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Digital SAT · Solve it in Desmos

How to Solve Scale Factor for Area and Volume on Desmos (Digital SAT)

The calculator carries the squares, cubes, and roots, which is welcome on grid-ins. Matching the power to the dimension is the decision it cannot make for you.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for scale factor for area and volume

The Desmos steps

  1. 1

    Type the scaled value

    Enter the base times the factor to the matching power, like A * 2^2 for an area.

  2. 2

    Root to go backward

    For a linear factor from an area ratio, take sqrt; from a volume ratio, the cube root.

  3. 3

    Read the result

    Desmos returns the scaled area or volume, or the recovered factor.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example
Two similar cubes have a linear scale factor of 2. The smaller cube has a volume of 7. What is the volume of the larger cube?
A
14
B
28
C
42
56

A: Incorrect. This multiplies the volume by 2 instead of by 232^{3}.

B: Incorrect. This multiplies the volume by 4 instead of by 232^{3}.

C: Incorrect. This multiplies the volume by 6 instead of by 232^{3}.

D: Correct. Volumes scale by k3=8k^{3} = 8, so the larger volume is 7×8=567 \times 8 = 56.

Explanation

Larger volume =7×23=56= 7 \times 2^{3} = 56.

Medium example
Two similar solids have a linear scale factor of 3. The smaller solid has a volume of 4. What is the volume of the larger solid?
A
12
B
36
108
D
324

A: Incorrect. This multiplies the volume by 3 instead of by 333^{3}.

B: Incorrect. This multiplies the volume by 323^{2}, the area factor, instead of by 333^{3}.

C: Correct. Volumes scale by k3=27k^{3} = 27, so the larger volume is 4×27=1084 \times 27 = 108.

D: Incorrect. This multiplies by 343^{4} instead of by 333^{3}.

Explanation

Larger volume =4×33=108= 4 \times 3^{3} = 108.

Detailed explanation

Since volumes scale by the cube of the linear factor, the larger volume is 4×33=4×27=1084 \times 3^{3} = 4 \times 27 = 108.

Hard example
A scale model is built at a ratio of 1 to 50. The model has a surface area of 8 square centimeters. What is the surface area, in square centimeters, of the actual object?
A
400
B
2,000
20,000
D
1,000,000

A: Incorrect. This multiplies by the linear factor 50 instead of by 50250^{2}.

B: Incorrect. This multiplies by 250 instead of by 50250^{2}.

C: Correct. Areas scale by 502=2,50050^{2} = 2{,}500, so the actual area is 8×2,500=20,0008 \times 2{,}500 = 20{,}000.

D: Incorrect. This multiplies by 50350^{3}, the volume factor, instead of by 50250^{2}.

Explanation

Actual area =8×502=20,000= 8 \times 50^{2} = 20{,}000.

Detailed explanation

Because areas scale by the square of the linear factor, the actual surface area is 8×502=8×2,500=20,0008 \times 50^{2} = 8 \times 2{,}500 = 20{,}000.

Try it with the calculator

Question 1easy
Similar figures
AB

Two similar rectangles have a linear scale factor of 3. The smaller rectangle has an area of 5. What is the area of the larger rectangle?

Question 2easy
Similar figures
AB

Two similar figures have a linear scale factor of 4. The larger figure has an area of 80. What is the area of the smaller figure?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Desmos questions

Can Desmos solve scale factor for area and volume questions on the SAT?

The calculator carries the squares, cubes, and roots, which is welcome on grid-ins. Matching the power to the dimension is the decision it cannot make for you.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master scale factor for area and volume, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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