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?

   • It holds more, because its slanted side is longer.
   • Because you use the slanted side as its height.
   • Because leaning cans are always smaller.
   • At every height, both have the same circle slice.
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.
   • They can't be equal: squares and circles are different shapes.
   • The cone holds more because it is round.
   • 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?

   • Because the stack is a cube.
   • Because the slanted edge is now the height.
   • Each card keeps its area, and the stack is the same height.
   • Leaning adds space between the cards, so it holds more.
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.
   • They don't: turned squares make different slices.
   • Each turned plate still has the same area at its height.
   • Twisting packs them tighter, so it holds less.
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. 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.

8. 7.

   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
9. 8.

   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
10. 9.

    Esme's leaning tower roof leans: radius 3 cm, height 10 cm straight up, and its longest side is 12 cm. What is its volume in cubic cm, in terms of π (like 14π)?

    30π

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