Why the volume formulas work (Cavalieri's principle)

Answer Key - Grade 10 Math Worksheet - Building Confidence

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

1. Read this first

   Sketch, not to scale

   The leaning stack with layers 9 cm by 6 cm, the straight-up height 4 cm, and the slanted edge 5 cm.

2. 1.

   Kira's stack of notebooks has 9 by 6 cm layers and stands 4 cm tall. Pushed so it leans, its slanted edge is 5 cm. What is its volume now, in cubic cm?

   216

3. Sketch, not to scale

   The leaning stack with layers 10 cm by 7 cm, the straight-up height 3 cm, and the slanted edge 5 cm.

4. 2.

   Luz's stack of paper has 10 by 7 cm layers and stands 3 cm tall. Pushed so it leans, its slanted edge is 5 cm. What is its volume now, in cubic cm?

   210
5. 3.

   Pia has two stacks of 20 equal coins. One stands straight; the other leans. How do their volumes compare?

   • The leaning stack is bigger.
   • You can only tell if both slices are the same shape.
   • They are equal.
   • The straight stack is bigger.
6. 4.

   “Sketch:” Ren stacks 12 equal books straight, then twists the next stack of 12 like a fan. How do the two stacks' volumes compare?

   • The twisted stack is bigger.
   • You can only tell if the stacks have the same outline.
   • They are equal.
   • The twisted stack is smaller.
7. 5.

   Tala's two clay towers are 10 cm tall. At every height, one has a square slice of 36 square cm, the other a circle slice of 36 square cm. How do their volumes compare?

   • The square tower is bigger.
   • They are equal.
   • The round tower is bigger because it leans out.
   • You can't compare a square and a circle.
8. 6.

   Zee stacks 8 equal slices of bread straight, and 8 more that slide sideways. How do the two stacks' volumes compare?

   • The straight stack is bigger.
   • They are equal.
   • The sliding stack is bigger.
   • You can only tell if both stacks have the same outline.
9. 7.

   The cube with edges 24 cm, split into three equal pyramids that share one corner. Ari cuts a foam cube with 24 cm edges into 3 equal pyramids, each with a cube face as its base. What is each pyramid's volume, in cubic cm?

   4608
10. 8.

    split into three equal pyramids that share one corner. Bo cuts a foam cube with 15 cm edges into 3 equal pyramids, each with a cube face as its base. What is each pyramid's volume, in cubic cm?

    1125
11. 9.

    Esme cuts a foam cube with 18 cm edges into 3 equal pyramids, each with a cube face as its base. What is each pyramid's volume, in cubic cm?

    1944
12. 10.

    Jun cuts a foam cube with 21 cm edges into 3 equal pyramids, each with a cube face as its base. What is each pyramid's volume, in cubic cm?

    3087