Quick Answer
SAT volume and surface area practice questions test rectangular prisms, cubes, cylinders, cones, pyramids, spheres, hemispheres, composite solids, missing dimensions, real-world containers, and scale factors. This guide includes 70 SAT-style questions that progress from basic formulas to surface-area models, composite solids, algebraic dimensions, and harder mixed geometry problems.
What Should You Know Before Practicing Volume and Surface Area?
- Rectangular prism volume is lwh.
- Prism and cylinder volume is Bh.
- Pyramid and cone volume is (1/3)Bh.
- Sphere volume is (4/3)πr³.
- Sphere surface area is 4πr².
- Cylinder total surface area is 2πr²+2πrh.
- Surface area uses square units, while volume uses cubic units.
- Similar-solid volume scales by k³ and surface area by k².
In This Guide – 70 SAT Volume and Surface Area Practice Questions
Start With SAT Math Topic-Wise Practice
Connect solid-geometry practice with weekly score planning, mock-test review, and targeted SAT Math correction.
Download the SAT Prep Guide E-Book
Organize solid-geometry formulas, volume models, surface-area practice, and weekly SAT Math preparation with a clear study plan.
Download SAT Prep E-Book GuideWhat Does the SAT Test in Volume and Surface Area Questions?
Students may need to apply a formula, convert diameter to radius, find a missing dimension, combine or subtract solids, interpret a real-world container, or analyze how scaling affects area and volume.
| Skill Area | What It Tests | Questions |
|---|---|---|
| Volume basics | Prisms, cubes, cylinders, cones, pyramids, spheres | Q1-Q15 |
| Surface area | Cubes, prisms, cylinders, cones, spheres, hemispheres | Q16-Q30 |
| Composite solids | Missing dimensions, joined solids, cutouts, partial fills | Q31-Q45 |
| Advanced applications | Scale factors, real-world models, algebraic dimensions | Q46-Q70 |
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How Are Volume Formula Questions Tested?
These questions focus on rectangular prisms, cubes, cylinders, cones, pyramids, spheres, and base-area models.
VOLUME FORMULA BASICS
EASY
Question 1
What is the volume of a rectangular prism with length 8, width 5, and height 3?
Which choice is correct?
A) 16
B) 40
C) 120
D) 240
Show full solution
Correct answer: C) 120
Volume=lwh=8×5×3=120 cubic units.
SAT trap: Volume uses three dimensions.
VOLUME FORMULA BASICS
EASY
Question 2
A rectangular prism has dimensions 12 by 4 by 6. What is its volume?
Which choice is correct?
A) 22
B) 72
C) 144
D) 288
Show full solution
Correct answer: D) 288
Volume=12×4×6=288 cubic units.
SAT trap: Do not add the dimensions.
VOLUME FORMULA BASICS
EASY
Question 3
What is the volume of a cube with side length 7?
Which choice is correct?
A) 49
B) 98
C) 294
D) 343
Show full solution
Correct answer: D) 343
Volume=s³=7³=343.
SAT trap: A cube volume uses the third power.
VOLUME FORMULA BASICS
EASY
Question 4
A cube has volume 216. What is its side length?
Which choice is correct?
A) 4
B) 6
C) 8
D) 12
Show full solution
Correct answer: B) 6
Since s³=216, s=6.
SAT trap: Take the cube root, not the square root.
VOLUME FORMULA BASICS
EASY
Question 5
What is the volume of a cylinder with radius 3 and height 10?
Which choice is correct?
A) 30π
B) 60π
C) 90π
D) 180π
Show full solution
Correct answer: C) 90π
Volume=πr²h=π(3²)(10)=90π.
SAT trap: Square the radius.
VOLUME FORMULA BASICS
EASY
Question 6
A cylinder has diameter 8 and height 5. What is its volume?
Which choice is correct?
A) 40π
B) 80π
C) 160π
D) 320π
Show full solution
Correct answer: B) 80π
The radius is 4, so V=π(4²)(5)=80π.
SAT trap: Convert diameter to radius first.
VOLUME FORMULA BASICS
EASY
Question 7
What is the volume of a cone with radius 6 and height 9?
Which choice is correct?
A) 36π
B) 54π
C) 108π
D) 324π
Show full solution
Correct answer: C) 108π
V=(1/3)πr²h=(1/3)π(36)(9)=108π.
SAT trap: Include the factor of one-third.
VOLUME FORMULA BASICS
EASY
Question 8
A cone has radius 4 and height 12. What is its volume?
Which choice is correct?
A) 16π
B) 32π
C) 64π
D) 192π
Show full solution
Correct answer: C) 64π
V=(1/3)π(4²)(12)=64π.
SAT trap: A cone is one-third of a matching cylinder.
VOLUME FORMULA BASICS
EASY
Question 9
What is the volume of a sphere with radius 3?
Which choice is correct?
A) 9π
B) 18π
C) 27π
D) 36π
Show full solution
Correct answer: D) 36π
V=(4/3)πr³=(4/3)π(27)=36π.
SAT trap: Cube the radius.
VOLUME FORMULA BASICS
EASY
Question 10
What is the volume of a sphere with diameter 10?
Which choice is correct?
A) 100π/3
B) 250π/3
C) 500π/3
D) 1000π/3
Show full solution
Correct answer: C) 500π/3
The radius is 5, so V=(4/3)π(125)=500π/3.
SAT trap: Use half the diameter.
VOLUME FORMULA BASICS
MEDIUM
Question 11
What is the volume of a square pyramid with base side 6 and height 10?
Which choice is correct?
A) 60
B) 120
C) 180
D) 360
Show full solution
Correct answer: B) 120
Base area=36, so V=(1/3)(36)(10)=120.
SAT trap: Use base area, not base perimeter.
VOLUME FORMULA BASICS
MEDIUM
Question 12
A rectangular pyramid has base area 54 and height 8. What is its volume?
Which choice is correct?
A) 72
B) 108
C) 144
D) 432
Show full solution
Correct answer: C) 144
V=(1/3)Bh=(1/3)(54)(8)=144.
SAT trap: Pyramid volume is one-third of Bh.
VOLUME FORMULA BASICS
MEDIUM
Question 13
A cylinder has base area 25π and height 7. What is its volume?
Which choice is correct?
A) 25π
B) 75π
C) 175π
D) 350π
Show full solution
Correct answer: C) 175π
Volume=base area×height=25π×7=175π.
SAT trap: The base area already includes πr².
VOLUME FORMULA BASICS
MEDIUM
Question 14
A prism has base area 18 and height 11. What is its volume?
Which choice is correct?
A) 29
B) 99
C) 198
D) 396
Show full solution
Correct answer: C) 198
V=Bh=18×11=198.
SAT trap: Prism volume is base area times height.
VOLUME FORMULA BASICS
MEDIUM
Question 15
A cone and a cylinder have the same base and height. How does the cone volume compare with the cylinder volume?
Which choice is correct?
A) It is equal
B) It is one-half
C) It is one-third
D) It is three times
Show full solution
Correct answer: C) It is one-third
A cone has volume (1/3)Bh, while a cylinder has volume Bh.
SAT trap: Match the base and height before comparing.
Build Stronger SAT Solid Geometry Skills
Connect volume formulas with algebra, unit analysis, and real SAT Math timing.
How Are Surface Area Questions Tested?
These questions use face areas, lateral area, total area, slant height, and curved surfaces.
SURFACE AREA FORMULAS
MEDIUM
Question 16
What is the total surface area of a cube with side length 4?
Which choice is correct?
A) 16
B) 64
C) 96
D) 128
Show full solution
Correct answer: C) 96
A cube has 6 square faces, so SA=6(4²)=96.
SAT trap: Use all six faces.
SURFACE AREA FORMULAS
MEDIUM
Question 17
A cube has side length 9. What is its total surface area?
Which choice is correct?
A) 81
B) 243
C) 486
D) 729
Show full solution
Correct answer: C) 486
SA=6s²=6(81)=486.
SAT trap: Surface area uses square units.
SURFACE AREA FORMULAS
MEDIUM
Question 18
What is the total surface area of a rectangular prism with dimensions 2, 3, and 5?
Which choice is correct?
A) 30
B) 31
C) 62
D) 90
Show full solution
Correct answer: C) 62
SA=2(lw+lh+wh)=2(6+10+15)=62.
SAT trap: Count each pair of opposite faces twice.
SURFACE AREA FORMULAS
MEDIUM
Question 19
A rectangular prism has length 10, width 4, and height 3. What is its total surface area?
Which choice is correct?
A) 82
B) 124
C) 164
D) 240
Show full solution
Correct answer: C) 164
SA=2(10×4+10×3+4×3)=2(82)=164.
SAT trap: Add the three face areas before doubling.
SURFACE AREA FORMULAS
MEDIUM
Question 20
What is the lateral surface area of a cylinder with radius 5 and height 8?
Which choice is correct?
A) 40π
B) 80π
C) 100π
D) 160π
Show full solution
Correct answer: B) 80π
Lateral area=2πrh=2π(5)(8)=80π.
SAT trap: Lateral area excludes the circular bases.
SURFACE AREA FORMULAS
MEDIUM
Question 21
What is the total surface area of a cylinder with radius 3 and height 7?
Which choice is correct?
A) 42π
B) 51π
C) 60π
D) 72π
Show full solution
Correct answer: C) 60π
SA=2πr²+2πrh=18π+42π=60π.
SAT trap: Include both circular bases.
SURFACE AREA FORMULAS
MEDIUM
Question 22
A cylinder has diameter 10 and height 4. What is its total surface area?
Which choice is correct?
A) 40π
B) 65π
C) 90π
D) 100π
Show full solution
Correct answer: C) 90π
Radius=5. SA=2π(25)+2π(5)(4)=50π+40π=90π.
SAT trap: Use radius 5, not diameter 10.
SURFACE AREA FORMULAS
MEDIUM
Question 23
What is the surface area of a sphere with radius 6?
Which choice is correct?
A) 36π
B) 72π
C) 144π
D) 288π
Show full solution
Correct answer: C) 144π
SA=4πr²=4π(36)=144π.
SAT trap: Sphere surface area uses r².
SURFACE AREA FORMULAS
MEDIUM
Question 24
A sphere has diameter 14. What is its surface area?
Which choice is correct?
A) 49π
B) 98π
C) 196π
D) 392π
Show full solution
Correct answer: C) 196π
Radius=7, so SA=4π(49)=196π.
SAT trap: Halve the diameter first.
SURFACE AREA FORMULAS
MEDIUM
Question 25
A cone has radius 5 and slant height 13. What is its lateral surface area?
Which choice is correct?
A) 25π
B) 50π
C) 65π
D) 130π
Show full solution
Correct answer: C) 65π
Lateral area=πrl=π(5)(13)=65π.
SAT trap: Use slant height, not vertical height.
SURFACE AREA FORMULAS
HARD
Question 26
A cone has radius 3 and slant height 5. What is its total surface area?
Which choice is correct?
A) 15π
B) 24π
C) 30π
D) 45π
Show full solution
Correct answer: B) 24π
Total area=πr²+πrl=9π+15π=24π.
SAT trap: Include the circular base.
SURFACE AREA FORMULAS
HARD
Question 27
A square pyramid has base side 8 and slant height 6. What is its lateral surface area?
Which choice is correct?
A) 48
B) 72
C) 96
D) 192
Show full solution
Correct answer: C) 96
Lateral area=(1/2)(base perimeter)(slant height)=(1/2)(32)(6)=96.
SAT trap: Use the base perimeter.
SURFACE AREA FORMULAS
HARD
Question 28
A square pyramid has base side 8 and slant height 6. What is its total surface area?
Which choice is correct?
A) 96
B) 128
C) 160
D) 192
Show full solution
Correct answer: C) 160
Add base area 64 to lateral area 96: total=160.
SAT trap: Total area includes the base.
SURFACE AREA FORMULAS
HARD
Question 29
A hemisphere has radius 4. What is its curved surface area?
Which choice is correct?
A) 16π
B) 32π
C) 48π
D) 64π
Show full solution
Correct answer: B) 32π
Curved hemisphere area=2πr²=2π(16)=32π.
SAT trap: Curved area excludes the circular base.
SURFACE AREA FORMULAS
HARD
Question 30
A hemisphere has radius 4. What is its total surface area, including the base?
Which choice is correct?
A) 32π
B) 40π
C) 48π
D) 64π
Show full solution
Correct answer: C) 48π
Total hemisphere area=3πr²=3π(16)=48π.
SAT trap: Add the circular base to the curved area.
How Are Missing Dimensions and Composite Solids Tested?
These questions combine algebra, hidden faces, cutouts, joined solids, and partial fills.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 31
A rectangular prism has volume 240, length 10, and width 6. What is its height?
Which choice is correct?
A) 2
B) 4
C) 6
D) 8
Show full solution
Correct answer: B) 4
240=10×6×h, so h=4.
SAT trap: Divide by both known dimensions.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 32
A cylinder has volume 144π and height 9. What is its radius?
Which choice is correct?
A) 2
B) 3
C) 4
D) 6
Show full solution
Correct answer: C) 4
144π=πr²(9), so r²=16 and r=4.
SAT trap: Take the positive square root.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 33
A cone has volume 48π and radius 4. What is its height?
Which choice is correct?
A) 3
B) 6
C) 9
D) 12
Show full solution
Correct answer: C) 9
48π=(1/3)π(16)h, so 144=16h and h=9.
SAT trap: Multiply by 3 before dividing by r².
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 34
A sphere has volume 288π. What is its radius?
Which choice is correct?
A) 4
B) 5
C) 6
D) 8
Show full solution
Correct answer: C) 6
288π=(4/3)πr³, so r³=216 and r=6.
SAT trap: Solve for r³ before taking the cube root.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 35
A cube has surface area 294. What is its side length?
Which choice is correct?
A) 5
B) 6
C) 7
D) 9
Show full solution
Correct answer: C) 7
6s²=294, so s²=49 and s=7.
SAT trap: Divide by 6 before taking the square root.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 36
A cylinder has total surface area 96π and radius 4. What is its height?
Which choice is correct?
A) 4
B) 6
C) 8
D) 10
Show full solution
Correct answer: C) 8
96π=2π(16)+2π(4)h=32π+8πh, so h=8.
SAT trap: Subtract the base areas first.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 37
A sphere and a cube both have a measure of 6: the sphere has radius 6 and the cube has side 6. Which has the greater volume?
Which choice is correct?
A) The sphere
B) The cube
C) They are equal
D) Cannot be determined
Show full solution
Correct answer: A) The sphere
Sphere volume=288π≈904.8, while cube volume=216.
SAT trap: Compare exact or approximate volumes.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 38
A solid consists of two identical cubes with side length 3 joined face to face. What is the volume?
Which choice is correct?
A) 27
B) 36
C) 54
D) 81
Show full solution
Correct answer: C) 54
Each cube has volume 27, so total volume=54.
SAT trap: Joining solids does not change total volume.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 39
Two cubes with side length 3 are joined face to face. What is the total exterior surface area?
Which choice is correct?
A) 72
B) 81
C) 90
D) 108
Show full solution
Correct answer: C) 90
Two cubes have 108 total face area before joining. The two hidden faces total 18, so exterior area=90.
SAT trap: Subtract both joined faces.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 40
A solid is made from a cylinder of radius 3 and height 8 topped by a hemisphere of radius 3. What is its volume?
Which choice is correct?
A) 72π
B) 81π
C) 90π
D) 99π
Show full solution
Correct answer: C) 90π
Cylinder volume=72π. Hemisphere volume=(2/3)π(27)=18π. Total=90π.
SAT trap: A hemisphere is half a sphere.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 41
A cube with side length 10 has a smaller cube with side length 4 removed. What volume remains?
Which choice is correct?
A) 936
B) 960
C) 984
D) 996
Show full solution
Correct answer: A) 936
Large cube volume=1000; small cube volume=64; remaining=936.
SAT trap: Subtract volumes, not side lengths.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 42
A rectangular tank measures 12 by 5 by 8 and is filled to half its height. What volume of water does it contain?
Which choice is correct?
A) 120
B) 240
C) 360
D) 480
Show full solution
Correct answer: B) 240
Half the height is 4, so water volume=12×5×4=240.
SAT trap: Use the filled height, not the full height.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 43
A cylinder has radius 4 and height 10. A cone with the same base and height is removed. What volume remains?
Which choice is correct?
A) 160π/3
B) 320π/3
C) 160π
D) 320π
Show full solution
Correct answer: B) 320π/3
Cylinder volume=160π; cone volume=160π/3; difference=320π/3.
SAT trap: A matching cone is one-third of the cylinder.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 44
A sphere of radius 5 is cut into two hemispheres. What is the combined volume of the two hemispheres?
Which choice is correct?
A) 250π/3
B) 500π/3
C) 250π
D) 500π
Show full solution
Correct answer: B) 500π/3
The two hemispheres together equal the original sphere: (4/3)π(125)=500π/3.
SAT trap: Cutting does not change total volume.
MISSING DIMENSIONS AND COMPOSITE SOLIDS
HARD
Question 45
A rectangular prism has surface area 166, length 7, and width 4. What is its height?
Which choice is correct?
A) 3
B) 4
C) 5
D) 6
Show full solution
Correct answer: C) 5
166=2(28+7h+4h)=56+22h, so 110=22h and h=5.
SAT trap: Use all three face-pair areas.
Practice More SAT Geometry and Measurement
Build accuracy with surface area, composite solids, and missing-dimension models.
How Are Volume and Surface Area Used in Real-World Models?
These questions use tanks, boxes, cans, fill percentages, and similarity scale factors.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 46
A shipping box measures 18 inches by 12 inches by 10 inches. What is its volume?
Which choice is correct?
A) 400 in³
B) 1,080 in³
C) 2,160 in³
D) 4,320 in³
Show full solution
Correct answer: C) 2,160 in³
Volume=18×12×10=2,160 cubic inches.
SAT trap: Volume units are cubic.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 47
A cylindrical can has radius 2.5 inches and height 6 inches. What is its volume?
Which choice is correct?
A) 15π in³
B) 25π in³
C) 37.5π in³
D) 75π in³
Show full solution
Correct answer: C) 37.5π in³
V=π(2.5²)(6)=37.5π.
SAT trap: Square 2.5 carefully.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 48
A water tank has a rectangular base 8 feet by 6 feet and height 5 feet. It is 75% full. How much water does it contain?
Which choice is correct?
A) 120 ft³
B) 160 ft³
C) 180 ft³
D) 240 ft³
Show full solution
Correct answer: C) 180 ft³
Full volume=8×6×5=240. Then 75% of 240 is 180.
SAT trap: Apply the fill percentage to the full volume.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 49
A spherical ball has radius 6 centimeters. Which expression gives its volume?
Which choice is correct?
A) 36π
B) 144π
C) 288π
D) 864π
Show full solution
Correct answer: C) 288π
V=(4/3)π(6³)=288π.
SAT trap: Use r³ in the sphere volume formula.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 50
A cylindrical tank has diameter 14 meters and height 10 meters. What is its volume?
Which choice is correct?
A) 490π m³
B) 980π m³
C) 1,400π m³
D) 1,960π m³
Show full solution
Correct answer: A) 490π m³
Radius=7, so V=π(49)(10)=490π.
SAT trap: Use half the diameter.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 51
All dimensions of a rectangular prism are doubled. By what factor does its volume change?
Which choice is correct?
A) 2
B) 4
C) 6
D) 8
Show full solution
Correct answer: D) 8
Volume scales by 2³=8.
SAT trap: Volume uses a cubic scale factor.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 52
All dimensions of a solid are multiplied by 3. By what factor does its surface area change?
Which choice is correct?
A) 3
B) 6
C) 9
D) 27
Show full solution
Correct answer: C) 9
Surface area scales by 3²=9.
SAT trap: Surface area uses the square of the scale factor.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 53
A sphere’s radius is doubled. By what factor does its volume change?
Which choice is correct?
A) 2
B) 4
C) 6
D) 8
Show full solution
Correct answer: D) 8
Sphere volume is proportional to r³, so doubling radius multiplies volume by 8.
SAT trap: Cube the radius scale factor.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 54
A cylinder’s radius is doubled while its height stays the same. By what factor does volume change?
Which choice is correct?
A) 2
B) 4
C) 6
D) 8
Show full solution
Correct answer: B) 4
Volume depends on r²h, so doubling r multiplies volume by 4.
SAT trap: Only the radius changes.
REAL-WORLD AND SCALE-FACTOR MODELS
HARD
Question 55
A cylinder’s radius and height are both doubled. By what factor does total volume change?
Which choice is correct?
A) 4
B) 6
C) 8
D) 16
Show full solution
Correct answer: C) 8
The factor is 2²×2=8.
SAT trap: Radius contributes a squared factor.
What Do Hard Mixed Volume and Surface Area Questions Look Like?
These questions combine algebraic dimensions, equal-volume conditions, surface-area-to-volume reasoning, and similar solids.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 56
A rectangular prism has dimensions x, x+3, and 4. Its volume is 160. What is x?
Which choice is correct?
A) 4
B) 5
C) 8
D) 10
Show full solution
Correct answer: B) 5
4x(x+3)=160, so x(x+3)=40. Then x²+3x-40=0=(x+8)(x-5), giving x=5.
SAT trap: Reject the negative solution.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 57
A cylinder has volume 200π and height 8. What is its diameter?
Which choice is correct?
A) 5
B) 10
C) 20
D) 25
Show full solution
Correct answer: B) 10
200π=πr²(8), so r²=25 and r=5. Diameter=10.
SAT trap: The question asks for diameter.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 58
A cone and a cylinder have equal volumes and equal radii. If the cylinder height is 4, what is the cone height?
Which choice is correct?
A) 4
B) 8
C) 12
D) 16
Show full solution
Correct answer: C) 12
Equal volumes give πr²(4)=(1/3)πr²h, so h=12.
SAT trap: The cone must be three times as tall.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 59
A sphere and a cylinder both have radius 3. The cylinder height is 4. How do their volumes compare?
Which choice is correct?
A) Sphere is larger
B) Cylinder is larger
C) They are equal
D) Cannot be determined
Show full solution
Correct answer: C) They are equal
Sphere volume=(4/3)π(27)=36π. Cylinder volume=π(9)(4)=36π.
SAT trap: Calculate both exactly.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 60
A cube has the same volume as a rectangular prism measuring 2 by 4 by 8. What is the cube side length?
Which choice is correct?
A) 2
B) 4
C) 6
D) 8
Show full solution
Correct answer: B) 4
The prism volume is 64, so cube side=∛64=4.
SAT trap: Use a cube root.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 61
A sphere has surface area 100π. What is its volume?
Which choice is correct?
A) 100π/3
B) 250π/3
C) 500π/3
D) 1,000π/3
Show full solution
Correct answer: C) 500π/3
4πr²=100π gives r=5. Then V=(4/3)π(125)=500π/3.
SAT trap: Find radius from surface area first.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 62
A cube has volume 512. What is its total surface area?
Which choice is correct?
A) 64
B) 128
C) 256
D) 384
Show full solution
Correct answer: D) 384
Side=∛512=8. Surface area=6(8²)=384.
SAT trap: Move from volume to side length first.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 63
A cone has radius 9, height 12, and slant height 15. What is its total surface area?
Which choice is correct?
A) 135π
B) 216π
C) 225π
D) 270π
Show full solution
Correct answer: B) 216π
Total area=πr²+πrl=81π+135π=216π.
SAT trap: Use slant height for the lateral area.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 64
A square pyramid has base side 10 and slant height 13. What is its total surface area?
Which choice is correct?
A) 130
B) 260
C) 360
D) 520
Show full solution
Correct answer: C) 360
Lateral area=(1/2)(40)(13)=260. Add base area 100 for total 360.
SAT trap: Include the base once.
HARD MIXED VOLUME AND SURFACE AREA
HARD
Question 65
A solid has scale factor 1/2 compared with a similar solid. What fraction of the original volume does it have?
Which choice is correct?
A) 1/2
B) 1/4
C) 1/6
D) 1/8
Show full solution
Correct answer: D) 1/8
Volume scales by (1/2)³=1/8.
SAT trap: Similar-solid volume uses the cube of the scale factor.
What Do Student-Produced Response Questions Look Like?
Enter the requested volume, side length, or coefficient of π.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 66
A rectangular prism measures 9 by 4 by 5. What is its volume?
Enter your answer.
Show full solution
Correct answer: 180
Volume=9×4×5=180.
SAT trap: Enter only the requested numerical value.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 67
A cube has volume 729. What is its side length?
Enter your answer.
Show full solution
Correct answer: 9
The cube root of 729 is 9.
SAT trap: Enter only the requested numerical value.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 68
A cylinder has radius 4 and height 6. What is the coefficient of π in its volume?
Enter your answer.
Show full solution
Correct answer: 96
V=π(4²)(6)=96π.
SAT trap: Enter only the requested numerical value.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 69
A sphere has radius 3. What is the coefficient of π in its surface area?
Enter your answer.
Show full solution
Correct answer: 36
SA=4π(3²)=36π.
SAT trap: Enter only the requested numerical value.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 70
A cone has radius 6 and height 12. What is the coefficient of π in its volume?
Enter your answer.
Show full solution
Correct answer: 144
V=(1/3)π(36)(12)=144π.
SAT trap: Enter only the requested numerical value.
Common SAT Volume and Surface Area Mistakes
| Mistake | Correction |
|---|---|
| Using diameter as radius | Divide the diameter by 2 first |
| Forgetting one-third for cones or pyramids | Use V=(1/3)Bh |
| Mixing lateral and total surface area | Check whether bases are included |
| Scaling volume by k² | Volume scales by k³ |
Two-Week SAT Volume and Surface Area Study Plan
| Days | Focus |
|---|---|
| 1-3 | Q1-Q15: volume formulas |
| 4-7 | Q16-Q30: surface area |
| 8-10 | Q31-Q45: missing dimensions and composite solids |
| 11-14 | Q46-Q70 and error review |
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