SAT Math: Area, Volume, and Scale Factors Practice Questions
Calculate volumes of common solids and track how length, area, and volume change under scaling. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
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1. A cylinder has radius 3 and height 8. Using V = pi r^2 h, what is its volume?
Explanation. Substitute r = 3 and h = 8: V = pi(3)^2(8) = 72pi cubic units. The radius is squared before multiplying by the height.
2. A cone has radius 6 and height 5. Using V = (1/3)pi r^2 h, what is its volume?
Explanation. The cylinder with the same radius and height has volume pi(6)^2(5) = 180pi. A cone has one third of that volume, or 60pi cubic units.
3. Two spheres have radii 3 and 6. What is the ratio of the larger sphere's volume to the smaller sphere's volume?
Explanation. Sphere volume is proportional to the cube of its radius. The radius ratio is 6/3 = 2, so the volume ratio is 2^3 = 8.
4. A rectangle is enlarged so that each side length is multiplied by 3/2. By what factor is its area multiplied?
Explanation. Area is length times width. Multiplying each dimension by 3/2 multiplies area by (3/2)(3/2) = 9/4.
5. A rectangular prism is 3 units long, 4 units wide, and 10 units high. What is its volume in cubic units?
Explanation. Multiply the three perpendicular dimensions: V = 3(4)(10) = 120 cubic units. Adding the dimensions would not measure volume.
6. A triangle has area 54 square units and base 12 units. What is its height to that base?
Explanation. Use A = (1/2)bh. Then 54 = (1/2)(12)h = 6h, so h = 9. The height is the perpendicular distance to the chosen base.
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FAQ
How does a scale factor change area and volume?
For similar figures with length scale factor k, area scales by k squared. For similar solids, volume scales by k cubed.
Which units should a result use?
Lengths use linear units, areas use square units, and volumes use cubic units. Check that the dimensions in your formula produce the requested units.