SAT Math: Area and volume
Area and volume (the reference sheet gives the formulas).
- Geometry and Trigonometry · Area and volume
- Both modules
- 14 practice questions
- Calculator allowed (Desmos)
What the test covers
- Composite areas; volume scaling by k^3
- Reference-sheet formulas
Key ideas
- Cylinder \(\pi r^2h\), cone \(\tfrac13\pi r^2h\), sphere \(\tfrac43\pi r^3\), prism = base area \(\times\) height.
- Scaling every length by \(k\) multiplies area by \(k^2\) and volume by \(k^3\).
- Composite shapes: add or subtract simple pieces.
Common mistakes
- Use the radius, not the diameter.
- Check whether the answer is wanted in terms of \(\pi\).
Do it in Desmos
Use Desmos only for the arithmetic; keep \(\pi\) symbolic if the options do.
Worked example
Worked example
A right circular cylinder has radius 3 and height 8. What is its volume?
- \(144\pi\)
- \(72\pi\)
- \(48\pi\)
- \(24\pi\)
Answer: B: \(72\pi\)
\(V=\pi r^2h=\pi(9)(8)=72\pi\).
Practice questions
Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.
Question 1
A cube has volume 343 cubic centimeters. What is its total surface area, in square centimeters?
Show the answer and solution
Answer: 294
The edge is \(\sqrt[3]{343}=7\). Surface area \(6\times7^2=294\).
Question 2
Each of the length, width and height of a rectangular box is multiplied by 3. By what factor is the volume multiplied?
- \(9\)
- \(27\)
- \(3\)
- \(6\)
Show the answer and solution
Answer: B: \(27\)
Volume is a product of three lengths, so it is multiplied by \(3\times3\times3=27\).
Question 3
A sector of a circle of radius 6 has a central angle of \(60^\circ\). What is the area of the sector?
- \(6\pi\)
- \(12\pi\)
- \(36\pi\)
- \(2\pi\)
Show the answer and solution
Answer: A: \(6\pi\)
\(\tfrac{60}{360}\times\pi\times6^2=6\pi\).
Question 4
A right circular cone has base radius 5 and height 9. What is its volume?
- \(225 \pi\)
- \(135 \pi\)
- \(\frac{225 \pi}{2}\)
- \(75 \pi\)
Show the answer and solution
Answer: D: \(75 \pi\)
\(V=\tfrac13\pi r^2h\) (on the reference sheet) gives \(75 \pi\).
Question 5
A sphere has radius 5. What is its volume?
- \(250 \pi\)
- \(\frac{100 \pi}{3}\)
- \(\frac{500 \pi}{3}\)
- \(100 \pi\)
Show the answer and solution
Answer: C: \(\frac{500 \pi}{3}\)
\(V=\tfrac43\pi r^3\) (on the reference sheet) gives \(\frac{500 \pi}{3}\).
Question 6
A right circular cone has base radius 6 and height 12. What is its volume?
- \(216 \pi\)
- \(432 \pi\)
- \(144 \pi\)
- \(288 \pi\)
Show the answer and solution
Answer: C: \(144 \pi\)
\(V=\tfrac13\pi r^2h\) (on the reference sheet) gives \(144 \pi\).
Question 7
A right circular cone has base radius 3 and height 12. What is its volume?
- \(36 \pi\)
- \(108 \pi\)
- \(144 \pi\)
- \(54 \pi\)
Show the answer and solution
Answer: A: \(36 \pi\)
\(V=\tfrac13\pi r^2h\) (on the reference sheet) gives \(36 \pi\).
Question 8
A right circular cylinder has radius 3 and height 12. What is its volume?
- \(72 \pi\)
- \(36 \pi\)
- \(108 \pi\)
- \(432 \pi\)
Show the answer and solution
Answer: C: \(108 \pi\)
\(V=\pi r^2h\) (on the reference sheet) gives \(108 \pi\).
Question 9
A solid has volume 12 cubic centimeters. Every length of the solid is multiplied by 3. What is the volume of the new solid, in cubic centimeters?
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Answer: 324
Volume scales by the cube of the length factor: \(12\times3^3=324\).
Question 10
A solid has volume 12 cubic centimeters. Every length of the solid is multiplied by 4. What is the volume of the new solid, in cubic centimeters?
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Answer: 768
Volume scales by the cube of the length factor: \(12\times4^3=768\).
Question 11
A solid has volume 20 cubic centimeters. Every length of the solid is multiplied by 5. What is the volume of the new solid, in cubic centimeters?
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Answer: 2500
Volume scales by the cube of the length factor: \(20\times5^3=2500\).
Question 12
A solid has volume 8 cubic centimeters. Every length of the solid is multiplied by 3. What is the volume of the new solid, in cubic centimeters?
Show the answer and solution
Answer: 216
Volume scales by the cube of the length factor: \(8\times3^3=216\).
Question 13
A solid has volume 12 cubic centimeters. Every length of the solid is multiplied by 5. What is the volume of the new solid, in cubic centimeters?
Show the answer and solution
Answer: 1500
Volume scales by the cube of the length factor: \(12\times5^3=1500\).
Keep going
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