Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 8

Question 141Question →

A shipping company uses rectangular boxes that measure $4$ feet in length, $3$ feet in width, and $2$ feet in height. The company needs to completely cover the outside of one box with a protective weather-resistant wrapping. How many square feet of wrapping are required?

14

24

26

52

Show answer & explanation

Answer: 52

Answer

52

The question asks for the amount of wrapping needed to completely cover the outside of the box, requiring the total surface area of a rectangular prism. Using the formula

SA = 2(lw + lh + wh)

and substituting the dimensions

l = 4

,

w = 3

, and

h = 2

yields

SA = 2((4)(3) + (4)(2) + (3)(2)) = 2(12 + 8 + 6) = 2(26) = 52

square feet.

Step-by-Step Solution

1

Identify the required measurement

Calculate the total surface area of the rectangular prism.

The problem asks for the amount of wrapping needed to 'completely cover the outside', which corresponds to surface area rather than volume.

2

Recall the surface area formula for a rectangular prism

SA = 2(lw + lh + wh)

A rectangular box has six faces, composed of three identical pairs of rectangles (top/bottom, front/back, left/right).

3

Substitute the given dimensions and solve

SA = 2((4)(3) + (4)(2) + (3)(2)) = 2(12 + 8 + 6) = 2(26) = 52

square feet.

Applying the measurements

l=4

,

w=3

, and

h=2

evaluates the total combined area of all six faces.

Key Concept

Calculating the total surface area of a rectangular prism.

Question 142Question →

In a right triangle, the length of the hypotenuse is denoted by $c$ , and the lengths of the two legs are denoted by $a$ and $b$ . If $a = 8\text{ cm}$ and $c = 17\text{ cm}$ , what is the length of leg $b$ ?

9\text{ cm}

225\text{ cm}

15\text{ cm}

\sqrt{353}\text{ cm}

Show answer & explanation

Answer: $15\text{ cm}$

Answer

The length of leg

b

is

15\text{ cm}

.

The Pythagorean theorem states that for any right triangle, the sum of the squares of the legs equals the square of the hypotenuse (

a^2 + b^2 = c^2

). By substituting

a = 8

and

c = 17

, the equation becomes

64 + b^2 = 289

. Subtracting

64

from both sides gives

b^2 = 225

. Taking the square root of

225

yields a leg length of

15\text{ cm}

.

Step-by-Step Solution

1

Identify the known and unknown values in the Pythagorean theorem formula.

Formula:

a^2 + b^2 = c^2

. Known:

a = 8

and

c = 17

. Unknown:

b

.

To ensure values are placed in the correct positions, specifically identifying

17

as the hypotenuse (

c

).

2

Substitute the known values into the equation and square them.

8^2 + b^2 = 17^2

becomes

64 + b^2 = 289

.

To simplify the equation before isolating the unknown variable.

3

Isolate

b^2

by subtracting

64

from both sides of the equation.

b^2 = 289 - 64

, which simplifies to

b^2 = 225

.

To get the squared unknown variable by itself on one side of the equals sign.

4

Take the square root of both sides to solve for

b

.

b = \sqrt{225} = 15\text{ cm}

.

To remove the exponent and find the final linear length of the leg.

Key Concept

Pythagorean Theorem and Variable Substitution

Estimated Time:1m 0s

Question 143Question →

The total surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters?

Show answer & explanation

Answer: 125

Answer

125

The correct answer is derived by first identifying the side length of the cube. Given the total surface area is

150

square centimeters, we divide by

6

to find the area of a single face (

25

square centimeters). The square root of

25

gives a side length of

5

centimeters. Finally, cubing this side length (

5 \times 5 \times 5

) results in a total volume of

125

cubic centimeters.

Step-by-Step Solution

1

Use the total surface area to find the area of a single face of the cube.

150 \div 6 = 25

square centimeters.

A cube has exactly

6

identical square faces. Dividing the total surface area by

6

isolates the area of just one face (

s^2

).

2

Determine the length of one edge of the cube.

s = \sqrt{25} = 5

centimeters.

Because the area of a square face is the side length squared (

s^2

), taking the square root gives the side length.

3

Calculate the total volume of the cube.

V = 5^3 = 125

cubic centimeters.

The formula for the volume of a cube is the side length cubed (

V = s^3

).

Key Concept

Converting between the surface area and volume of a 3D figure by finding the missing dimension.

Estimated Time:45s

Question 144Question →

A painter leans a $25\text{-foot}$ ladder against a vertical wall. The base of the ladder is placed exactly $7\text{ feet}$ away from the base of the wall on level ground. How high up the wall, in feet, does the top of the ladder reach?

Show answer & explanation

Answer: 24

Answer

The top of the ladder reaches 24 feet up the wall.

The correct answer is derived by modeling the scenario as a right triangle. The ladder acts as the hypotenuse (

c = 25

) and the distance along the ground acts as one leg (

a = 7

). Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we solve for the missing height (

b

). Substituting the values gives

7^2 + b^2 = 25^2

, which simplifies to

49 + b^2 = 625

. Subtracting 49 yields

b^2 = 576

. The square root of 576 is 24, meaning the ladder reaches 24 feet up the wall.

Step-by-Step Solution

1

Set up the right triangle relationship.

a^2 + b^2 = c^2

, where

c = 25

and

a = 7

The wall is perpendicular to the ground, forming a right triangle with the ladder acting as the hypotenuse.

2

Substitute known values into the Pythagorean theorem.

7^2 + b^2 = 25^2

We know the hypotenuse (

25

) and one leg (

7

), and we need to find the other leg (

b

).

3

Square the known numbers.

49 + b^2 = 625

Evaluating the exponents is required before isolating the variable term.

4

Isolate the variable term.

b^2 = 625 - 49 = 576

Subtracting

49

from both sides leaves

b^2

by itself.

5

Solve for the height.

b = \sqrt{576} = 24

Taking the square root of both sides gives the length of the missing side.

Key Concept

Using the Pythagorean theorem to find a missing leg of a right triangle in a real-world scenario.

Question 145Question →

A solid cylindrical metal rod with a radius of $3$ inches and a length of $8$ inches is melted down and completely recast into a new cylindrical rod. If the new rod must have a radius of $2$ inches and no material is lost, what is its length in inches?

12

72

18

36

Show answer & explanation

Answer: 18

Answer

The length of the new cylindrical rod is 18 inches.

The correct answer is 18. Because no material is lost, the volume of the new rod must strictly equal the volume of the original rod. The original volume is calculated using the formula

V = \pi r^2 h

, resulting in

\pi(3)^2(8) = 72\pi

. Setting this equal to the new rod's volume expression,

72\pi = \pi(2)^2 h

, simplifies the equation to

72\pi = 4\pi h

. Dividing both sides by

4\pi

yields a final length of 18 inches.

Step-by-Step Solution

1

Calculate the volume of the original cylindrical rod.

V_{1} = \pi r^2 h = \pi(3)^2(8) = \pi(9)(8) = 72\pi

cubic inches.

Since the rod is being melted down, the total amount of material (volume) remains constant. We first need the total volume of the original rod.

2

Set up the volume expression for the new cylindrical rod.

V_{2} = \pi r^2 h = \pi(2)^2 h = 4\pi h

.

The new rod is also a cylinder, but with a different radius (2 inches) and an unknown length (height).

3

Equate the two volumes and solve for the unknown length.

72\pi = 4\pi h \rightarrow h = \frac{72\pi}{4\pi} = 18

inches.

Because no material is lost during recasting, the initial volume must equal the final volume.

Key Concept

Conservation of volume in 3D cylinders

Question 146Question →

An architectural firm is analyzing the material requirements and interior spaces of several proposed structures. Match each structural description to its correct calculated capacity (volume) or exterior area (surface area).

Click a left item, then click its matching right item

Show answer & explanation

Answer

The rectangular room matches

960

cubic feet, the cubic pod matches

600

square feet, the cylindrical tank matches

500\pi

cubic feet, and the spherical sculpture matches

400\pi

square feet.

Each structural description correctly aligns with its corresponding formula and numerical result as detailed in the solution steps.

Step-by-Step Solution

1

Calculate the volume of the rectangular room.

V = l \times w \times h = 10 \times 12 \times 8 = 960

cubic feet.

The capacity of a rectangular room is found using the volume formula for a rectangular prism.

2

Calculate the surface area of the cubic storage pod.

SA = 6 \times s^2 = 6 \times (10)^2 = 6 \times 100 = 600

square feet.

A cube has 6 identical square faces, each with an area of

s^2

.

3

Calculate the volume of the cylindrical water tank.

V = \pi \times r^2 \times h = \pi \times (5)^2 \times 20 = 25 \times 20 \times \pi = 500\pi

cubic feet.

The volume of a cylinder is the area of its circular base times its height.

4

Calculate the surface area of the spherical sculpture.

SA = 4 \times \pi \times r^2 = 4 \times \pi \times (10)^2 = 4 \times 100 \times \pi = 400\pi

square feet.

The surface area formula for a sphere is

4

times the area of a circle with the same radius.

Key Concept

Applying appropriate volume and surface area formulas for fundamental 3D figures: rectangular prisms, cubes, cylinders, and spheres.

Question 147Question →

A drone takes off from a launch pad and flies $6\text{ kilometers}$ due south. It then turns and flies due west. If the drone's straight-line distance from the launch pad is now exactly $10\text{ kilometers}$ , how many kilometers did it fly due west?

4

8

16

64

Show answer & explanation

Answer: 8

Answer

The drone flew 8 kilometers due west.

The drone's path forms a right triangle with a known leg of

6\text{ km}

, an unknown leg

x

, and a hypotenuse of

10\text{ km}

. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known values:

6^2 + x^2 = 10^2

. This simplifies to

36 + x^2 = 100

. Subtracting

36

from both sides gives

x^2 = 64

. Taking the square root of both sides yields

x = 8

. Therefore, the drone flew

8\text{ kilometers}

due west.

Step-by-Step Solution

1

Identify the geometric shape formed by the drone's flight path.

The flight path forms a right-angled triangle where the southward and westward paths are the legs, and the straight-line distance is the hypotenuse.

The cardinal directions South and West are perpendicular to each other, forming a 90-degree angle.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

6^2 + b^2 = 10^2

The southward distance is one leg (

a = 6

), the westward distance is the unknown leg (

b

), and the straight-line distance is the hypotenuse (

c = 10

).

3

Solve the equation for the unknown leg.

36 + b^2 = 100 \rightarrow b^2 = 64 \rightarrow b = 8

Squaring the known sides and subtracting gives the square of the unknown leg. Taking the square root gives the final distance.

Key Concept

Using the Pythagorean theorem to find a missing leg of a right triangle in a real-world navigation scenario.

Estimated Time:1m 0s

Question 148Question →

A farmer needs to completely fill a cylindrical grain silo with wheat. If the silo has a diameter of $12$ meters and a height of $10$ meters, what is the maximum capacity of the silo in cubic meters?

60\pi

120\pi

360\pi

1440\pi

Show answer & explanation

Answer: $360\pi$

Answer

The maximum capacity of the silo is

360\pi

cubic meters.

The question asks for the maximum capacity to completely fill the silo, which means finding its volume. The volume of a cylinder is found using the formula

V = \pi r^2 h

. First, the diameter of

12

meters must be halved to find the radius of

6

meters. Substituting the radius and the height (

10

meters) into the formula gives

V = \pi(6)^2(10) = \pi(36)(10) = 360\pi

cubic meters.

Step-by-Step Solution

1

Determine the required measurement and the appropriate formula.

The problem asks for the capacity to 'fill' a cylinder, which requires the volume formula:

V = \pi r^2 h

.

Capacity refers to the interior space (volume) of a 3D figure.

2

Find the radius of the circular base.

The radius is half of the diameter:

12 \div 2 = 6

meters.

The volume formula requires the radius, but the problem provides the diameter.

3

Substitute the values into the formula and solve.

V = \pi(6)^2(10) = \pi(36)(10) = 360\pi

cubic meters.

Executing the calculation according to the order of operations yields the final volume.

Key Concept

Volume of a Cylinder

Estimated Time:1m 0s

Question 149Question →

Determine the missing measurement to complete the statement below.

Fill in the blanks below

A museum exhibit features a custom glass panel shaped exactly like a right triangle. The longest edge of the glass panel measures

29\text{ inches}

, and the shortest edge measures

20\text{ inches}

. The length of the third edge is inches.

Show answer & explanation

Answer

21

The correct answer is

21

. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), the square of the hypotenuse (

29^2 = 841

) minus the square of the known leg (

20^2 = 400

) equals the square of the unknown leg (

441

). The square root of

441

is

21

.

Step-by-Step Solution

1

Identify the given parts of the right triangle.

Hypotenuse

c = 29

, and leg

a = 20

.

The longest edge of a right triangle is always the hypotenuse.

2

Set up the Pythagorean theorem equation.

20^2 + b^2 = 29^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) connects the lengths of the legs to the hypotenuse.

3

Square the known side lengths.

400 + b^2 = 841

20 \times 20 = 400

and

29 \times 29 = 841

.

4

Isolate the unknown variable

b^2

.

b^2 = 841 - 400 = 441

Subtract

400

from both sides to find the square of the missing leg.

5

Solve for

b

by taking the square root.

b = \sqrt{441} = 21

The square root of

441

gives the length of the third edge in inches.

Key Concept

The Pythagorean Theorem (

a^2 + b^2 = c^2

) allows you to find an unknown leg of a right triangle by subtracting the square of the known leg from the square of the hypotenuse.

Question 150Question →

A community center is repainting the inside of an empty rectangular swimming pool. The pool is $25$ meters long, $10$ meters wide, and has a uniform depth of $2$ meters. If the entire bottom and all four interior walls are to be painted, what is the total area, in square meters, that needs to be painted?

Show answer & explanation

Answer: 390

Answer

390

The correct total area is found by adding the area of the bottom (

250

m

^2

) and the areas of the four vertical walls (

100

m

^2

and

40

m

^2

). Because it is a swimming pool, there is no top to paint, making the total area

390

square meters.

Step-by-Step Solution

1

Determine which surfaces of the rectangular pool need to be painted.

One bottom surface and four side walls (two long and two short). The top is open.

A pool is shaped like an open-top rectangular prism, so we only need to calculate the area of 5 faces instead of 6.

2

Calculate the area of the bottom surface.

Area =

25 \times 10 = 250

square meters.

The bottom is a rectangle bounded by the length and width of the pool.

3

Calculate the combined area of the four side walls.

Long walls =

2 \times (25 \times 2) = 100

square meters. Short walls =

2 \times (10 \times 2) = 40

square meters.

There are two identical walls along the length and two identical walls along the width. The area of each wall is its base dimension multiplied by the depth.

4

Sum the areas of all the painted surfaces.

250 + 100 + 40 = 390

square meters.

The total area to be painted is the sum of the areas of the individual painted faces.

Key Concept

Surface area of an open rectangular prism

Question 151Question →

A rectangular television screen has a horizontal width of $20\text{ inches}$ and a vertical height of $15\text{ inches}$ . What is the length of the diagonal of the screen?

5\text{ inches}

\sqrt{175}\text{ inches}

25\text{ inches}

35\text{ inches}

Show answer & explanation

Answer: $25\text{ inches}$

Answer

The correct length of the diagonal is

25\text{ inches}

.

The diagonal of the screen forms the hypotenuse of a right triangle with the width and height acting as the two legs. By applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we calculate

20^2 + 15^2 = 400 + 225 = 625

. Taking the square root of 625 gives a diagonal length of 25 inches.

Step-by-Step Solution

1

Identify the shape and the relationship between the given measurements.

The width (

20\text{ inches}

) and height (

15\text{ inches}

) form the legs of a right triangle, and the diagonal represents the hypotenuse.

A rectangle's corners are right angles, so drawing a diagonal line creates two identical right triangles.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing hypotenuse.

15^2 + 20^2 = c^2

The Pythagorean theorem relates the lengths of the legs (

a

and

b

) to the hypotenuse (

c

) in any right triangle.

3

Square the lengths of the legs and add them together.

225 + 400 = 625

Following the order of operations, exponents must be evaluated before addition.

4

Take the square root of the sum to find the length of the diagonal.

\sqrt{625} = 25\text{ inches}

To solve for

c

from

c^2

, we must find the principal square root.

Key Concept

Using the Pythagorean theorem to find the hypotenuse (diagonal) of a right triangle formed within a rectangle.

Estimated Time:1m 0s

Question 152Question →

The entrance of a camping tent is shaped like an isosceles triangle. The base of the tent resting on the ground is $16\text{ feet}$ wide, and the two slanted sides are each $17\text{ feet}$ long. What is the maximum height of the tent, in feet, from the ground to the peak?

Show answer & explanation

Answer: 15

Answer

15

By dividing the isosceles triangle in half, we form a right triangle with a hypotenuse of 17 and a base leg of 8. Applying the Pythagorean theorem (

8^2 + h^2 = 17^2

) gives

64 + h^2 = 289

, resulting in

h^2 = 225

. The square root of 225 is 15, which is the maximum height of the tent.

Step-by-Step Solution

1

Divide the base of the isosceles triangle in half to find the base of the inner right triangle.

16 \div 2 = 8\text{ feet}

The height (altitude) of an isosceles triangle bisects the base, creating a right angle necessary for the Pythagorean theorem.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

h^2 + 8^2 = 17^2

The vertical height (

h

) and half of the base (

8

) are the legs of the right triangle, while the slanted side (

17

) is the hypotenuse.

3

Square the known values.

h^2 + 64 = 289

Simplifying the terms makes it possible to solve for the unknown.

4

Subtract 64 from both sides to isolate

h^2

.

h^2 = 225

Standard algebraic process for solving a one-variable equation.

5

Take the square root of both sides to find

h

.

h = 15

The inverse operation of squaring gives the final linear dimension of the height.

Key Concept

Using the properties of isosceles triangles alongside the Pythagorean theorem to find an unknown altitude

Question 153Question →

A right-angled triangular warning sign is being designed for a factory floor. The longest side of the sign measures $17\text{ inches}$ , and one of its perpendicular sides measures $8\text{ inches}$ . What is the total area of the front face of the sign?

36\text{ square inches}

40\text{ square inches}

60\text{ square inches}

68\text{ square inches}

Show answer & explanation

Answer: $60\text{ square inches}$

Answer

60\text{ square inches}

To find the area of a right triangle, the lengths of both perpendicular legs must be known. Using the Pythagorean theorem, the missing leg is calculated as

15\text{ inches}

(since

\sqrt{17^2 - 8^2} = 15

). The area is then found by multiplying one half by the product of the two legs (

0.5 \times 8 \times 15 = 60

).

Step-by-Step Solution

1

Identify the given sides of the right triangle.

The hypotenuse (longest side,

c

) is

17\text{ inches}

and one leg (

a

) is

8\text{ inches}

.

The area formula for a right triangle requires the lengths of both perpendicular legs, not the hypotenuse.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing leg (

b

).

8^2 + b^2 = 17^2

, which simplifies to

64 + b^2 = 289

. Solving for

b^2

gives

225

, so

b = 15\text{ inches}

.

Both perpendicular legs are needed to establish the base and height of the triangle.

3

Calculate the area of the triangle using the formula Area =

0.5 \times \text{base} \times \text{height}

.

Area =

0.5 \times 8 \times 15 = 60\text{ square inches}

.

The two legs of a right triangle serve as its base and height.

Key Concept

Pythagorean Theorem and Triangle Area

Question 154Question →

Fill in the blanks with the correct numerical values based on the geometric properties described.

Fill in the blanks below

A city park features a right-angled triangular garden. The longest side of the garden measures

34\text{ yards}

, and one of the shorter sides is

16\text{ yards}

. The length of the other shorter side is yards, and the area of the garden is square yards.

Show answer & explanation

Answer

The missing shorter side is 30 yards, and the area of the garden is 240 square yards.

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) with a hypotenuse of

34

and a leg of

16

, the missing leg is

\sqrt{34^2 - 16^2} = \sqrt{1156 - 256} = \sqrt{900} = 30\text{ yards}

. The area is found by multiplying half of the base by the height:

\frac{1}{2} \times 16 \times 30 = 240\text{ square yards}

.

Step-by-Step Solution

1

Identify the given sides in relation to a right-angled triangle.

The hypotenuse (longest side)

c = 34\text{ yards}

, and one leg

a = 16\text{ yards}

. The other leg

b

is unknown.

Correctly identifying the hypotenuse is necessary to set up the Pythagorean theorem.

2

Use the Pythagorean theorem to find the length of the unknown leg.

a^2 + b^2 = c^2 \Rightarrow 16^2 + b^2 = 34^2 \Rightarrow 256 + b^2 = 1156 \Rightarrow b^2 = 900 \Rightarrow b = 30

.

The Pythagorean theorem relates the lengths of all three sides in a right triangle.

3

Calculate the area of the triangular garden.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16 \times 30 = 240

.

The area of a right triangle is half the product of the lengths of its two perpendicular legs.

Key Concept

Using the Pythagorean theorem to find a missing leg, and applying the triangle area formula.

Question 155Question →

A naval patrol vessel leaves port and sails due South for $10\text{ nautical miles}$ . It then changes course and sails due West. After completing this leg of the journey, the vessel's radar indicates it is exactly $26\text{ nautical miles}$ in a straight line from the port. How many nautical miles did the vessel travel due West?

16

24

36

576

Show answer & explanation

Answer: $24$

Answer

The vessel traveled

24

nautical miles due West.

The path of the vessel forms a right triangle. The straight-line distance from the port is the hypotenuse (

26\text{ nautical miles}

), and the southward distance is one leg (

10\text{ nautical miles}

). Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known values:

10^2 + b^2 = 26^2

. This simplifies to

100 + b^2 = 676

. Subtracting

100

from both sides gives

b^2 = 576

. Taking the square root of

576

yields

24\text{ nautical miles}

.

Step-by-Step Solution

1

Identify that the vessel's path forms a right triangle.

The southward path (

10

) and westward path (

x

) are the legs, and the straight-line distance (

26

) is the hypotenuse.

South and West directions are perpendicular, creating a 90-degree angle.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

10^2 + x^2 = 26^2

This formula relates the lengths of the legs to the hypotenuse in a right triangle.

3

Square the known values.

100 + x^2 = 676

To begin isolating the unknown variable

x

.

4

Subtract

100

from both sides of the equation.

x^2 = 576

To completely isolate the

x^2

term.

5

Take the square root of both sides.

x = 24

To find the final distance traveled due West.

Key Concept

Applying the Pythagorean theorem to find the missing leg of a right triangle.

Estimated Time:1m 0s

Question 156Question →

The floor of a rectangular storage room has a diagonal measurement of $39$ feet and a width of $15$ feet. What is the total area of the floor in square feet?

Show answer & explanation

Answer: 540

Answer

540

The storage room floor forms a right triangle with the 39-foot diagonal as the hypotenuse and the 15-foot width as a leg. The missing length is calculated to be 36 feet using the Pythagorean theorem. Multiplying this 36-foot length by the 15-foot width yields a total area of 540 square feet.

Step-by-Step Solution

1

Identify the right triangle formed by the floor dimensions

A right triangle is formed with the diagonal (39 feet) as the hypotenuse and the width (15 feet) as one leg.

The diagonal of any rectangle splits it into two identical right triangles.

2

Find the missing length of the floor

The length is 36 feet.

Using the Pythagorean theorem (

a^2 + b^2 = c^2

),

15^2 + b^2 = 39^2

. This simplifies to

225 + b^2 = 1521

, so

b^2 = 1296

and

b = 36

. Alternatively, recognize the 5-12-13 Pythagorean triple multiplied by 3.

3

Calculate the area of the rectangle

Area = 540 square feet.

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

36 \times 15 = 540

).

Key Concept

Using the Pythagorean theorem to find unknown dimensions for area calculations.

Alternative Method

Instead of calculating

39^2

and

15^2

manually, recognize that the given sides are multiples of the common 5-12-13 Pythagorean triple (

3 \times 5 = 15

and

3 \times 13 = 39

). This immediately gives the missing length as

3 \times 12 = 36

.

Estimated Time:1m 0s

Question 157Question →

A canvas sail is shaped like a right triangle. The longest side of the sail measures $15\text{ feet}$ , and the bottom edge measures $9\text{ feet}$ . What is the total area of the canvas sail?

54\text{ square feet}

67.5\text{ square feet}

108\text{ square feet}

27\text{ square feet}

Show answer & explanation

Answer: $54\text{ square feet}$

Answer

54\text{ square feet}

The area of a right triangle is half the product of its legs. First, the Pythagorean theorem (

a^2 + b^2 = c^2

) must be used to find the missing perpendicular height (the second leg). The longest side is the hypotenuse (

15

), and the bottom edge is a leg (

9

). Solving for the missing height gives

\sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12\text{ feet}

. The area is then found by using the formula

\frac{1}{2} \times \text{base} \times \text{height}

, which equals

\frac{1}{2} \times 9 \times 12 = 54\text{ square feet}

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The hypotenuse (longest side) is

15\text{ feet}

and the base (one leg) is

9\text{ feet}

.

The Pythagorean theorem requires knowing two sides of a right triangle to find the third.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing height (the other leg).

Substituting the known values gives

9^2 + b^2 = 15^2

, which simplifies to

81 + b^2 = 225

. Solving for

b^2

gives

144

, meaning the height is

\sqrt{144} = 12\text{ feet}

.

The height of the right triangle is necessary to calculate its total area.

3

Calculate the area of the right triangle.

Area =

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 = 54\text{ square feet}

.

The standard triangle area formula provides the final answer.

Key Concept

Using the Pythagorean theorem to find a missing side length before calculating the area of a right triangle.

Geometry

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes