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Math: Geometry and Trigonometry, 3 original questions

SAT Area and Volume Practice: Questions and Answers

Practice cylinders, similar solids and rectangle measurements in three original SAT-style questions. Track squared and cubed units with fully explained answers.

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The short answer

Area measures a two-dimensional surface, while volume measures three-dimensional space. Identify the shape and the requested quantity before selecting a formula. A diagram may supply a diameter where a formula needs a radius, or a perimeter where you must find a side first. Keep units consistent. When similar shapes scale by a length factor, areas scale by its square and volumes by its cube. Estimate the size of your result and distinguish a length from an area or volume.

Method at a glance

  1. 1Identify the shape and quantity
  2. 2Find any missing measurement
  3. 3Calculate with consistent units
  4. 4Check the dimension and scale

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Identify the shape and quantity

    Choose whether the problem asks for length, area, surface area or volume. List the measurements required by that formula.

  2. Step 2: Find any missing measurement

    Convert a diameter to radius or use a perimeter relationship to obtain a side. Use perpendicular height when the formula requires it.

  3. Step 3: Calculate with consistent units

    Square lengths for area and cube the relevant scale factor for volume. Do not combine centimeters with meters without conversion.

  4. Step 4: Check the dimension and scale

    Report square or cubic units as appropriate. For similar figures, verify whether the change should follow k, k² or k³.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

A right circular cylinder has radius 3 centimeters and height 10 centimeters. What is its volume in cubic centimeters?

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Correct answer

Choice C: 90π

Explanation

A cylinder's volume is πr²h. With r = 3 and h = 10, V = π × 9 × 10 = 90π cubic centimeters. The radius is squared before multiplying by the height.

Why each choice is right or wrong

  1. Choice A

    This uses r instead of r² in the volume formula.

  2. Choice B

    This calculates circumference times height, not the cylinder's volume.

  3. Choice C

    Correct answer

    Correct: π × 3² × 10 = 90π.

  4. Choice D

    This adds an unnecessary factor of 2 to the volume formula.

Question 2 of 3

Two solid figures are similar. Every length in the larger figure is 3 times its corresponding length in the smaller figure. The smaller volume is 8 cubic units. What is the larger volume?

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Correct answer

Choice C: 216

Explanation

A length scale factor of 3 gives a volume scale factor of 3³ = 27. Multiply the smaller volume by 27: 8 × 27 = 216 cubic units. The cube is needed because all three dimensions scale.

Why each choice is right or wrong

  1. Choice A

    This applies the length factor 3 to volume.

  2. Choice B

    This applies the area factor 3² to volume.

  3. Choice C

    Correct answer

    Correct: 8 × 3³ = 216.

  4. Choice D

    This applies an extra factor of 3 beyond the three scaled dimensions.

Question 3 of 3

A rectangle has perimeter 34 units and width 6 units. What is its area in square units?

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Correct answer

Choice C: 66

Explanation

The perimeter equation is 2L + 2 × 6 = 34, so 2L = 22 and L = 11. The area is length times width: 11 × 6 = 66 square units. The perimeter determines a length before the area is calculated.

Why each choice is right or wrong

  1. Choice A

    11 is the length, not the area.

  2. Choice B

    22 is twice the length, not the area.

  3. Choice C

    Correct answer

    Correct: the rectangle's sides are 11 and 6, giving area 66.

  4. Choice D

    This multiplies half the perimeter by the width without subtracting the width to find the length.

Common mistakes

  • Using diameter in place of radius. Halve the diameter before using a radius-based formula.

  • Scaling volume by the length factor. Similar solids scale in all three dimensions, so their volumes change by the cube.

  • Using perimeter as area. First solve for the required side lengths, then multiply those lengths.

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Common questions

How do area and volume scale for similar figures?

If corresponding lengths multiply by k, areas multiply by k² and volumes by k³. The similarity condition is important.

Can I use a slanted length as height?

Only when it is the height required by the formula. For a triangle or prism, distinguish a perpendicular height from an unrelated slanted edge.

Should an area answer use square units?

Yes. A volume answer uses cubic units. Checking units helps detect a formula that calculates the wrong quantity.

Sources

  1. College Board: Geometry and Trigonometrysatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides