420 feet). After subtracting the 15-foot gate, 405 feet of fencing is needed. Converting this to yards by dividing by 3 gives 135 yards. Multiplying 135 yards by the

25 per yard cost results in

3,375.

Step-by-Step Solution

1

Calculate the lengths of the two missing inner boundaries.

Inner horizontal border = 120 - 75 = 45 feet. Inner vertical border = 90 - 45 = 45 feet.

All six sides of the L-shape are needed to find the total perimeter. The missing sides are found by subtracting the known parallel outer edges.

2

Calculate the total perimeter of the equipment depot.

120 + 90 + 75 + 45 + 45 + 45 = 420 feet.

The perimeter is the total length of the continuous boundary line enclosing the facility.

3

Subtract the width of the entrance gate from the total perimeter.

420 - 15 = 405 feet.

The 15-foot gate opening does not require fencing material, so it must be removed from the total.

4

Convert the required fencing from feet to yards.

405 / 3 = 135 yards.

The fencing cost is provided in yards, so the perimeter measurement must match the pricing unit.

5

Calculate the total cost of the fencing material.

135 \times 25 = 3,375 dollars.

Multiplying the total yards of fencing by the cost per yard gives the final price.

Key Concept

Calculating the perimeter of composite rectilinear shapes, subtracting non-fenced gaps, and applying unit conversions for total cost.

Question 14Question →

A contractor is installing a security perimeter around a rectangular warehouse facility. The facility is $40$ yards long and $25$ yards wide. The fence will include three $10$ -foot gates that do not require fencing material. If the fencing costs $\$18$ per yard, what is the total cost of the fencing material required?

$990

$1,800

$2,160

$18,000

Show answer & explanation

Answer: $2,160

Answer

The correct total cost of the fencing material is $2,160.

The total perimeter is found using the formula

2 \times (L + W)

, yielding

130

yards. The three

10

-foot gates have a combined width of

30

feet, which converts to

10

yards (

30 \div 3

). Subtracting the gate width leaves

120

yards of actual fencing needed. Multiplying this by the cost of

\$18

per yard yields the total of

\$2,160

.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular facility in yards.

2 \times (40 + 25) = 130

yards

The fencing will go around the entire boundary of the rectangle, which requires the perimeter formula

P = 2L + 2W

.

2

Convert the total width of the gates from feet to yards.

3 \text{ gates} \times 10 \text{ feet/gate} = 30 \text{ feet}

. Since

3 \text{ feet} = 1 \text{ yard}

,

30 \text{ feet} = 10 \text{ yards}

.

The perimeter and the fencing cost are both in yards, so the gate measurements must match the primary unit.

3

Subtract the gate width from the total perimeter to find the required fencing.

130 \text{ yards} - 10 \text{ yards} = 120 \text{ yards}

The sections where the gates are located do not require any fencing material.

4

Calculate the total cost of the fencing material.

120 \text{ yards} \times \$18/\text{yard} = \$2,160

Multiplying the total required yards by the price per yard provides the final monetary cost.

Key Concept

Calculating perimeter of a rectangle, performing unit conversions (feet to yards), and subtracting openings to find total material cost.

Question 15Question →

An airport authority is installing a security barrier around a new rectangular radar facility. The facility measures $40$ yards in length and $32$ yards in width. The design for the security barrier requires a heavy-duty steel post at each of the four corners, with additional posts placed exactly every $8$ feet along all four sides. How many steel posts are required in total to enclose the facility?

Show answer & explanation

Answer: 54

Answer

54 posts are required in total.

The correct answer is found by first ensuring all units match. Converting the dimensions to feet gives a length of 120 feet and a width of 96 feet. The total perimeter is 432 feet. Dividing this continuous perimeter by the 8-foot spacing requirement yields exactly 54 posts.

Step-by-Step Solution

1

Convert the given dimensions from yards to feet.

Length = 40 * 3 = 120 feet. Width = 32 * 3 = 96 feet.

The post spacing is provided in feet, so all measurements must be converted to the same unit before calculating the perimeter.

2

Calculate the total perimeter of the rectangular facility.

Perimeter = 2 * (120 + 96) = 2 * 216 = 432 feet.

The security barrier encloses the entire outer boundary of the facility, which corresponds to its perimeter.

3

Determine the total number of posts required.

Total posts = 432 / 8 = 54.

For any closed loop, the total number of equally spaced posts equals the total perimeter divided by the distance between the posts.

Key Concept

Calculating perimeter of a rectangle, unit conversions, and dividing a perimeter into equal segments.

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