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Effects of Scaling on Area and Volume — Practice Set

Work through each section in order. For multiple-choice problems, select the best answer. Show your work where space is provided.

Grade 10·22 problems·~35 min·Digital SAT Math·topic·sat-geotrig-area-scale
Work through problems with immediate feedback
A

Recall / Warm-Up

1.

What is the area of a circle with radius 6 cm?

2.

What is the volume of a sphere with radius 3 cm? Use V=43πr3V = \frac{4}{3}\pi r^3.

3.

Evaluate k3k^3 when k=2k = 2.

B

Fluency Practice

Side-by-side rectangles showing a 3×5 original scaled to 6×10, with four small copies fitting inside the large one to illustrate that area scales by k² = 4.
1.

A rectangle has dimensions 3 cm by 5 cm. Both dimensions are doubled. By what factor does the area increase?

2.

A square has side length 8 m. All dimensions are multiplied by 12\frac{1}{2}. What is the area of the new square in m²?

Two spheres with radii 2 cm and 6 cm side by side, showing the k=3 scale factor and illustrating that the large sphere contains 27 times the volume of the small one.
3.

A sphere has radius 2 cm. A second sphere has radius 6 cm. How many times greater is the volume of the second sphere than the first?

4.

A cylinder has radius 4 cm and height 5 cm. All dimensions are tripled. What is the volume of the new cylinder in cm³? (Use π≈3.14\pi \approx 3.14. Round to the nearest whole number.)

5.

A cylinder has radius rr and height hh. The radius is doubled, but the height stays the same. By what factor does the volume change?

C

Varied Practice

1.

A circle has area AA. If the radius is tripled, the new area is:

2.

A rectangle has dimensions 10 cm by 4 cm. Its dimensions are multiplied by 12\frac{1}{2}. Fill in the blanks: the new area equals [original area]\text{[original area]} × [scale factor for area]\text{[scale factor for area]} = [answer in cm²].

original area (cm²):
scale factor for area:
new area (cm²):
3.

A cube has side length 2 m. A second cube has side length 6 m. What is the ratio of the volume of the first cube to the volume of the second cube?

Two cylinders: the original with r=3, h=10 and the scaled version with r=9, h=10 (height unchanged). Labels show that multiplying by k³=27 is wrong and k²=9 is correct when only the radius scales.
4.

A cylinder has radius 3 cm and height 10 cm. Only the radius is multiplied by 3; the height stays 10 cm. Which expression gives the new volume?

5.

Two similar rectangular prisms have volumes in the ratio 1:641:64. What is the ratio of their corresponding side lengths?

D

Word Problems

Two similar triangles: small (base 5, height 4, area 10) and large (base 15, height 12, area 90), showing the k=3 scale factor and area multiplier of 9.
1.

Two similar triangles are shown. The smaller triangle has base 5 cm and height 4 cm. The larger triangle has its base scaled to 15 cm (all dimensions tripled).

How many times greater is the area of the larger triangle than the area of the smaller triangle?

2.

A cylindrical storage tank has radius 5 ft and height 20 ft. A larger tank is built with all dimensions doubled.

The volume of the larger tank is how many times the volume of the original tank?

3.

Priya is painting two similar rectangular murals. The smaller mural is 2 m wide and 3 m tall (area = 6 m²). The larger mural has all dimensions multiplied by a factor of kk.

1.

If k=4k = 4, what is the area of the larger mural?

2.

The larger mural has area 150 m². What is the scale factor kk? (Round to the nearest tenth if needed.)

4.

Two similar spheres have volumes in the ratio 1:81:8. The radius of the smaller sphere is 3 cm.

What is the radius of the larger sphere?

E

Error Analysis

1.

A student solved the following problem:

"A square has side length 5 cm. All dimensions are tripled. What is the area of the new square?"

Student's work:
New area = original area × kk = 25×3=7525 \times 3 = 75 cm²

What error did the student make?

2.

A student solved the following problem:

"Two similar cylinders have volumes in the ratio 1:271:27. Find the ratio of their radii."

Student's work:
Volume ratio = 27, so area ratio =27≈5.2= \sqrt{27} \approx 5.2.
Therefore the radii are in ratio 1 ⁣: ⁣5.21\!:\!5.2.

Identify all errors in the student's work and state the correct ratio of the radii.

F

Challenge / Extension

1.

A cone has radius 6 cm and height 8 cm. A smaller, similar cone is made by reducing all dimensions by a factor of 13\frac{1}{3}. What fraction of the original cone's volume does the smaller cone have? Express as a simplified fraction.

2.

A sphere's radius is increased so that its volume is 125 times the original volume. By what factor was the radius increased?

0 of 22 answered