Home / Digital SAT Math Practice Questions -Medium : Area and volume

# Digital SAT Math Practice Questions -Medium : Area and volume

## SAT MAth Practice questions – all topics

• Geometry and Trigonometry Weightage: 15% Questions: 5-7
• Area and volume
• Lines, angles, and triangles
• Right triangles and trigonometry
• Circles

## SAT MAth and English  – full syllabus practice tests

[Calc]  Question   Medium

Oocytes are a type of cell that can be modeled as a sphere. The table shows the surface area, in square micrometers $$\left(\mu \mathrm{m}^2\right)$$, and volume, in cubic micrometers $$\left(\mu \mathrm{m}^3\right)$$, based on the average radius for oocytes at the same stage of development in four types of mammals.
(The surface area of a sphere with a radius of $$r$$ is $$4 \pi r^2$$, and the volume of a sphere with a radius of $$r$$ is equal to $$\frac{4}{3} \pi r^3$$.

Based on the information in the table, what is the average radius, in micrometers, of a hamster oocyte?
A) 68.4
B) 20.7
C) 14.2
D) 11.7

Ans:D

To find the average radius of a hamster oocyte, we use the given surface area for the hamster:

Surface area of hamster oocyte: $$1,720.2 \, \mu \text{m}^2$$

The surface area $$A$$ of a sphere is given by the formula:
$A = 4\pi r^2$

Rearrange to solve for $$r$$:
$r^2 = \frac{A}{4\pi}$
$r = \sqrt{\frac{A}{4\pi}}$

Substitute the given surface area:
$r = \sqrt{\frac{1,720.2}{4\pi}}$

Calculate the value:
$r = \sqrt{\frac{1,720.2}{4 \times 3.14159}} = \sqrt{\frac{1,720.2}{12.56636}} \approx \sqrt{136.897} \approx 11.7$

[No calc]  Question  medium

In the figure shown, ABCD is a parallelogram and EBFD is a square. The area of ABCD is 112 square meters $$(m^2)$$, and the area of EBFD is 64 $$m^2$$. What is the length, in meters, of line segment AE ?

A) 6

B) 8

C) 14

D) 23

A) 6

$\text{Area of parallelogram}= \text{AD}\times \text{BE}$

$\text{Area of Square}= \text{DE}\times \text{BE}$ Since, in Square all sides are equal .

$\text{Area of Square}=64$

$\text{DE}^2~\text{or}~\text{BE}^2=64\Rightarrow 8$

Now, $\text{Area of parallelogram}=112$

$112= \text{AD}\times \text{BE}$

$\text{AD}=\frac{112}{8}\Rightarrow 14$

$\text{AD}=\rm{AE+ED}$ And DE=8

So, AE= 6 m

[Calc]  Question  Medium

A right circular cylinder has a height of 6 inches. The radius of the base of the cylinder is 5 inches. What is the volume, in cubic inches, of the cylinder?

A. 10$$\pi$$
B. 30$$\pi$$
C. 50$$\pi$$
D. 150$$\pi$$

Ans: D

The volume $$V$$ of a right circular cylinder is given by the formula:

$V = \pi r^2 h$

where $$r$$ is the radius of the base and $$h$$ is the height of the cylinder.

Given that the radius $$r$$ is $$5$$ inches and the height $$h$$ is $$6$$ inches,

$V = \pi (5)^2 (6)$
$V = \pi (25)(6)$
$V = 150\pi$

So, the volume of the cylinder is $$150\pi$$ cubic inches.

[Calc]  Question  Medium

The area of a rectangular region is increasing at a rate of 20 square feet per hour. Which of the following is closest to this rate in square meters per minute? (Use 1 meter = 3.28 feet.)
A. 0.03
B. 0.10
C. 1.09
D. 2.03

Ans: A

Step 1: Convert square feet per hour to square feet per minute

Given:
$20 \text{ square feet per hour}$

Since there are 60 minutes in an hour, we convert square feet per hour to square feet per minute by dividing by 60:
$20 \text{ square feet per hour} \times \frac{1 \text{ hour}}{60 \text{ minutes}} = \frac{20}{60} \text{ square feet per minute} = \frac{1}{3} \text{ square feet per minute}$

Step 2: Convert square feet to square meters

The conversion factor is:
$1 \text{ meter} = 3.28 \text{ feet}$

Therefore, $$1 \text{ square meter} = (3.28 \text{ feet})^2 = 10.7584 \text{ square feet}$$

To convert from square feet to square meters:
$1 \text{ square foot} = \frac{1 \text{ square meter}}{10.7584 \text{ square feet}} = \frac{1}{10.7584} \text{ square meters}$

Now, convert $$\frac{1}{3}$$ square feet per minute to square meters per minute:
$\frac{1}{3} \text{ square feet per minute} \times \frac{1 \text{ square meter}}{10.7584 \text{ square feet}} = \frac{1}{3} \times \frac{1}{10.7584} \text{ square meters per minute}$

Simplify the calculation:
$\frac{1}{3} \times \frac{1}{10.7584} = \frac{1}{3 \times 10.7584} = \frac{1}{32.2752} \approx 0.031 \text{ square meters per minute}$

$\boxed{0.03}$

[Calc]  Question  Medium

The volume of the right triangular prism shown is 96 cubic centimeters $$\left(\mathrm{cm}^3\right)$$. What is the area, in $$\mathrm{cm}^2$$, of one of the triangular bases of the prism?
A) 4
B) 8
C) 16
D) 42

Ans:B

The volume $$V$$ of a prism is given by the formula:
$V = \text{Base Area} \times \text{Height}$
where the height is the length between the two triangular bases, which in this case is 12 cm.

$V = 96 \, \text{cm}^3$
$\text{Height} = 12 \, \text{cm}$

$\text{Base Area} = \frac{V}{\text{Height}}$
$\text{Base Area} = \frac{96 \, \text{cm}^3}{12 \, \text{cm}}$
$\text{Base Area} = 8 \, \text{cm}^2$

[Calc]  Question  medium

The area, in square inches, of a certain right triangle is given by the equation $$A=\frac{1}{2}b(2b)$$, where b is the length, in inches, of one of the legs of the triangle. Which expression represents the length, in inches, of the shortest leg of the triangle?

A) $$\frac{1}{2} b$$

B) b

C) 2b

D) $$2b^2$$

B) b

Given the area of a right triangle as $$\frac{1}{2} b(2b)$$, where $$b$$ is the length of one of the legs of the triangle, we need to find the expression representing the length of the shortest leg.

We know that the area of a right triangle is given by the formula $$\frac{1}{2} \times \text{base} \times \text{height}$$. Here, $$b$$ represents one of the legs, so the length of the other leg is also $$b$$.

So, the expression representing the length of the shortest leg is simply $$b$$.

[Calc]  Question  Medium

In the figure shown, all angles formed by adjacent sides are right angles.

What is the perimeter of the figure?
A) 25
B) 39
C) 42
D) 46

D

Sum of all sides length will be the perimeter of the figure.

$7+2++4.5+9+5+x+2.5+8+2.5+x+2.5+2$

$45+2x\Rightarrow 46$

[Calc]  Question    Medium

The total $$\operatorname{cost} C$$, in dollars to tile a square floor is represented by the equation $$C=16 L^2$$, where $$L$$ is the length of one side of the floor, in feet. Which of the following represents the cost, in dollars per square foot, to tile the floor?
A) $$L$$
B) 4
C) 16
D) $$16 L$$

Ans:C

The cost $$C$$, in dollars, to tile a square floor is given by the equation $$C = 16L^2$$, where $$L$$ is the length of one side of the floor in feet.

We need to determine the cost in dollars per square foot to tile the floor.

1. The area of the square floor is:
$L^2 \text{ square feet}$

2. The total cost to tile the floor is:
$C = 16L^2 \text{ dollars}$

3. To find the cost per square foot, we divide the total cost by the area:
$\text{Cost per square foot} = \frac{C}{L^2} = \frac{16L^2}{L^2} = 16$

[Calc]  Question   medium

Scientists took 94 ice core sections from a glacier. Each section was in the shape of a right circular cylinder and had a length of 1 meter and a diameter of 0.1 meter. Which of the following is closest to the total volume, in cubic meters, of the 94 sections?

A)30

B)7

C)3

D)0.7

D)0.7

To find the total volume of the ice core sections, we first need to find the volume of one section and then multiply it by the total number of sections.

The volume $$V$$ of a right circular cylinder can be calculated using the formula:
$V = \pi r^2 h$
Where $$r$$ is the radius and $$h$$ is the height (or length in this case) of the cylinder.

Given:
Length $$h = 1$$ meter
Diameter $$d = 0.1$$ meter

We can find the radius $$r$$ by dividing the diameter by $$2$$:
$r = \frac{d}{2} = \frac{0.1}{2} = 0.05 \text{ meters}$

Now, we can calculate the volume of one section:
$V = \pi \times (0.05)^2 \times 1$
$V = \pi \times 0.0025$
$V \approx 0.00785 \text{ cubic meters}$

Now, to find the total volume of 94 sections, we multiply the volume of one section by the number of sections:
$\text{Total volume} = 0.00785 \times 94$
$\text{Total volume} \approx 0.7379 \text{ cubic meters}$

Given the options, the closest value to $$0.7379$$ is D) 0.7.

[No- Calc]  Question  Medium

Two beach balls are each in the shape of a sphere. The larger beach ball has a diameter of 3x, and the smaller beach ball has a diameter of x. What is the ratio of the volume of the larger beach ball to the
volume of the smaller beach ball?

A) 3 to 1
B) 6 to 1
C) 9 to 1
D) 27 to 1

Ans: D

The volume of a sphere is given by the formula $$V = \frac{4}{3}\pi r^3$$, where $$r$$ is the radius.

The ratio of the volumes of the larger beach ball to the smaller beach ball is:

$\frac{V_{\text{larger}}}{V_{\text{smaller}}} = \frac{\frac{4}{3}\pi (3x)^3}{\frac{4}{3}\pi x^3}$

$= \frac{27x^3}{x^3}$

$= 27$

So, the ratio of the volume of the larger beach ball to the volume of the smaller beach ball is D) 27 to 1.

[Calc]  Question Medium

What is the area, in square units, of the figure shown?
A) 20
B) 22
C) 24
D) 28

B

[Calc]  Question  Medium

In the figure above, $A B=A D, B C=C D, B E=2$, $B C=4$, and $A C=10$. What is the area of triangle $A B D$ ?
A) $40-8 \sqrt{3}$
B) $30-6 \sqrt{3}$
C) $20-4 \sqrt{3}$
D) $10-2 \sqrt{3}$

C

Question

Crawford County, Iowa, is shaped like a rectangle. The length of the county is 6 miles longer than the width of the county. If $$A(x)$$ is the area of the county, in square miles, and $$x$$ is the width, in miles, which equation best models the area of the county? 2.13

1. $$A(x) = x(6x)$$
2. $$A(x) = x(6 – x)$$
3. $$A(x) = x(x – 6)$$
4. $$A(x) = x(x + 6)$$

D

Questions

The length of a rectangular tile is 4 times the width of the tile. If the area of the tile is 144 square inches, what is the width of the tile, in inches?
A. 6
B. 12
C. 24
D. 36

Ans: A

Question

The result of two random samples of votes for a proposition are shown above. The samples were selected from the same population, and the margins of error were calculated using the same method. Which of the following is the most appropriate reason that the margin of error for sample A is greater than the margin of error for sample B?

1. Sample A had a smaller number of votes that could not be recorded.
2. Sample A had a higher percent of favorable responses.
3. Sample A had a larger sample size.
4. Sample A had a smaller sample size

Ans: D

Question

A graphic designer is creating a logo for a company. The logo is shown in the figure above. The logo is in the shape of a trapezoid and consists of three congruent equilateral triangles. If the perimeter of the logo is 20 centimeters, what is the combined area of the shaded regions, in square centimeters, of the logo?

1. $$2\sqrt{3}$$
2. $$4\sqrt{3}$$
3. $$8\sqrt{3}$$
4. 16

Ans: C

Questions

The diagram above represents Edward T, Hall’s concept of space surrounding a person defined by four nonoverlapping regions. Intimate space is the region inside a circle of radius 1 foot. Personal space is the region within a circle of radius 4 feet but outside intimate space. Social space is the region within a circle of radius 12 feet but outside personal space. Public space is the region within a circle of radius 25 feet but outside social space. What is the area, in square feet, of the shaded region representing a person’s social space?
A. $127 \pi$
B. $128 \pi$
C. $144 \pi$
D. $625 \pi$

Ans: B

Questions

Kelly enlarged the area of a photograph to $250 \%$ of its original size. The original dimensions of the photograph were 5 inches by 7 inches. What is the area of the enlarged photograph, in square inches?
A. 71.25
B. 87.5
C. 218.75
D. 3,000

Ans: B

Questions

A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. Which of the following expresses the volume, $V$, in cubic inches, of such containers, where $r$ is the radius, in inches?
A. $V=4 \pi r^3$
B. $V=\pi(2 r)^3$
C. $V=\pi r^2+4 \pi r$
D. $V=\pi r^2+4 \pi r^2$

Ans: D

Questions

The figure above represents a rectangular painting with a frame that is 2 inches wide. The expression $$2x^2 – (x – 4)(2x – 4)$$ represents the area of the frame, in square inches. What does the quantity $$(x – 4) (2x – 4)$$ in the expression represent?

1. The width of the painting, in inches
2. The height of the frame, in inches
3. The area, in square inches, of the inner rectangle
4. The combined area, in square inches, of the frame and painting