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. 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?

   • At every height, both have the same circle slice.
   • Because you use the slanted side as its height.
   • Because leaning cans are always smaller.
   • It holds more, because its slanted side is longer.
2. 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?

   • At every height, their slices have equal areas.
   • The cone holds more because it is round.
   • They can't be equal: squares and circles are different shapes.
   • The pyramid holds more because it has corners.
3. 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.
   • Because the slanted edge is now the height.
   • Leaning adds space between the cards, so it holds more.
   • Because the stack is a cube.
4. 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 adds corners, so it holds more.
   • Twisting packs them tighter, so it holds less.
   • They don't: turned squares make different slices.
   • Each turned plate still has the same area at its height.
5. 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. 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. 7.

   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

8. 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.

9. 8.

   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
10. 9.

    Bo's leaning ice cream cone leans: radius 6 cm, height 5 cm straight up, and its longest side is 8 cm. What is its volume in cubic cm, in terms of π (like 14π)?

    60π

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. 10.

    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