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Area, Perimeter & Volume for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about area, perimeter, and volume in one place: perimeter formulas, area formulas, volume formulas, composite figures, and the effect of scaling — with step-by-step examples, worked problems, and a free practice quiz.
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Free · No signupArea, Perimeter & Volume
Perimeter, area, and volume formulas, plus composite figures.
What's given vs. what to memorize
The PSAT provides a reference sheet with several area and volume formulas at the start of every Math section. Even so, the fastest students memorize the core formulas, since flipping back to the reference sheet on every question costs valuable time — and perimeter formulas aren't included on the reference sheet at all.
Perimeter questions rarely need a formula at all — just add up the side lengths. The real skill is figuring out any missing side lengths first, especially in irregular figures.
Perimeter formulas
| Shape | Formula |
|---|---|
| Rectangle | P = 2(length + width) |
| Square | P = 4 · side |
| Triangle | P = side₁ + side₂ + side₃ |
| Circle (circumference) | C = 2πr |
A figure is an L-shape with all right angles: outer dimensions 10 by 8, with a 4 by 3 rectangular notch cut from one corner. Find the perimeter.
For an L-shaped figure formed by cutting a rectangular notch from a corner, the perimeter equals the perimeter of the original full rectangle — the notch's cut edges add exactly as much length as they remove.
Area formulas
| Shape | Formula |
|---|---|
| Rectangle | A = length · width |
| Triangle | A = ½ · base · height |
| Trapezoid | A = ½ · (base₁ + base₂) · height |
| Circle | A = πr² |
A triangle has a base of 14 and a height of 9. Find its area.
Volume formulas
| Solid | Formula |
|---|---|
| Rectangular prism | V = length · width · height |
| Cylinder | V = πr²h |
| Cone | V = ⅓πr²h |
| Sphere | V = &frac43;πr³ |
A box measures 5 by 4 by 3. Find its volume.
The cone formula includes a factor of ⅓ — a cone is exactly one-third the volume of a cylinder with the same base and height, not the same volume.
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.
A rectangle measures 12 by 6. A semicircle with diameter 6 is attached to one side. Find the total area.
The effect of scaling
A cube has a volume of 8. If every side length is doubled, what is the new volume?
Doubling every side length of a figure does not double its area or volume — area scales by k² and volume scales by k³. This is one of the most frequently missed relationships in PSAT geometry.
Common PSAT traps
- Confusing radius and diameter: area and volume formulas use radius — halve a given diameter before substituting.
- Dropping the ⅓ in the cone formula: always check whether a solid is a cone before using the cylinder formula.
- Linear scaling assumption: area scales by k² and volume scales by k³, never by k alone.
- Miscounting sides in an irregular perimeter figure: trace the full outline carefully, including any notches or cutouts.
Test day strategy for area, perimeter & volume
| Question signal | Fastest approach |
|---|---|
| Basic shape given directly | Apply the matching formula; double check radius vs. diameter |
| Irregular figure, perimeter asked | Add all outer side lengths, filling in any missing ones using given dimensions |
| Cone, pyramid, or sphere involved | Watch for the ⅓ or &frac43; factor in the formula |
| Combined or irregular shape, area asked | Break into simple shapes; add or subtract their areas |
| "If every dimension is scaled by ___" | Area scales by k², volume scales by k³ |
Now put it to work
One comprehensive quiz covering the full topic — perimeter, area, and volume formulas, plus composite figures.
Area, Perimeter & Volume
Perimeter, area, and volume formulas, plus composite figures.