Worksheet

Why the volume formulas work (Cavalieri's principle)

Grade 10 Math Worksheet

NameDate

Circle the best answer, or write your answer on the line.

  1. Kira leans a stack of coins into a slanted can shape. Why does it hold the same as a straight can with the same base and height?

    • It holds more, because its slanted side is longer.
    • Because leaning cans are always smaller.
    • At every height, both have the same circle slice.
    • Because you use the slanted side as its height.

  2. A pyramid and a cone have the same height and bases of the same area. Luz says their volumes are equal. Which reason is right?

    • The cone holds more because it is round.
    • The pyramid holds more because it has corners.
    • They can't be equal: squares and circles are different shapes.
    • At every height, their slices have equal areas.

  3. Milo pushes a box-shaped stack of cards so it leans. Why is its volume still the base area times the height?

    • Each card keeps its area, and the stack is the same height.
    • Leaning adds space between the cards, so it holds more.
    • Because the slanted edge is now the height.
    • Because the stack is a cube.

  4. Noah stacks 10 square paper plates straight, then twists another 10 so their corners turn. Why do the two stacks have the same volume?

    • Twisting packs them tighter, so it holds less.
    • Twisting adds corners, so it holds more.
    • They don't: turned squares make different slices.
    • Each turned plate still has the same area at its height.

  5. Pia's clay can has radius and height 6 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 2 cm up, in terms of π (like 14π)?

  6. Ren's clay can has radius and height 7 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 3 cm up, in terms of π (like 14π)?

  7. Tala's clay can has radius and height 9 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 5 cm up, in terms of π (like 14π)?

  8. Zee's clay can has radius and height 5 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 2 cm up, in terms of π (like 14π)?

  9. Read this first

    Sketch, not to scale

    The leaning cone with radius 3 cm, the height 11 cm drawn straight up from the base to the tip's level, and its longest side 12 cm.

  10. Ari's clay volcano leans: radius 3 cm, height 11 cm straight up, and its longest side is 12 cm. What is its volume in cubic cm, in terms of π (like 14π)?

  11. Use this

    Sketch, not to scale: the leaning cone with radius 9 cm, the height 4 cm drawn straight up from the base to the tip's level, and its longest side 15 cm.

  12. Jun's leaning sand volcano leans: radius 9 cm, height 4 cm straight up, and its longest side is 15 cm. What is its volume in cubic cm, in terms of π (like 14π)?

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Why the volume formulas work (Cavalieri's principle)

Answer Key - Grade 10 Math Worksheet

Circle the best answer, or write your answer on the line.

  1. Kira leans a stack of coins into a slanted can shape. Why does it hold the same as a straight can with the same base and height?

    • It holds more, because its slanted side is longer.
    • Because leaning cans are always smaller.
    • At every height, both have the same circle slice.
    • Because you use the slanted side as its height.

  2. A pyramid and a cone have the same height and bases of the same area. Luz says their volumes are equal. Which reason is right?

    • The cone holds more because it is round.
    • The pyramid holds more because it has corners.
    • They can't be equal: squares and circles are different shapes.
    • At every height, their slices have equal areas.

  3. Milo pushes a box-shaped stack of cards so it leans. Why is its volume still the base area times the height?

    • Each card keeps its area, and the stack is the same height.
    • Leaning adds space between the cards, so it holds more.
    • Because the slanted edge is now the height.
    • Because the stack is a cube.

  4. Noah stacks 10 square paper plates straight, then twists another 10 so their corners turn. Why do the two stacks have the same volume?

    • Twisting packs them tighter, so it holds less.
    • Twisting adds corners, so it holds more.
    • They don't: turned squares make different slices.
    • Each turned plate still has the same area at its height.

  5. Pia's clay can has radius and height 6 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 2 cm up, in terms of π (like 14π)?

    32π

  6. Ren's clay can has radius and height 7 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 3 cm up, in terms of π (like 14π)?

    40π

  7. Tala's clay can has radius and height 9 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 5 cm up, in terms of π (like 14π)?

    56π

  8. Zee's clay can has radius and height 5 cm, with a cone scooped out, tip at the bottom center. What is the area of its slice 2 cm up, in terms of π (like 14π)?

    21π

  9. Read this first

    Sketch, not to scale

    The leaning cone with radius 3 cm, the height 11 cm drawn straight up from the base to the tip's level, and its longest side 12 cm.

  10. Ari's clay volcano leans: radius 3 cm, height 11 cm straight up, and its longest side is 12 cm. What is its volume in cubic cm, in terms of π (like 14π)?

    33π

  11. Use this

    Sketch, not to scale: the leaning cone with radius 9 cm, the height 4 cm drawn straight up from the base to the tip's level, and its longest side 15 cm.

  12. Jun's leaning sand volcano leans: radius 9 cm, height 4 cm straight up, and its longest side is 15 cm. What is its volume in cubic cm, in terms of π (like 14π)?

    108π