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Math foundations
SAT area and volume: what happens when you scale a shape
Double the side of a cube and its volume grows eightfold. Explore scale factors with an interactive model, review the formulas on the SAT reference sheet, and practice the questions that trip students up.
SPSAT Practice Editorial8 min read
The short version
What you'll take away
- If every length is multiplied by k, area is multiplied by k² and volume by k³.
- The SAT reference sheet in Bluebook lists the main area and volume formulas, so focus on choosing and applying them.
- Change one dimension at a time by tracking where it appears in the formula: doubling a cylinder’s radius quadruples its volume.
In this article
Scale every length, watch everything else
When a shape is scaled by a factor k, every length is multiplied by k. Area measures two dimensions, so it is multiplied by k twice. Volume measures three, so it is multiplied by k three times. Move the slider to see the squares and cubes add up.
Try it yourself · interactive
Scale a unit square and a unit cube
Area: 2 × 2 unit squares
Volume: 2 layers of 4 unit cubes
- Length
- × 2
- Area
- × 4
- Volume
- × 8
Formulas you will be given
Bluebook includes a reference sheet with the common formulas. The SAT rarely asks you to memorize them. It asks you to choose the right one, substitute carefully, and interpret the result.
- Circle: area \(\displaystyle \pi r^2\), circumference \(\displaystyle 2\pi r\).
- Rectangle \(\displaystyle \ell w\) and triangle \(\displaystyle \frac{1}{2}bh\).
- Rectangular prism \(\displaystyle \ell w h\) and cylinder \(\displaystyle \pi r^2 h\).
- Sphere \(\displaystyle \frac{4}{3}\pi r^3\), cone \(\displaystyle \frac{1}{3}\pi r^2 h\), pyramid \(\displaystyle \frac{1}{3}\ell w h\).
Changing one dimension at a time
When only one measurement changes, look at its exponent in the formula. In \(\displaystyle V = \pi r^2 h\), the radius is squared and the height is not. Doubling the radius multiplies the volume by 4, doubling the height multiplies it by 2, and doubling both multiplies it by 8.
Use the same reasoning in reverse. If a sphere’s volume is multiplied by 27, its radius was multiplied by 3, because \(\displaystyle 3^3 = 27\).
Units square and cube too
One yard is 3 feet, so one square yard is \(\displaystyle 3 \times 3 = 9\) square feet, and one cubic yard is \(\displaystyle 3 \times 3 \times 3 = 27\) cubic feet. Converting area or volume with the linear factor alone is a common mistake.
Your turn
Quick practice
A cube has a volume of 27 cubic centimeters. Each edge is tripled. What is the volume of the new cube, in cubic centimeters?
Hint: Volume uses three dimensions. What is 3 × 3 × 3?
Practice: try these before checking the answers
Write the formula, then substitute.
- 1. Find the volume of a cylinder with radius 3 and height 5.
- 2. A 6-by-4 rectangle is scaled by a factor of 2.5. What is the new area?
- 3. Find the volume of a sphere with radius 3.
- 4. Two similar prisms have heights 2 and 6. What is the ratio of their volumes?
Sources & further reading
One step from here
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