Perimeter and Area of 2D Figures Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

A rectangular swimming pool measures $40$ feet in length and $30$ feet in width. A concrete deck of uniform width is built around the entire outer edge of the pool. If the area of the concrete deck alone is $296$ square feet, what is the width of the deck, in feet?

Show answer & explanation

Answer: 2

Answer

2

The correct width is found by setting up an equation for the total area. The deck adds its width to both sides of the pool's length and width, making the overall dimensions

40 + 2w

and

30 + 2w

. The area of the deck alone is the total area minus the pool's area. Solving the algebraic equation

(40 + 2w)(30 + 2w) - 1200 = 296

yields a positive root of

w = 2

.

Step-by-Step Solution

1

Calculate the area of the pool.

40 \times 30 = 1200

square feet.

You need the baseline area of the inner rectangle before the deck is added.

2

Write algebraic expressions for the outer dimensions of the pool and deck combined.

Length

= 40 + 2w

, Width

= 30 + 2w

.

The deck has a uniform width

w

, which adds to both the top/bottom and left/right sides of the pool.

3

Formulate an equation for the area of the deck.

(40 + 2w)(30 + 2w) - 1200 = 296

.

The area of the deck alone is equal to the total area (pool plus deck) minus the area of the pool.

4

Expand the binomials and simplify into a standard quadratic equation.

1200 + 140w + 4w^2 - 1200 = 296 \implies 4w^2 + 140w - 296 = 0

.

Standard quadratic form (

ax^2 + bx + c = 0

) is required to solve for the unknown variable

w

.

5

Solve the quadratic equation for

w

.

Divide by

4

to get

w^2 + 35w - 74 = 0

. Factoring yields

(w + 37)(w - 2) = 0

, so

w = 2

.

Finding the positive root gives the actual physical width of the concrete deck.

Key Concept

Calculating the area of composite rectangular figures and solving quadratic equations derived from expanding geometric boundaries.

Question 2Question →

A rectangular room has a length of $12$ feet and a width of $10$ feet. What is the perimeter of the room?

22

feet

34

feet

44

feet

120

feet

Show answer & explanation

Answer: $44$ feet

Answer

44

feet

The perimeter is the total distance around the boundary of the rectangle. Since a rectangle has two equal lengths and two equal widths, the perimeter is

2 \times 12 + 2 \times 10 = 24 + 20 = 44

feet.

Step-by-Step Solution

1

Identify the formula for the perimeter of a rectangle.

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

or

P = 2L + 2W

The perimeter is the total distance around the outside of the shape, which includes two lengths and two widths.

2

Substitute the given dimensions into the formula.

P = 2 \times (12 + 10)

The length is given as

12

feet and the width is

10

feet.

3

Calculate the final perimeter.

P = 2 \times 22 = 44

Performing the addition inside the parentheses first, then multiplying by

2

yields the total distance.

Key Concept

The perimeter of a rectangle is the total distance around its edges, calculated as twice the sum of its length and width.

Estimated Time:45s

Question 3Question →

A triangular pennant has a base of $8$ inches and a height of $15$ inches. What is the area of the pennant, in square inches?

Show answer & explanation

Answer: 60

Answer

60

The correct area is found by multiplying the base by the height, and then dividing by two according to the standard triangle area formula.

Step-by-Step Solution

1

Identify the formula for the area of a triangle.

A = \frac{1}{2}bh

The object is described as a triangular pennant.

2

Substitute the given base and height into the formula.

A = \frac{1}{2}(8)(15)

The base is given as

8

inches and the height is

15

inches.

3

Calculate the final product.

60

Multiplying the terms:

\frac{1}{2} \times 8 = 4

, and

4 \times 15 = 60

.

Key Concept

Area of a Triangle

Estimated Time:45s

Question 4Question →

A circular target has a diameter of $16$ inches. Which of the following represents the area of the target, in square inches?

64\pi

256\pi

16\pi

32\pi

Show answer & explanation

Answer: $64\pi$

Answer

64\pi

The correct answer accurately divides the diameter of

16

inches by

2

to find the radius of

8

inches. It then squares the radius (

8^2 = 64

) and multiplies by

\pi

to find the area of

64\pi

square inches.

Step-by-Step Solution

1

Find the radius of the circle.

r = 16 / 2 = 8

inches

The area formula requires the radius, which is exactly half of the diameter.

2

Apply the area formula for a circle (

A = \pi r^2

).

A = \pi(8)^2

Substituting the calculated radius into the standard area formula.

3

Simplify the expression.

A = 64\pi

square inches

Squaring the radius (

8 \times 8 = 64

) provides the final area in terms of

\pi

.

Key Concept

The area of a circle is found using the formula

A = \pi r^2

. When given the diameter, it must first be divided by 2 to find the radius.

Question 5Question →

The perimeter of a square courtyard is $36$ meters. What is the area of the courtyard, in square meters?

Show answer & explanation

Answer: 81

Answer

81

A square has four sides of equal length. Given a perimeter of

36

meters, each side measures

36 \div 4 = 9

meters. The area of a square is its side length squared, which is

9^2 = 81

square meters.

Step-by-Step Solution

1

Find the length of one side of the square.

36 \div 4 = 9

meters.

The perimeter of a square is the sum of its four equal sides, so dividing the total perimeter by 4 gives the length of a single side.

2

Calculate the area of the square.

9 \times 9 = 81

square meters.

The area of a square is calculated by squaring the length of one of its sides (

A = s^2

).

Key Concept

Perimeter and Area of 2D Figures

Question 6Question →

A circular stage has a radius of $5$ meters. Which of the following represents the total area of the stage, in square meters?

10\pi

5\pi

25\pi

100\pi

Show answer & explanation

Answer: $25\pi$

Answer

The area of the stage is

25\pi

square meters.

The correct answer is found by using the formula for the area of a circle,

A = \pi r^2

. Substituting the radius of 5 meters into the formula gives

A = \pi (5)^2 = 25\pi

.

Step-by-Step Solution

1

Identify the correct formula for the area of a circle.

A = \pi r^2

The question asks for the total area of a circular surface.

2

Substitute the given radius into the area formula.

A = \pi (5)^2

The problem states that the radius of the stage is 5 meters.

3

Calculate the final area.

A = 25\pi

Squaring 5 yields 25, resulting in an area of

25\pi

.

Key Concept

Area of a Circle

Estimated Time:45s

Question 7Question →

A rectangular community garden has an area of $120$ square feet. If the length of the garden is $15$ feet, what is the width of the garden, in feet?

Show answer & explanation

Answer: 8

Answer

8

The area of a rectangle is calculated by multiplying its length by its width (

A = l \times w

). To find the missing width when the area and length are known, you divide the total area by the length (

w = A \div l

). Dividing

120

by

15

results in exactly

8

feet.

Step-by-Step Solution

1

Identify the given values from the problem

Area = 120, Length = 15

To know which values to substitute into the formula for the area of a rectangle.

2

Apply the area formula for a rectangle

120 = 15 \times w

The formula relates the total area to the length and width: Area = length \times width.

3

Solve for the unknown width

w = 8

Dividing the area by the length isolates the width variable, giving the final dimension.

Key Concept

Calculating a missing dimension of a rectangle given its area and one other dimension.

Estimated Time:45s

Question 8Question →

A newly constructed rectangular parking lot is $40$ yards long and $25$ yards wide. What is the total perimeter of the parking lot, in yards?

65

105

130

1,000

Show answer & explanation

Answer: $130$

Answer

The total perimeter of the parking lot is

130

yards.

To find the perimeter of a rectangle, you must add the lengths of all four sides. Using the formula

P = 2(l + w)

, we substitute the given length (

40

) and width (

25

). Adding those gives

65

, and multiplying by

2

results in a total perimeter of

130

yards.

Step-by-Step Solution

1

Identify the shape and the given dimensions.

Shape: Rectangle. Length (

l

) =

40

yards, Width (

w

) =

25

yards.

Understanding the shape and dimensions is necessary to select the correct formula.

2

Recall the formula for the perimeter of a rectangle.

P = 2l + 2w

or

P = 2(l + w)

Perimeter is the total distance around the outside of the shape.

3

Substitute the dimensions into the formula and solve.

P = 2(40 + 25) = 2(65) = 130

Calculating the expression yields the final perimeter in yards.

Key Concept

Perimeter of Rectangles

Alternative Method

Instead of using the formula

P = 2(l + w)

, you can simply add all four sides together:

40 + 40 + 25 + 25 = 130

.

Estimated Time:45s

Question 9Question →

A competitive swimmer is walking around the outer edge of a rectangular pool to warm up. If the pool has a length of $50$ meters and a width of $25$ meters, what is the total perimeter of the pool, in meters?

Show answer & explanation

Answer: 150

Answer

The total perimeter of the pool is 150 meters.

The perimeter of a rectangle is found by adding the lengths of all four sides. Using the formula

P = 2l + 2w

, we calculate

2(50) + 2(25) = 150

meters.

Step-by-Step Solution

1

Identify the shape and the goal.

The shape is a rectangle and we need to find its perimeter.

Understanding what needs to be calculated determines the correct formula.

2

Recall the perimeter formula for a rectangle.

P = 2l + 2w

or

P = 2(l + w)

The perimeter is the total distance around the outside of the shape.

3

Substitute the given dimensions into the formula.

P = 2(50) + 2(25)

The length (

l

) is

50

and the width (

w

) is

25

.

4

Perform the arithmetic operations.

100 + 50 = 150

Multiplying the terms and adding them together yields the final perimeter.

Key Concept

Perimeter of Rectangles

Question 10Question →

A rectangular metallic sheet has a length of $28$ inches and a width of $15$ inches. A triangular metallic sheet has the exact same area. If the base of the triangular sheet is $24$ inches, what is its height, in inches?

Show answer & explanation

Answer: 35

Answer

35

The correct height is found by first determining the area of the rectangle (

28 \times 15 = 420

). Since the triangle shares this exact area and has a given base of

24

, solving the triangle area formula

420 = \frac{1}{2}(24)h

yields

12h = 420

, which simplifies to

h = 35

.

Step-by-Step Solution

1

Calculate the area of the rectangular sheet.

Area = 420 square inches

The area of the rectangle is required to find the area of the triangle, as the problem explicitly states their areas are equal.

2

Set up the area equation for the triangular sheet.

420 = 0.5 * 24 * h

Substituting the known area and base into the triangle area formula allows us to solve for the unknown height.

3

Solve the algebraic equation for the height (h).

h = 35

Simplifying 420 = 12h by dividing both sides by 12 isolates the height variable.

Key Concept

Calculating and equating the area of different 2D figures using standard area formulas.

Estimated Time:1m 0s

Question 11Question →

A rectangular floor is $18$ feet long and $12$ feet wide. A square rug with a side length of $9$ feet is placed on the floor. What is the area of the floor that is NOT covered by the rug, in square feet?

180

24

135

297

Show answer & explanation

Answer: $135$

Answer

The area of the floor not covered by the rug is

135

square feet.

The correct answer is found by calculating the total area of the floor (

18 \times 12 = 216

sq ft) and subtracting the area of the square rug (

9 \times 9 = 81

sq ft). This gives an uncovered area of

216 - 81 = 135

square feet.

Step-by-Step Solution

1

Calculate the total area of the rectangular floor.

\text{Area of floor} = 18 \times 12 = 216 \text{ square feet}

The area of a rectangle is found by multiplying its length by its width.

2

Calculate the area of the square rug.

\text{Area of rug} = 9 \times 9 = 81 \text{ square feet}

The area of a square is found by squaring its side length.

3

Subtract the area of the rug from the total area of the floor.

216 - 81 = 135 \text{ square feet}

The uncovered area is the difference between the total floor area and the area covered by the rug.

Key Concept

Calculating and subtracting areas of 2D figures

Estimated Time:1m 0s

Question 12Question →

A graphic designer creates a rectangular product label that is $5.5$ centimeters long and $4.0$ centimeters wide. What is the area of the label, in square centimeters?

19

9.5

22

220

Show answer & explanation

Answer: $22$

Answer

The correct area of the label is

22

square centimeters.

The area of a rectangle is found by multiplying its length by its width. The length of the label is

5.5

cm and the width is

4.0

cm. Multiplying these together gives

5.5 \times 4.0 = 22

square centimeters.

Step-by-Step Solution

1

Identify the shape and the goal from the word problem.

The shape is a rectangle and the goal is to find its area.

Identifying the shape ensures the correct mathematical formula is applied.

2

Set up the area formula for a rectangle.

\text{Area} = \text{length} \times \text{width}

This is the standard geometric formula for the area of a rectangle.

3

Substitute the given values into the formula and multiply.

\text{Area} = 5.5 \times 4.0 = 22

Multiplying the given dimensions provides the final area in square centimeters.

Key Concept

The area of a rectangle is found by multiplying its length by its width (

A = l \times w

).

Question 13Question →

A painter is marking a new parallelogram-shaped crosswalk on a street. The base of the parallelogram is $45$ feet long, and the perpendicular height across the street is $20$ feet. What is the total area of the crosswalk, in square feet?

Show answer & explanation

Answer: 900

Answer

900

The area of a parallelogram is calculated by multiplying its base by its perpendicular height. Given a base of

45

feet and a height of

20

feet, the area is

45 \times 20 = 900

square feet.

Step-by-Step Solution

1

Identify the relevant formula

A = b \times h

The area of a parallelogram is the product of its base and perpendicular height.

2

Substitute the given values into the formula

A = 45 \times 20

The base (

b

) is given as

45

feet and the height (

h

) is

20

feet.

3

Calculate the final area

900

Multiplying

45

by

20

gives

900

.

Key Concept

Area of a Parallelogram

Estimated Time:30s

Question 14Question →

An artist is framing a rectangular painting that measures $10$ inches in length and $8$ inches in width. The frame has a uniform width of $2$ inches on all sides. What is the total area of the frame itself, in square inches?

Show answer & explanation

Answer: 88

Answer

88

To find the area of the frame, subtract the area of the painting from the total area of the painting with the frame included. Because the frame is 2 inches wide on all sides, the overall dimensions increase by 4 inches in both directions. The outer rectangle is 14 inches by 12 inches (168 square inches). The inner painting is 10 inches by 8 inches (80 square inches). Subtracting the inner area from the outer area gives 168 - 80 = 88 square inches.

Step-by-Step Solution

1

Determine the outer dimensions of the framed painting.

Outer length = 14 inches, Outer width = 12 inches

The frame adds 2 inches to both the left and right sides (total of 4 inches to length), and 2 inches to both the top and bottom (total of 4 inches to width).

2

Calculate the total combined area of the painting and the frame.

168 square inches

Area of a rectangle is length multiplied by width (14 * 12).

3

Calculate the area of the inner painting.

80 square inches

The painting dimensions are 10 by 8 (10 * 8).

4

Subtract the inner area from the total area to find the area of the frame.

88 square inches

The frame is the region between the outer and inner rectangles.

Key Concept

The area of a border or frame is found by calculating the total outer area and subtracting the inner area.

Question 15Question →

A circular swimming pool is surrounded by a concrete deck of uniform width. The circumference of the outside edge of the deck is $44\pi$ feet, and the circumference of the pool itself is $36\pi$ feet. What is the area of the concrete deck in square feet?

8\pi

16\pi

160\pi

640\pi

Show answer & explanation

Answer: $160\pi$

Answer

The area of the concrete deck is

160\pi

square feet.

To find the area of the concrete deck, we must calculate the area of the large circle (pool plus deck) and subtract the area of the small circle (pool). First, find the radius of the large circle by dividing its circumference by

2\pi

:

44\pi / 2\pi = 22

. Its area is

\pi(22)^2 = 484\pi

. Next, find the radius of the pool:

36\pi / 2\pi = 18

. Its area is

\pi(18)^2 = 324\pi

. Finally, subtract the inner area from the outer area:

484\pi - 324\pi = 160\pi

.

Step-by-Step Solution

1

Find the radius of the entire circular area (pool plus deck).

2\pi R = 44\pi

gives

R = 22

feet.

The area formula requires the radius, which is found by dividing the circumference by

2\pi

.

2

Find the radius of the pool itself.

2\pi r = 36\pi

gives

r = 18

feet.

We need the inner radius to calculate the area of the pool.

3

Calculate the total area of the pool and deck combined.

Area_{total} = \pi (22)^2 = 484\pi

square feet.

This is the area of the larger outer circle.

4

Calculate the area of the pool.

Area_{pool} = \pi (18)^2 = 324\pi

square feet.

This is the area of the inner circle.

5

Subtract the pool's area from the total area to find the area of the deck.

484\pi - 324\pi = 160\pi

square feet.

The deck is the ring-shaped region between the outer and inner circles.

Key Concept

Area of a Circular Ring (Annulus)

Question 16Question →

A rectangular window glass measures $24$ inches in length and $18$ inches in width. What is the perimeter of the window glass, in inches?

42

66

84

432

Show answer & explanation

Answer: 84

Answer

84

The correct answer is

84

because the perimeter is the total distance around the boundary of the rectangle. Adding all four sides (

24 + 24 + 18 + 18

) or using the standard formula

P = 2L + 2W

gives

2(24) + 2(18) = 48 + 36 = 84

inches.

Step-by-Step Solution

1

Identify the correct formula for the perimeter of a rectangle.

P = 2L + 2W

or

P = 2(L + W)

The perimeter measures the total continuous distance around the outside edge of a shape.

2

Substitute the given length (

L = 24

) and width (

W = 18

) into the formula.

P = 2(24) + 2(18)

Plugging the known values into the equation sets up the calculation.

3

Perform the multiplication and addition to find the total distance.

48 + 36 = 84

Evaluating the mathematical expression yields the final perimeter in inches.

Key Concept

Perimeter of Rectangles

Question 17Question →

An architect is designing a circular observation deck with a circumference of $100\pi$ feet. The floor of the deck will be covered entirely with weather-resistant tiles. If one box of tiles covers $50\pi$ square feet, how many boxes are required to completely cover the floor?

2

50

100

200

Show answer & explanation

Answer: 50

Answer

50

The radius of the deck is found by dividing the circumference by

2\pi

(

100\pi \div 2\pi = 50

feet). The total area is then calculated using the formula

A = \pi r^2

, resulting in

2500\pi

square feet. Finally, dividing the total area by the coverage per box (

50\pi

) gives the required number of boxes, which is

50

.

Step-by-Step Solution

1

Calculate the radius of the circular deck using the given circumference.

r = 50

feet

The formula for circumference is

C = 2\pi r

. Substituting

100\pi

for

C

gives

100\pi = 2\pi r

, which simplifies to

r = 50

.

2

Calculate the total area of the observation deck.

Area = 2500\pi

square feet

The formula for the area of a circle is

A = \pi r^2

. Substituting

r = 50

gives

A = \pi(50)^2 = 2500\pi

.

3

Determine the number of boxes of tiles needed.

50

boxes

Divide the total area by the coverage of one box:

2500\pi \div 50\pi = 50

.

Key Concept

Perimeter and Area of 2D Figures

Estimated Time:1m 0s

Question 18Question →

A municipal park features a garden in the shape of a trapezoid. The parallel sides of the garden measure $18$ meters and $12$ meters. If the total area of the garden is $150$ square meters, what is the perpendicular distance between the parallel sides?

5

meters

10

meters

30

meters

50

meters

Show answer & explanation

Answer: $10$ meters

Answer

The perpendicular distance between the parallel sides is

10

meters.

The correct answer accurately applies the trapezoid area formula. By setting

150 = \frac{1}{2}(18 + 12)h

, the sum of the bases is calculated as

30

. Multiplying by

\frac{1}{2}

results in

15

. Dividing the total area (

150

) by

15

yields a perpendicular height of

10

meters.

Step-by-Step Solution

1

Identify the formula for the area of a trapezoid.

A = \frac{1}{2}(b_1 + b_2)h

This formula relates the known area and bases to the unknown height.

2

Substitute the known values into the formula.

150 = \frac{1}{2}(18 + 12)h

The given area is

150

, and the bases are

18

and

12

.

3

Simplify the sum inside the parentheses.

150 = \frac{1}{2}(30)h

Adding the lengths of the bases gives

30

.

4

Multiply

30

by

\frac{1}{2}

.

150 = 15h

This isolates the variable term

h

.

5

Divide both sides by

15

to solve for

h

.

h = 10

Dividing by

15

determines the perpendicular height.

Key Concept

Using the area formula of a trapezoid to solve for an unknown dimension.

Question 19Question →

A rectangular parking lot has a perimeter of $200$ yards. The length of the parking lot is $40$ yards greater than its width. What is the area of the parking lot, in square yards?

Show answer & explanation

Answer: 2100

Answer

2100

The correct answer is derived by first finding the individual dimensions using the perimeter formula and the given relationship (

l = w + 40

), which gives a width of 30 yards and a length of 70 yards. Multiplying these dimensions yields an area of 2100 square yards.

Step-by-Step Solution

1

Write the perimeter formula for a rectangle.

P = 2l + 2w

The perimeter is the total distance around the rectangle.

2

Translate the given information into equations.

2l + 2w = 200

and

l = w + 40

The problem states the perimeter is 200 yards and the length is 40 yards greater than the width.

3

Substitute the expression for length into the perimeter equation and solve for width.

2(w + 40) + 2w = 200 \Rightarrow 2w + 80 + 2w = 200 \Rightarrow 4w = 120 \Rightarrow w = 30

Substitution allows us to create an equation with only one variable.

4

Calculate the length.

l = 30 + 40 = 70

Now that we know the width is 30 yards, we can add 40 to find the length.

5

Calculate the area of the rectangle.

Area = l \times w = 70 \times 30 = 2100

The area of a rectangle is found by multiplying its length by its width.

Key Concept

Using algebraic equations to find the dimensions of a rectangle from its perimeter, and then calculating its area.

Question 20Question →

A circle is perfectly inscribed within a square that has a perimeter of $48$ centimeters. What is the area of the region located inside the square but outside the circle, in square centimeters?

144\pi - 144

48 - 36\pi

144 - 36\pi

144 - 12\pi

Show answer & explanation

Answer: $144 - 36\pi$

Answer

The correct answer is

144 - 36\pi

.

To find the area of the specified region, the area of the inscribed circle must be subtracted from the area of the square. A perimeter of 48 gives the square a side length of 12 and an area of 144. The inscribed circle's diameter matches the square's side length (12), meaning its radius is 6. Its area is calculated as pi times the radius squared, which is 36pi. Subtracting these gives the final area of 144 - 36pi.

Step-by-Step Solution

1

Find the side length of the square.

The side length is

48 \div 4 = 12

cm.

The perimeter of a square is equal to

4

times its side length.

2

Calculate the area of the square.

The area is

12 \times 12 = 144

cm

^2

.

The area of a square is the side length squared.

3

Determine the radius of the inscribed circle.

The radius is

12 \div 2 = 6

cm.

Because the circle is perfectly inscribed, its diameter equals the side length of the square (

12

cm). The radius is half of the diameter.

4

Calculate the area of the circle.

The area is

\pi \times 6^2 = 36\pi

cm

^2

.

The formula for the area of a circle is

A = \pi r^2

.

5

Find the area of the region outside the circle but inside the square.

The area is

144 - 36\pi

cm

^2

.

Subtracting the circle's area from the total square's area leaves the remaining corner regions.

Key Concept

Area of composite figures and inscribed shapes

Perimeter and Area of 2D Figures

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