SAT Area and volume
Apply area/volume formulas — forward and in reverse.
How to score it
- The reference sheet has every formula — use it; don't memorize-and-misremember.
- Scaling: area grows by k², volume by k³.
- Inverse problems: substitute into the formula and solve for the unknown dimension.
Common traps
- Scales area linearly (×k) instead of ×k².
- Mixes radius and diameter.
- Forgets π or a fractional constant in a volume formula.
The 4 question types, with real examples
Direct formula application
“What is the [area / volume] of the [solid]?”
A circular pool has a diameter of 20 feet. What is the area, in square feet, of the pool?
- A
- B
- C✓
- D
The radius is half the diameter, so feet. The area of a circle is , which gives square feet.
Scaling area / volume
“Side lengths are k times larger. The area is how many times larger?”
A rectangular field has a length of 120 meters and a width of 80 meters. If both the length and width are increased by 50%, what is the area, in square meters, of the enlarged field?
- A
- B
- C
- D✓
Increasing each dimension by 50% means multiplying by 1.5. The new length is 120 × 1.5 = 180 meters and the new width is 80 × 1.5 = 120 meters. The area is 180 × 120 = 21600 square meters.
Inverse: measure → dimension
“The volume of a cube is V. What is the edge length?”
A circle has an area of square feet. What is the radius, in feet, of the circle?
- A
- B✓
- C
- D
The area of a circle is . Setting equal to and solving for r yields = 64, so feet.
Composite figure
“[FIGURE: composite shape, e.g. rectangle union semicircle]. What is the area of the figure?”
A rectangular garden has a length of 18 feet and a width of 15 feet. A concrete path 2 feet wide is built around the entire perimeter of the garden. What is the area, in square feet, of the concrete path?
- A
- B
- C✓
- D
The outer dimensions including the path are (18 + 2 + 2) = 22 feet by (15 + 2 + 2) = 19 feet. The outer area is 22(19) = 418 square feet. The inner garden area is 18(15) = 270 square feet. The path area is 418 - 270 = 148 square feet.
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