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

Question 21Question →

A rectangular community garden is $3$ times as long as it is wide. If the total area of the garden is $108$ square feet, what is the perimeter of the garden, in feet?

48

108

42

24

Show answer & explanation

Answer: $48$

Answer

The perimeter of the garden is

48

feet.

The correct answer is found by first determining the dimensions of the garden. Since the length is

3

times the width, the area can be represented as

(3w)(w) = 3w^2

. Setting this equal to the given area of

108

square feet gives

3w^2 = 108

. Dividing by

3

yields

w^2 = 36

, so the width is

6

feet. The length is

3 \times 6 = 18

feet. The perimeter formula for a rectangle is

P = 2(length + width)

. Substituting the dimensions gives

P = 2(18 + 6) = 2(24) = 48

feet.

Step-by-Step Solution

1

Let the width of the garden be

w

. The length is

3

times the width, so the length is

3w

.

Width =

w

, Length =

3w

Translating the word problem into algebraic expressions allows us to set up an equation.

2

Set up an equation using the formula for the area of a rectangle (

Area = length \times width

).

3w \times w = 108

We are given the total area, which links our variable expressions to a known value.

3

Solve the equation for

w

.

3w^2 = 108 \Rightarrow w^2 = 36 \Rightarrow w = 6

Isolating the variable determines the numerical value for the width.

4

Substitute

w = 6

back into the expression for length (

3w

).

Length =

3(6) = 18

feet

Both dimensions are needed to calculate the perimeter.

5

Calculate the perimeter using the formula

P = 2(length + width)

.

P = 2(18 + 6) = 2(24) = 48

feet

The perimeter represents the total distance around the outside of the rectangular garden.

Key Concept

Perimeter and Area of 2D Figures

Estimated Time:1m 0s

Question 22Question →

A piece of wire is bent to form a square with an area of $144$ square centimeters. If the same wire is straightened and then bent to form a rectangle whose length is $4$ centimeters longer than its width, what is the area of this new rectangle?

48

square centimeters

140

square centimeters

144

square centimeters

572

square centimeters

Show answer & explanation

Answer: $140$ square centimeters

Answer

140

square centimeters

First, find the side length of the square. Since the area is

144

, the side length is

\sqrt{144} = 12

. The total length of the wire is the perimeter of the square, which is

12 \times 4 = 48

. This wire is then used to form a rectangle, so the rectangle's perimeter is also

48

. Let the width of the rectangle be

W

. The length is

W + 4

. The perimeter formula is

2(L + W) = P

. Substituting the values gives

2(W + W + 4) = 48

, which simplifies to

4W + 8 = 48

. Solving for

W

yields

4W = 40

, so

W = 10

. The length is

10 + 4 = 14

. The area of the new rectangle is

10 \times 14 = 140

square centimeters.

Step-by-Step Solution

1

Calculate the total length of the wire using the square's area.

The side of the square is

\sqrt{144} = 12

cm. The perimeter is

12 \times 4 = 48

cm.

The total length of the wire remains constant and equals the perimeter of both shapes.

2

Set up an equation for the rectangle's perimeter.

Let width be

W

. Length is

W + 4

. The perimeter is

2(W + (W + 4)) = 48

.

The rectangle is made from the same wire, so its perimeter is

48

cm, and its length is

4

cm longer than its width.

3

Solve for the width (

W

) and length (

L

) of the rectangle.

2(2W + 4) = 48 \Rightarrow 4W + 8 = 48 \Rightarrow 4W = 40 \Rightarrow W = 10

cm. The length is

10 + 4 = 14

cm.

Solving the linear equation provides the dimensions needed to find the area.

4

Calculate the area of the rectangle.

Area =

10 \times 14 = 140

square centimeters.

The area of a rectangle is its length multiplied by its width.

Key Concept

Perimeter and Area of 2D Figures

Question 23Question →

A kite designer is creating a kite in the shape of a rhombus. The two crossing wooden dowels that form the internal diagonals of the kite measure $16$ inches and $30$ inches. What is the total perimeter of the kite's outer edge, in inches?

Show answer & explanation

Answer: 68

Answer

68

The correct answer is found by recognizing that a rhombus's diagonals bisect each other perpendicularly. This allows the use of the Pythagorean theorem with half-diagonal lengths (

8

and

15

) to find a side length of

17

. Multiplying this single side length by the four equal sides of the rhombus yields a total perimeter of

68

inches.

Step-by-Step Solution

1

Determine the properties of the diagonals in a rhombus.

The diagonals bisect each other at right angles, forming four identical right triangles.

This breaks down the rhombus into simpler geometric shapes (right triangles), allowing for side length calculation.

2

Calculate the lengths of the legs for one of the right triangles.

The legs are

8

inches (

16 \div 2

) and

15

inches (

30 \div 2

).

The diagonals cut each other exactly in half at the intersection.

3

Use the Pythagorean theorem to find the length of the rhombus's outer edge.

The side length is

\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17

inches.

The outer edge forms the hypotenuse of the right triangle created by the half-diagonals.

4

Calculate the total perimeter.

17 \times 4 = 68

inches.

A rhombus has four equal sides, so the total perimeter is four times the length of one side.

Key Concept

Perimeter of a Rhombus and Pythagorean Theorem

Question 24Question →

A square metal plate has an area of $144$ square inches. A rectangular metal plate is cut so that it has the exact same width as the square plate, but its length is twice the width of the square. What is the perimeter of the rectangular plate in inches?

288

72

48

36

Show answer & explanation

Answer: 72

Answer

72

The area of the square is

144

square inches, which means each side of the square is

12

inches. The new rectangular plate has the same width as the square (

12

inches) and a length that is twice that width (

24

inches). To find the perimeter of the rectangle, add the length and width (

12 + 24 = 36

) and multiply by

2

to account for all four sides, resulting in

72

inches.

Step-by-Step Solution

1

Find the side length of the original square metal plate.

Side =

12

inches.

The area of a square is the side length squared (

s^2 = 144

), so the side is the square root of

144

.

2

Determine the dimensions of the rectangular plate.

Width =

12

inches, Length =

24

inches.

The problem states the rectangle's width equals the square's side (

12

), and its length is twice that amount (

2 \times 12 = 24

).

3

Calculate the perimeter of the rectangular plate.

Perimeter =

2 \times (12 + 24) = 72

inches.

The perimeter of a rectangle is found by adding the length and width, then multiplying by

2

.

Key Concept

Perimeter and Area of 2D Figures

Question 25Question →

A cross-section of an irrigation canal has the shape of an isosceles trapezoid. The flat bottom of the canal is $10$ meters wide, the open water surface at the top is $20$ meters wide, and the vertical depth of the canal is $12$ meters. To prevent seepage, a waterproof lining is applied only to the flat bottom and the two slanted sides of the canal. What is the total length, in meters, of the canal's cross-section that receives this lining?

Show answer & explanation

Answer: 36

Answer

36

The lined portion consists of the flat bottom and the two slanted sides. By dropping perpendicular altitude lines from the top corners to the bottom plane, two identical right triangles are formed on the left and right edges. The horizontal base of each right triangle is half of the difference between the top and bottom widths:

(20 - 10) / 2 = 5

meters. The vertical height of the canal is given as

12

meters. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the length of each slanted side is

\sqrt{5^2 + 12^2} = \sqrt{169} = 13

meters. The total length of the waterproof lining is the sum of the bottom width and the two slanted sides:

10 + 13 + 13 = 36

meters.

Step-by-Step Solution

1

Calculate the horizontal difference on each side of the trapezoid.

5 meters

To form a right triangle to calculate the slanted side, we must distribute the difference between the top and bottom widths equally to both sides.

2

Apply the Pythagorean theorem to find the length of the slanted sides.

13 meters

Each slanted side acts as the hypotenuse of a right triangle with legs of 5 meters (horizontal) and 12 meters (vertical height).

3

Sum the lengths of the sections that receive the lining.

36 meters

The lining is only applied to the bottom (10 meters) and the two slanted sides (13 meters each). The top is open and excluded.

Key Concept

Calculating the partial perimeter of a composite/trapezoidal figure using the Pythagorean theorem.

Question 26Question →

A city park is shaped like a trapezoid with parallel sides measuring $140$ meters and $100$ meters. The perpendicular distance between these parallel sides is $60$ meters. Inside the park, there is a rectangular concrete plaza that measures $40$ meters by $25$ meters. The rest of the park is covered in grass. What is the total area covered in grass, in square meters?

Show answer & explanation

Answer: 6200

Answer

6200

The total area of the trapezoid is found by multiplying the average of the two bases by the height, resulting in

7200

square meters. The rectangular plaza has an area of

40 \times 25 = 1000

square meters. Subtracting the inner plaza's area from the total area yields

6200

square meters for the grass.

Step-by-Step Solution

1

Calculate the total area of the trapezoidal park.

The area of the park is

7200

square meters.

The formula for the area of a trapezoid is

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

. Substituting the given dimensions yields

A = \frac{1}{2}(140 + 100) \times 60 = 7200

.

2

Calculate the area of the rectangular concrete plaza.

The area of the plaza is

1000

square meters.

The area of a rectangle is length times width. Calculating

40 \times 25

yields

1000

.

3

Subtract the area of the plaza from the total area of the park to find the grass area.

The total area covered in grass is

6200

square meters.

The grass covers the portion of the park that is not the plaza. Subtracting

1000

from

7200

leaves

6200

.

Key Concept

Calculating and subtracting areas of composite 2D figures

Question 27Question →

A circular security zone has a circumference of $16\pi$ meters. What is the area of this zone in square meters?

16\pi

64\pi

128\pi

256\pi

Show answer & explanation

Answer: $64\pi$

Answer

64\pi

To find the area, the radius of the circle must first be determined using the circumference formula (

C = 2\pi r

). Setting the given circumference equal to the formula (

16\pi = 2\pi r

) gives a radius of

8

meters. The area is then calculated using the formula

A = \pi r^2

, which results in

\pi(8)^2 = 64\pi

square meters.

Step-by-Step Solution

1

Identify the formula for circumference to solve for the radius.

C = 2\pi r = 16\pi

The radius is required to calculate the area of the circle.

2

Solve for the radius (

r

).

r = 8

meters

Dividing both sides of the equation by

2\pi

isolates the radius.

3

Apply the area formula for a circle using the found radius.

A = \pi r^2 = \pi(8)^2 = 64\pi

square meters

Substituting the radius (

8

) into the area formula yields the final required value.

Key Concept

Calculating the area of a circle by first deriving its radius from a given circumference.

Estimated Time:45s

Question 28Question →

A municipal park has a flat rectangular section and a semi-circular section attached to one of the rectangle's shorter sides. The overall length of the park, measured from the straight edge of the rectangle to the farthest curved point of the semi-circle, is $200$ meters. The width of the rectangular section is $100$ meters. What is the outer perimeter of the entire park in meters? (Use $\pi \approx 3.14$ )

657

714

557

18,925

Show answer & explanation

Answer: $557$

Answer

The outer perimeter of the park is

557

meters.

The perimeter is the total distance around the outside of the shape. First, find the radius of the semi-circle, which is half of its

100

-meter diameter (

50

m). Subtract this radius from the total

200

-meter length to find the straight edge of the rectangle (

150

m). The outer boundary consists of one

100

-meter straight edge, two

150

-meter straight edges, and the semi-circular arc. The arc length is half the circumference of a full circle (

0.5 \times 3.14 \times 100 = 157

m). Adding these outer edges together gives

100 + 150 + 150 + 157 = 557

meters.

Step-by-Step Solution

1

Identify the diameter and radius of the semi-circle.

Diameter =

100

m, Radius =

50

m

The semi-circle is attached to the

100

-meter wide side of the rectangle, so its diameter matches that width.

2

Determine the length of the rectangular section's straight sides.

200 - 50 = 150

m

The overall length (

200

m) includes both the rectangle's length and the semi-circle's radius (

50

m).

3

Calculate the arc length of the semi-circular section.

\frac{1}{2} \times \pi \times d = 0.5 \times 3.14 \times 100 = 157

m

The outer curved edge is exactly half of a full circle's circumference.

4

Sum the lengths of all outer boundary segments.

100 + 150 + 150 + 157 = 557

m

The total perimeter includes one short straight edge (

100

m), two long straight edges (

150

m each), and the curved arc (

157

m). The internal dashed line is not part of the outer perimeter.

Key Concept

Calculating unknown dimensions and outer perimeters of composite figures.

Estimated Time:1m 0s

Question 29Question →

A trapezoidal wooden sign has a height of $18$ inches. Its parallel top and bottom edges measure $24$ inches and $36$ inches, respectively. If a rectangular hole measuring $8$ inches by $6$ inches is cut out of the sign to mount a digital clock, what is the area of the remaining wooden surface in square inches?

Show answer & explanation

Answer: 492

Answer

492

The remaining area is found by subtracting the area of the inner rectangle from the area of the outer trapezoid. The trapezoid area is

\frac{1}{2}(24 + 36) \times 18 = 540

. The rectangle area is

8 \times 6 = 48

. The difference is

540 - 48 = 492

.

Step-by-Step Solution

1

Calculate the area of the full trapezoidal sign.

540 square inches

This gives the total area before the hole is cut, using the formula 0.5 * (base1 + base2) * height.

2

Calculate the area of the rectangular cutout.

48 square inches

This represents the area that is removed from the sign, calculated as length * width.

3

Subtract the cutout area from the total area.

492 square inches

Subtracting the removed area from the initial total area gives the area of the remaining solid surface.

Key Concept

Area of composite 2D figures involving trapezoids and rectangles.

Estimated Time:1m 0s

Question 30Question →

A circular fountain has a diameter of $12$ meters and is surrounded by a paved walkway that is $3$ meters wide. Which of the following represents the area of the walkway in square meters?

180\pi

81\pi

45\pi

6\pi

Show answer & explanation

Answer: $45\pi$

Answer

The correct answer is

45\pi

.

The correct answer is found by subtracting the area of the inner circle from the area of the larger outer circle. The inner radius is

6

meters (half of the

12

-meter diameter), giving an area of

36\pi

. The outer radius is

9

meters (

6

meters plus the

3

-meter walkway), giving a total area of

81\pi

. The difference is

81\pi - 36\pi = 45\pi

.

Step-by-Step Solution

1

Find the radius of the inner fountain.

12 \div 2 = 6

meters.

The area formula for a circle requires the radius, which is half of the diameter.

2

Calculate the area of the fountain.

Area =

\pi \times 6^2 = 36\pi

square meters.

This establishes the area of the space not covered by the walkway.

3

Find the radius of the total circular area (fountain plus walkway).

6 + 3 = 9

meters.

The walkway adds

3

meters to the radius of the inner circle.

4

Calculate the total area.

Area =

\pi \times 9^2 = 81\pi

square meters.

This gives the combined area of both the fountain and the walkway.

5

Subtract the fountain's area from the total area.

81\pi - 36\pi = 45\pi

square meters.

Subtracting the inner area isolates the area of just the paved walkway.

Key Concept

To find the area of a ring (annulus), calculate the area of the outer circle and subtract the area of the inner circle.

Question 31Question →

A rectangular landing pad is designed so that its length is exactly three times its width. If the total area of the landing pad is $147$ square meters, what is the perimeter of the landing pad in meters?

Show answer & explanation

Answer: 56

Answer

56

By setting the width as

w

and length as

3w

, the area equation is

3w^2 = 147

. Solving for

w

yields a width of

7

meters, making the length

21

meters. The perimeter is the sum of all four sides:

2(21 + 7) = 56

meters.

Step-by-Step Solution

1

Define variables and set up the area equation.

w \times 3w = 147

The area of a rectangle is width times length, and the length is given as three times the width.

2

Solve for the width (

w

).

w^2 = 49 \Rightarrow w = 7

Dividing both sides by

3

isolates

w^2

, and taking the square root gives the width.

3

Calculate the length and the perimeter.

Length =

21

, Perimeter =

56

Length is

3 \times 7

. Perimeter is

2(21 + 7)

.

Key Concept

Calculating the perimeter of a rectangle when given its area and the proportional relationship between its sides.

Question 32Question →

A canvas tarp is cut into the shape of a right triangle. The area of the tarp is $60$ square feet, and its shortest leg measures $8$ feet. What is the perimeter of the tarp in feet?

23

40

46

60

Show answer & explanation

Answer: $40$

Answer

40

To find the perimeter, all three sides of the right triangle are needed. Using the given area (

60

) and shortest leg (

8

), the length of the other leg is found with the area formula

A = \frac{1}{2}bh

, yielding

15

feet. Next, the Pythagorean theorem (

8^2 + 15^2 = c^2

) reveals the hypotenuse is

17

feet. Summing the three sides (

8 + 15 + 17

) gives the correct perimeter of

40

feet.

Step-by-Step Solution

1

Find the length of the other leg using the area formula for a triangle,

A = \frac{1}{2}bh

.

60 = \frac{1}{2}(8)b \Rightarrow 60 = 4b \Rightarrow b = 15

feet.

The perimeter requires all three side lengths, so the missing leg must be calculated first.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the hypotenuse.

8^2 + 15^2 = c^2 \Rightarrow 64 + 225 = c^2 \Rightarrow 289 = c^2 \Rightarrow c = 17

feet.

The hypotenuse is the third side needed to find the complete outer boundary of the right triangle.

3

Calculate the perimeter by summing the three side lengths.

P = 8 + 15 + 17 = 40

feet.

The perimeter of any polygon is the total length of its outer boundary.

Key Concept

Calculating missing dimensions of a right triangle using area and the Pythagorean theorem to find the perimeter.

Estimated Time:1m 0s

Question 33Question →

A rectangular garden has a perimeter of $56$ meters. The length of the garden is $4$ meters greater than its width. If a $3$ -meter wide paved path is built entirely inside the garden along all four of its borders, what is the area, in square meters, of the remaining unpaved section of the garden?

Show answer & explanation

Answer: 60

Answer

The area of the remaining unpaved section of the garden is 60 square meters.

The original dimensions of the garden are found to be 12 meters and 16 meters based on the given perimeter and side relationships. The inner unpaved section has dimensions of 12 - 6 = 6 meters and 16 - 6 = 10 meters because the 3-meter path cuts into both sides of each dimension. Multiplying these inner dimensions yields a final area of 60 square meters.

Step-by-Step Solution

1

Use the perimeter formula to find the outer length and width of the garden.

Width = 12 meters, Length = 16 meters.

The relationship between length and width (L = W + 4) and the total perimeter formula

2L + 2W = 56

allows us to find the exact dimensions.

2

Calculate the dimensions of the inner unpaved area by subtracting the path's width.

Inner width = 6 meters, Inner length = 10 meters.

The 3-meter path runs along all four borders, meaning the path's width must be subtracted twice (3 + 3 = 6 meters) from both the outer length and the outer width.

3

Calculate the final area of the inner unpaved section.

60 square meters.

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

10 \times 6

).

Key Concept

Solving for unknown dimensions using perimeter, and determining the area of a smaller inscribed rectangle by correctly subtracting border widths.

Question 34Question →

A rectangular swimming pool is twice as long as it is wide. It is surrounded by a tiled deck that has a uniform width of $2$ meters. If the area of the deck alone is $136$ square meters, what is the perimeter of the swimming pool in meters?

50

60

132

200

Show answer & explanation

Answer: 60

Answer

60

The correct answer accurately models the outer dimensions of the pool plus deck as

(w + 4)

and

(2w + 4)

. Subtracting the inner pool area

(2w^2)

leaves the deck area equation

12w + 16 = 136

. Solving this yields a width of

10

meters and a length of

20

meters, giving a perimeter of

60

meters.

Step-by-Step Solution

1

Define variables for the dimensions of the pool and the outer boundaries including the deck.

Let width =

w

and length =

2w

. The outer dimensions, which add the 2-meter deck to both sides, are width =

w + 4

and length =

2w + 4

.

This establishes algebraic expressions to represent the areas.

2

Set up an equation for the deck area by subtracting the pool's area from the total outer area.

(w + 4)(2w + 4) - (w)(2w) = 136

The area of the deck alone is given as 136 square meters.

3

Expand the expressions and solve the equation for

w

.

(2w^2 + 12w + 16) - 2w^2 = 136 \rightarrow 12w + 16 = 136 \rightarrow 12w = 120 \rightarrow w = 10

meters.

Simplifying isolates the variable

w

to find the width of the pool.

4

Calculate the pool's length and then its perimeter.

Length =

2(10) = 20

meters. Perimeter =

2(10 + 20) = 60

meters.

The question asks for the perimeter of the swimming pool.

Key Concept

Solving for unknown geometric dimensions using the area of a surrounding path, followed by perimeter calculation.

Question 35Question →

A landscaping company is designing a parallelogram-shaped flower bed that will cover exactly $288$ square feet. If the design requires the base of the flower bed to be twice its height, what must the length of the base be in feet?

Show answer & explanation

Answer: 24

Answer

24

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

A = b \times h

). By substituting

2h

for the base, the equation becomes

288 = 2h^2

. Solving for

h

yields

12

. The base is twice the height, so

2 \times 12 = 24

.

Step-by-Step Solution

1

Set up the area formula

A = b \times h

To relate the given area to the dimensions of the parallelogram.

2

Express the base in terms of height

b = 2h

The problem states the base is twice the height.

3

Substitute the expressions into the area equation

288 = (2h) \times h

To create an equation with only one variable.

4

Solve for the height (

h

)

2h^2 = 288 \Rightarrow h^2 = 144 \Rightarrow h = 12

To find the numerical value of the height.

5

Calculate the base (

b

)

b = 2 \times 12 = 24

The question asks for the length of the base, not the height.

Key Concept

Area of a parallelogram and algebraic substitution

Estimated Time:1m 0s

Question 36Question →

A circular metal plate has a total area of $100\pi \text{ cm}^2$ . A smaller circular hole is drilled completely out of its exact center. If the area of the remaining metal plate is $64\pi \text{ cm}^2$ , what is the radius of the drilled hole?

6 \text{ cm}

12 \text{ cm}

18 \text{ cm}

36 \text{ cm}

Show answer & explanation

Answer: $6 \text{ cm}$

Answer

The radius of the drilled hole is

6 \text{ cm}

.

The area of the removed hole is the difference between the original total area and the remaining area, which is

100\pi - 64\pi = 36\pi \text{ cm}^2

. Using the circle area formula (

A = \pi r^2

), setting

\pi r^2 = 36\pi

yields

r^2 = 36

. Taking the square root of both sides confirms the radius is

6 \text{ cm}

.

Step-by-Step Solution

1

Find the area of the drilled hole by subtracting the remaining area from the total area.

100\pi - 64\pi = 36\pi \text{ cm}^2

.

The drilled hole accounts for the area that was removed from the total plate.

2

Set up the area formula for the drilled hole to solve for the unknown radius.

A = \pi r^2 \rightarrow 36\pi = \pi r^2

.

The area of a circle is calculated using the formula

A = \pi r^2

.

3

Solve the equation for the radius

r

.

r^2 = 36

, so

r = 6 \text{ cm}

.

Dividing both sides by

\pi

leaves

r^2 = 36

, and taking the square root of both sides gives the radius.

Key Concept

Perimeter and Area of 2D Figures

Estimated Time:1m 0s

Perimeter and Area of 2D Figures

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