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Area & Volume for the SAT: Complete Study Guide + Free Practice Problems
Everything the SAT tests about area and volume in one place: area formulas, volume formulas, composite figures, the effect of scaling, and word problems — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat's given vs. what to memorize
The SAT provides a reference sheet with several area and volume formulas at the start of every Math section. Even so, the fastest students still memorize the core formulas, since flipping back to the reference sheet on every question costs valuable time.
Even though formulas are provided, the SAT rarely tests a formula in isolation. Most questions wrap the formula inside a word problem, a composite figure, or a scaling scenario — memorizing the formula is only the first step.
Area formulas
| Shape | Formula |
|---|---|
| Rectangle | A = length · width |
| Triangle | A = ½ · base · height |
| Parallelogram | A = base · height |
| Trapezoid | A = ½ · (base₁ + base₂) · height |
| Circle | A = πr² |
A trapezoid has parallel sides of 8 and 14, with a height of 5. Find its area.
Composite figures
A composite figure combines two or more basic shapes into one. To find its area, break it into simpler pieces, calculate each piece separately, then add or subtract.
Adding pieces
Used when the shape is made of separate sections joined together, like a rectangle with a triangular roof.
Subtracting pieces
Used when a shape has a piece removed from it, like a square with a circular hole cut out.
A rectangle measures 10 by 6. A semicircle with diameter 6 is cut out of one side. Find the remaining area.
Volume formulas
| Solid | Formula |
|---|---|
| Rectangular prism | V = length · width · height |
| Cylinder | V = πr²h |
| Cone | V = ⅓πr²h |
| Sphere | V = &frac43;πr³ |
| Pyramid | V = ⅓ · base area · height |
A cylinder has a radius of 3 and a height of 10. Find its volume in terms of π.
The cone and pyramid formulas both include a factor of ⅓, which is easy to drop under time pressure. A cone is not simply a cylinder's formula — it's exactly one-third the volume of a cylinder with the same base and height.
The effect of scaling on area & volume
When every linear dimension of a figure is multiplied by a scale factor k, area and volume don't scale by that same factor — they scale by a power of it.
A cube has a volume of 27. If every side length is tripled, what is the new volume?
Doubling every dimension of a solid does not double its volume — it multiplies the volume by 2³ = 8. This is one of the most frequently missed relationships on the SAT's geometry section.
Area & volume word problems
Many SAT area and volume questions are wrapped in a real-world context — paint needed to cover a wall, water filling a tank, material needed to build a container. Identify the correct shape and formula before doing any arithmetic.
A cylindrical water tank has a radius of 4 feet and a height of 12 feet. Approximately how many cubic feet of water can it hold? (Use π ≈ 3.14)
Common SAT traps
- Dropping the ⅓ in cone and pyramid formulas: always check whether the solid is a cone/pyramid before using the prism or cylinder formula.
- Confusing radius and diameter: area and volume formulas use radius — halve a given diameter before substituting.
- Linear scaling assumption: area scales by k² and volume scales by k³, never by k alone.
- Mismatched units: convert all measurements to the same unit before calculating area or volume, especially in word problems mixing feet and inches.
Test day strategy for area & volume
| Question signal | Fastest approach |
|---|---|
| Basic shape given directly | Apply the matching formula; double check radius vs. diameter |
| Cone, pyramid, or sphere involved | Watch for the ⅓ or &frac43; factor in the formula |
| Irregular or combined shape | Break into simple shapes; add or subtract their areas/volumes |
| "If every dimension is scaled by ___" | Area scales by k², volume scales by k³ |
| Real-world container or space problem | Identify the shape first, then match it to the correct formula |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Area & Volume 1
Area formulas and composite figures.
Start quiz → Quiz 2Area & Volume 2
Volume formulas, scaling, and word problems.
Start quiz →