Why the volume formulas work (Cavalieri's principle)

Answer Key - Grade 10 Math Worksheet - Challenge Me

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

1. 1.

   Kira matches a half-ball of radius 3 cm, slice by slice, to a can (radius and height 3 cm) with a cone scooped out. What is the half-ball's volume, in terms of π (like 14π)?

   18π
2. 2.

   Luz matches a half-ball of radius 6 cm, slice by slice, to a can (radius and height 6 cm) with a cone scooped out. What is the half-ball's volume, in terms of π (like 14π)?

   144π
3. 3.

   Noah matches a half-ball of radius 12 cm, slice by slice, to a can (radius and height 12 cm) with a cone scooped out. What is the half-ball's volume, in terms of π (like 14π)?

   1152π
4. 4.

   Pia's sand pyramid has a 6 by 6 cm base and is 10 cm tall. A cone just as tall has a base of the same area. What is the cone's volume in cubic cm?

   120
5. 5.

   Tala's pyramid roof has a 4 by 4 cm base and is 12 cm tall. A cone just as tall has a base of the same area. What is the cone's volume in cubic cm?

   64
6. 6.

   Zee's pyramid box has a 12 by 12 cm base and is 7 cm tall. A cone just as tall has a base of the same area. What is the cone's volume in cubic cm?

   336
7. 7.

   A half-ball and a can with a cone scooped out have the same radius and height. Ari says they have equal volumes. Which reason is right?

   • The half-ball equals the can plus the cone.
   • They can't be equal: a ring and a circle are different shapes.
   • The half-ball is bigger because it is round.
   • At every height, the circle slice and the ring slice have equal areas.
8. 8.

   “Sketch:” Bo builds two towers 9 cm tall: a leaning stack of squares and a straight stack of circles. Every slice is 25 square cm. How do their volumes compare, and why?

   • Equal: same slice area at every height.
   • Can't tell: squares and circles are different shapes.
   • The leaning tower is bigger, because it leans.
   • The round tower is bigger, because circles are wider.
9. 9.

   Esme compares a leaning cone and a straight cone with the same base and the same height straight up. Which statement is right?

   • The straight cone holds more, because it is taller.
   • Use the leaning cone's longest side as its height.
   • The leaning cone holds more, because it leans.
   • Equal volumes: their slices match at every height.
10. 10.

    A ball fits snugly in a can. Jun wants to show the ball is two thirds of the can. Which idea works?

    • The ball touches the can in three places.
    • The ball is the can plus two cones.
    • Each half-ball is a half-can minus a cone, so the ball is the can minus two cones.
    • No rule links them: a ball's slices are circles but the can's are rings.