Worksheet

Volume of combined solids

Grade 10 Math Worksheet

NameDate

Circle the best answer.

  1. Sketch, not to scale. A cone is 12 cm tall with radius 6 cm. Bo cuts off the top 4 cm parallel to the base and says the top cone is 1/27 of the whole. What licenses 1/27?

    • The top cone's base area is 1/9 of the whole, so its volume is also 1/9 of it
    • It is similar with scale 1/3, so volume scales by (1/3)³
    • The top looks small on the page, about one tenth of the whole cone by eye
    • The top cone is similar with scale 1/3, so its volume is 1/3 of the whole cone

  2. A cone and a square pyramid both have base area 30 cm² and height 12 cm. Esme wants to show equal volumes without using volume formulas. Which argument works?

    • Both have base area 30, and the base area alone decides the volume of a solid
    • Both come to a point at the top, so their volumes must be equal by that shape
    • Equal slice areas at every height, so Cavalieri gives equal volumes
    • Both have the same slant height, so their volumes are equal because slant decides

  3. A rock is lowered into a cylinder 10 cm across. The water level rises 2 cm. Sketch, not to scale. A rock sinks into a cylinder 10 cm across and the water rises 2 cm. Jun says the rock's volume is the base area times the rise. What licenses that?

    • The rock's volume equals one third of the base area times the rise, as for a cone
    • It pushes aside equal water: the tank's base times the rise
    • The rock's volume equals the base area times the new depth of the water
    • The rock's volume equals the area of a circle 10 cm in radius times the rise

  4. Kira says a bucket with radii 6 and 3 cm and height 8 cm has the same volume as a can of average radius 4.5 cm and height 8 cm. Which reply is right?

    • Yes: both have straight sides and the same height, so they hold the same
    • Yes: the halfway radius gives the halfway volume of the two cones
    • No: the bucket is 168π and the can is 162π, since volume follows r²
    • No: the can is 192π, the volume of the big cone alone, and the bucket is less

  5. Milo says two solids with equal height and equal base area always have equal volume. Which reply is right?

    • Yes: each is one third of base times height, so they must agree anyway
    • No: they match only if their slant heights match as well as the base
    • No: Cavalieri needs equal slices at every height; cylinder and cone differ 3 to 1
    • Yes: the same base and the same height make the same volume for every solid

  6. Noah says a ball of radius 3 cm is exactly two thirds of the snug cylinder around it (radius 3, height 6). Which reply is right?

    • No: the ball is π/4 of the cylinder, like a circle in a square, so under 80%
    • Yes: the ball is half the cylinder, which looks right
    • Yes: the ball is 36π, the cylinder is 54π, and 36π/54π = 2/3
    • No: the ball is one third of the cylinder, like a cone in its can, so less

  7. Pia finds a pyramid's volume (base 6 cm square, slant height 5 cm): Line 1: height = 5 − 3 = 2. Line 2: volume = (1/3)(36)(2) = 24. Which line is wrong?

    • Line 1: the height is √(5² − 3²) = 4 by Pythagoras, not 5 − 3
    • Neither line is wrong; just the units are missing from the final answer
    • Line 2: the pyramid formula has no one third, so the volume is 72
    • Line 2: the base area should be 6 × 3 = 18, using half of one side

  8. Ren finds a silo's volume (cylinder radius 3 m, height 10 m, half-ball top): Line 1: cylinder 90π. Line 2: half-ball (4/3)π·27 = 36π. Line 3: total 126π. Which line is wrong?

    • Line 2: a half-ball is half a ball, (2/3)π·27 = 18π
    • Line 3: add another 36π for a second half-ball at the bottom of the silo
    • No line is wrong; just the units are missing from the total
    • Line 1: the cylinder should use radius 6 because the silo is 6 m across

  9. Tala finds a rock's volume (cylinder 10 cm across, water from 10 cm to 12 cm): Line 1: rise = 2 cm. Line 2: volume = π·5²·12. Which line is wrong?

    • Line 2: the volume uses the rise, 2 cm, not the depth, 12 cm
    • Line 2: use π·5·2, with the radius not squared, since the rock is solid
    • Line 2: the radius should be 10 cm, because the cylinder is 10 cm across
    • Line 1: the rise should be 12 cm, the depth of the water after the rock

  10. Zee finds a cone's height (radius 5 cm, volume 100π cm³): Line 1: (1/3)π·25·h = 100π. Line 2: 25h = 100. Line 3: h = 4. Which line is wrong?

    • Line 1: the cone formula has no one third, so the equation is wrong from the start
    • Line 2: times 3 gives 25h = 300, so h = 300 ÷ 25 = 12
    • Line 1: the radius should be 10 cm, as the cone base is wide
    • Line 3: divide 25h = 100 by 25π, because the π was not removed
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Volume of combined solids

Answer Key - Grade 10 Math Worksheet

Circle the best answer.

  1. Sketch, not to scale. A cone is 12 cm tall with radius 6 cm. Bo cuts off the top 4 cm parallel to the base and says the top cone is 1/27 of the whole. What licenses 1/27?

    • The top cone's base area is 1/9 of the whole, so its volume is also 1/9 of it
    • It is similar with scale 1/3, so volume scales by (1/3)³
    • The top looks small on the page, about one tenth of the whole cone by eye
    • The top cone is similar with scale 1/3, so its volume is 1/3 of the whole cone

  2. A cone and a square pyramid both have base area 30 cm² and height 12 cm. Esme wants to show equal volumes without using volume formulas. Which argument works?

    • Both have base area 30, and the base area alone decides the volume of a solid
    • Both come to a point at the top, so their volumes must be equal by that shape
    • Equal slice areas at every height, so Cavalieri gives equal volumes
    • Both have the same slant height, so their volumes are equal because slant decides

  3. A rock is lowered into a cylinder 10 cm across. The water level rises 2 cm. Sketch, not to scale. A rock sinks into a cylinder 10 cm across and the water rises 2 cm. Jun says the rock's volume is the base area times the rise. What licenses that?

    • The rock's volume equals one third of the base area times the rise, as for a cone
    • It pushes aside equal water: the tank's base times the rise
    • The rock's volume equals the base area times the new depth of the water
    • The rock's volume equals the area of a circle 10 cm in radius times the rise

  4. Kira says a bucket with radii 6 and 3 cm and height 8 cm has the same volume as a can of average radius 4.5 cm and height 8 cm. Which reply is right?

    • Yes: both have straight sides and the same height, so they hold the same
    • Yes: the halfway radius gives the halfway volume of the two cones
    • No: the bucket is 168π and the can is 162π, since volume follows r²
    • No: the can is 192π, the volume of the big cone alone, and the bucket is less

  5. Milo says two solids with equal height and equal base area always have equal volume. Which reply is right?

    • Yes: each is one third of base times height, so they must agree anyway
    • No: they match only if their slant heights match as well as the base
    • No: Cavalieri needs equal slices at every height; cylinder and cone differ 3 to 1
    • Yes: the same base and the same height make the same volume for every solid

  6. Noah says a ball of radius 3 cm is exactly two thirds of the snug cylinder around it (radius 3, height 6). Which reply is right?

    • No: the ball is π/4 of the cylinder, like a circle in a square, so under 80%
    • Yes: the ball is half the cylinder, which looks right
    • Yes: the ball is 36π, the cylinder is 54π, and 36π/54π = 2/3
    • No: the ball is one third of the cylinder, like a cone in its can, so less

  7. Pia finds a pyramid's volume (base 6 cm square, slant height 5 cm): Line 1: height = 5 − 3 = 2. Line 2: volume = (1/3)(36)(2) = 24. Which line is wrong?

    • Line 1: the height is √(5² − 3²) = 4 by Pythagoras, not 5 − 3
    • Neither line is wrong; just the units are missing from the final answer
    • Line 2: the pyramid formula has no one third, so the volume is 72
    • Line 2: the base area should be 6 × 3 = 18, using half of one side

  8. Ren finds a silo's volume (cylinder radius 3 m, height 10 m, half-ball top): Line 1: cylinder 90π. Line 2: half-ball (4/3)π·27 = 36π. Line 3: total 126π. Which line is wrong?

    • Line 2: a half-ball is half a ball, (2/3)π·27 = 18π
    • Line 3: add another 36π for a second half-ball at the bottom of the silo
    • No line is wrong; just the units are missing from the total
    • Line 1: the cylinder should use radius 6 because the silo is 6 m across

  9. Tala finds a rock's volume (cylinder 10 cm across, water from 10 cm to 12 cm): Line 1: rise = 2 cm. Line 2: volume = π·5²·12. Which line is wrong?

    • Line 2: the volume uses the rise, 2 cm, not the depth, 12 cm
    • Line 2: use π·5·2, with the radius not squared, since the rock is solid
    • Line 2: the radius should be 10 cm, because the cylinder is 10 cm across
    • Line 1: the rise should be 12 cm, the depth of the water after the rock

  10. Zee finds a cone's height (radius 5 cm, volume 100π cm³): Line 1: (1/3)π·25·h = 100π. Line 2: 25h = 100. Line 3: h = 4. Which line is wrong?

    • Line 1: the cone formula has no one third, so the equation is wrong from the start
    • Line 2: times 3 gives 25h = 300, so h = 300 ÷ 25 = 12
    • Line 1: the radius should be 10 cm, as the cone base is wide
    • Line 3: divide 25h = 100 by 25π, because the π was not removed