Find Volume With Fractional Edge Lengths - 6th Grade Math

1. Find the missing length $l$ if $V=\frac{1}{6}$, $w=\frac{1}{2}$, and $h=\frac{2}{3}$.

   $l=\frac{1}{2}$. Solve $\frac{1}{6} = l \times \frac{1}{2} \times \frac{2}{3}$; $l = \frac{1}{2}$.

2. What is the volume of a prism with $l=1\frac{1}{2}$, $w=\frac{2}{3}$, $h=\frac{1}{2}$?

   $\frac{1}{2}$. Convert: $1\frac{1}{2} = \frac{3}{2}$, then $\frac{3}{2} \times \frac{2}{3} \times \frac{1}{2} = \frac{1}{2}$.

3. Find the base area: $l=\frac{5}{6}$ and $w=\frac{3}{10}$.

   $\frac{1}{4}$ square units. Base area = length × width = $\frac{5}{6} \cdot \frac{3}{10} = \frac{15}{60} = \frac{1}{4}$.

4. What is the volume of a prism with $l=\frac{5}{6}$, $w=\frac{3}{5}$, $h=\frac{2}{3}$?

   $\frac{1}{3}$. Multiply: $\frac{5}{6} \times \frac{3}{5} \times \frac{2}{3} = \frac{30}{90} = \frac{1}{3}$.

5. Choose the correct expression for volume: $l=\frac{2}{5}$, $w=\frac{3}{4}$, $h=\frac{1}{2}$.

   $V=\frac{2}{5}\cdot\frac{3}{4}\cdot\frac{1}{2}$. The volume formula requires multiplying all three dimensions together.

6. What unit should be used for volume if edge lengths are measured in inches?

   cubic inches, $\text{in}^3$. Volume units are always the cube of the linear measurement unit.

7. Find the missing width: $V=\frac{1}{6}$, $l=\frac{1}{2}$, $h=\frac{2}{3}$.

   $w=\frac{1}{2}$. Solve: $\frac{1}{6} = \frac{1}{2} \cdot w \cdot \frac{2}{3}$; $w = \frac{ \frac{1}{6} }{ \frac{1}{3} } = \frac{1}{2}$

8. Which expression correctly represents the volume of a prism with edges $\frac{2}{3}$, $\frac{3}{5}$, and $\frac{5}{2}$?

   $V=\frac{2}{3}\times\frac{3}{5}\times\frac{5}{2}$. Volume formula multiplies all three edge lengths.

9. What is the volume of a prism with $l=\frac{3}{4}$, $w=\frac{2}{3}$, $h=\frac{1}{2}$?

   $\frac{1}{4}$. Multiply: $\frac{3}{4} \times \frac{2}{3} \times \frac{1}{2} = \frac{6}{24} = \frac{1}{4}$.

10. Find the missing height $h$ if $V=\frac{3}{10}$, $l=\frac{3}{5}$, and $w=\frac{1}{2}$.

    $h=1$. Solve $\frac{3}{10} = \frac{3}{5} \times \frac{1}{2} \times h$; $h = 1$.

11. How many $\left(\frac{1}{4}\right)$-edge unit cubes fit in a $\left(\frac{1}{2}\right)\times 1\times 2$ prism?

    $64$ cubes. Dimensions fit $2 \times 4 \times 8 = 64$ cubes of size $\frac{1}{4}$.

12. Identify the number of $\left(\frac{1}{2}\right)$-edge unit cubes that fit in a $2\times 1\times 1$ prism.

    $16$ cubes. Each dimension fits $2 \div \frac{1}{2} = 4$ cubes, so $4 \times 2 \times 2 = 16$.

13. Find the volume of a box with $l=\frac{4}{5}$ ft, $w=\frac{3}{2}$ ft, and $h=\frac{5}{6}$ ft.

    $1 \, \mathrm{ft}^3$. Apply $V = lwh = \frac{4}{5} \times \frac{3}{2} \times \frac{5}{6} = 1$.

14. State the volume formula for a right rectangular prism using length, width, and height.

    $V=lwh$. Volume equals length times width times height for rectangular prisms.

15. Identify the number of $\left(\frac{1}{3}\right)$-edge unit cubes that fit in a $1\times 1\times 1$ prism.

    $27$ cubes. Each dimension fits $1 \div \frac{1}{3} = 3$ cubes, so $3^3 = 27$.

16. Identify the volume of a prism $1 \times \frac{1}{2} \times \frac{3}{2}$ in cubic units.

    $\frac{3}{4}$ cubic unit. Apply $V = lwh = 1 \times \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$.

17. What units are used for volume if edge lengths are measured in units (not squared units)?

    Cubic units (units$^3$). Volume is 3-dimensional, so units are cubed.

18. Find and correct the error: A student wrote $V=lw+h$ for a right rectangular prism.

    Correct: $V=lwh$. Volume requires multiplication, not addition, of the three dimensions.

19. How many $\left(\frac{1}{3}\right)^3$ unit cubes fill a $1\times 1\times 1$ cube?

    $27$ cubes. Each dimension fits $\frac{1}{\frac{1}{3}} = 3$ cubes, so $3^3 = 27$ total.

20. What is the volume of a prism with $l=\frac{1}{2}$, $w=\frac{1}{2}$, and $h=\frac{1}{2}$?

    $\frac{1}{8}$ cubic unit. Apply $V = lwh = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.

21. Identify the volume of a prism with $l=\frac{5}{2}$, $w=\frac{2}{5}$, $h=\frac{3}{4}$.

    $\frac{3}{4}$ cubic unit. Apply $V = lwh = \frac{5}{2} \times \frac{2}{5} \times \frac{3}{4} = \frac{3}{4}$.

22. Find $V$ if base area is $b=\frac{3}{8}$ and height is $h=\frac{4}{9}$.

    $\frac{1}{6}$. Use $V = bh$: $\frac{3}{8} \times \frac{4}{9} = \frac{12}{72} = \frac{1}{6}$.

23. What is the volume of a prism with $l=\frac{1}{2}$, $w=\frac{1}{3}$, $h=\frac{1}{4}$?

    $\frac{1}{24}$. Multiply: $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24}$.

24. Find the volume using $V=Bh$: $B=\frac{7}{12}$ and $h=\frac{2}{7}$.

    $\frac{1}{6}$ cubic units. Apply $V=Bh$: $\frac{7}{12} \cdot \frac{2}{7} = \frac{14}{84} = \frac{1}{6}$.

25. A box is $\frac{2}{3}$ ft by $\frac{3}{4}$ ft by $\frac{1}{2}$ ft. Find its volume.

    $\frac{1}{4}\,\text{ft}^3$. Apply $V=lwh$: $\frac{2}{3} \cdot \frac{3}{4} \cdot \frac{1}{2} = \frac{6}{24} = \frac{1}{4}$.

26. How many $\left(\frac{1}{2}\right)^3$ unit cubes fill a $1\times 1\times 1$ cube?

    $8$ cubes. Each dimension fits $\frac{1}{\frac{1}{2}} = 2$ cubes, so $2^3 = 8$ total.

27. Identify the correct volume for a prism with $l=\frac{3}{2}$, $w=\frac{2}{5}$, $h=\frac{5}{6}$.

    $V=\frac{1}{2}$. Calculate: $\frac{3}{2} \times \frac{2}{5} \times \frac{5}{6} = \frac{30}{60} = \frac{1}{2}$.

28. Identify the volume using $V=bh$ if $b=\frac{5}{6}$ and $h=\frac{3}{5}$.

    $\frac{1}{2}$ cubic unit. Apply $V = bh = \frac{5}{6} \times \frac{3}{5} = \frac{1}{2}$.

29. What is the total number of $\frac{1}{2}$-edge cubes in a prism $1 \times \frac{1}{2} \times \frac{3}{2}$?

    $6$ cubes. Count: $2 \times 1 \times 3 = 6$ cubes fit in the prism.

30. What does $b$ represent in the formula $V = bh$ for a right rectangular prism?

    $b$ is the area of the base. In this formula, base area replaces length times width.

31. Identify the missing height $h$ if $V=\frac{3}{4}$ and $b=\frac{1}{2}$ for a right rectangular prism.

    $h=\frac{3}{2}$. Solve $V = bh$: $\frac{3}{4} = \frac{1}{2} \times h$, so $h = \frac{3}{2}$.

32. What does $B$ represent in the formula $V=Bh$ for a right rectangular prism?

    $B$ is the area of the base. In $V=Bh$, $B$ represents the two-dimensional area of the prism's base.

33. Find the volume: $l=\frac{3}{4}$, $w=\frac{2}{3}$, $h=\frac{1}{2}$.

    $\frac{1}{4}$ cubic units. Multiply: $\frac{3}{4} \cdot \frac{2}{3} \cdot \frac{1}{2} = \frac{6}{24} = \frac{1}{4}$.

34. How many $\left(\frac{1}{4}\right)^3$ cubes fill a prism $\frac{1}{2}\times\frac{3}{4}\times 1$?

    $24$ cubes. $(\frac{1/2}{1/4}) \cdot (\frac{3/4}{1/4}) \cdot (\frac{1}{1/4}) = 2 \cdot 3 \cdot 4 = 24$.

35. Find the missing width $w$ if $V=\frac{2}{9}$, $l=\frac{2}{3}$, and $h=\frac{1}{2}$.

    $w=\frac{2}{3}$. Solve $\frac{2}{9} = \frac{2}{3} \times w \times \frac{1}{2}$; $w = \frac{2}{3}$.

36. Identify the base area $b$ if a prism has $l=\frac{3}{4}$ and $w=\frac{2}{3}$.

    $\frac{1}{2}$ square unit. Base area $b = l \times w = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$.

37. State the volume formula for a right rectangular prism using length, width, and height.

    $V=lwh$. Volume equals length times width times height for rectangular prisms.

38. What is the volume of one unit cube with edge length $\frac{1}{5}$?

    $\frac{1}{125}$ cubic unit. Volume of a cube = edge³ = $(\frac{1}{5})^3 = \frac{1}{125}$.

39. How many cubes of edge length $\frac{1}{2}$ fit in a prism $1 \times 1 \times 1$?

    $8$ cubes. Each dimension fits $1 \div \frac{1}{2} = 2$ cubes, so $2^3 = 8$.

40. If each unit cube has edge length $\frac{1}{2}$, what is the volume of one cube?

    $\frac{1}{8}$ cubic unit. Volume of a cube is edge cubed: $(\frac{1}{2})^3 = \frac{1}{8}$.

41. What is the volume of a prism with $l=\frac{2}{5}$, $w=\frac{3}{4}$, $h=\frac{5}{6}$?

    $\frac{1}{4}$. Multiply: $\frac{2}{5} \times \frac{3}{4} \times \frac{5}{6} = \frac{30}{120} = \frac{1}{4}$.

42. Find $V$ if base area is $b=\frac{5}{12}$ and height is $h=\frac{3}{5}$.

    $\frac{1}{4}$. Use $V = bh$: $\frac{5}{12} \times \frac{3}{5} = \frac{15}{60} = \frac{1}{4}$.

43. Identify the base area expression for a prism with base sides $l$ and $w$.

    $b=lw$. Base area of a rectangle is length times width.

44. Identify the volume of a prism with $l=\frac{3}{2}$, $w=2$, and $h=\frac{1}{2}$.

    $\frac{3}{2}$ cubic units. Apply $V = lwh = \frac{3}{2} \times 2 \times \frac{1}{2} = \frac{3}{2}$.

45. Find the missing height: $V=\frac{3}{10}$, $l=\frac{3}{5}$, $w=\frac{1}{2}$.

    $h=1$. Solve: $\frac{3}{10} = \frac{3}{5} \cdot \frac{1}{2} \cdot h$; $h = \frac{3/10}{3/10} = 1$.

46. State the volume formula for a right rectangular prism using base area and height.

    $V=bh$. When base area is known, multiply by height to find volume.

47. What is the volume of a prism with $l=\frac{7}{8}$, $w=\frac{4}{7}$, $h=\frac{1}{2}$?

    $\frac{1}{4}$. Multiply: $\frac{7}{8} \times \frac{4}{7} \times \frac{1}{2} = \frac{28}{112} = \frac{1}{4}$.

48. Find the volume: $l=\frac{1}{2}$, $w=\frac{1}{3}$, $h=\frac{1}{4}$.

    $\frac{1}{24}$ cubic units. Multiply: $\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{24}$.

49. Identify the unit cube edge length: a cube has volume $\frac{1}{27}$ cubic unit.

    edge length $=\frac{1}{3}$. Since $V = s^3$, if $s^3 = \frac{1}{27}$, then $s = \frac{1}{3}$.

50. Identify the volume of a prism with $l=2$, $w=3$, and $h=\frac{1}{2}$.

    $3$ cubic units. Apply $V = lwh = 2 \times 3 \times \frac{1}{2} = 3$.

51. What units are used for volume if edge lengths are measured in units?

    Cubic units, written as $\text{units}^3$. Volume is 3-dimensional, so units are cubed.

52. How many cubes of edge length $\frac{1}{2}$ fit along an edge of length $\frac{3}{2}$?

    $3$ cubes. Divide edge length by cube size: $\frac{3}{2} div \frac{1}{2} = 3$.

53. How many cubes of edge $\frac{1}{4}$ fit along an edge of length $\frac{3}{4}$?

    $3$ cubes. Divide edge length by cube size: $\frac{3/4}{1/4} = \frac{3}{4} \cdot 4 = 3$.

54. Identify the volume of a prism with $l=\frac{2}{3}$, $w=\frac{3}{4}$, $h=2$.

    $1$ cubic unit. Apply $V = lwh = \frac{2}{3} \times \frac{3}{4} \times 2 = 1$.

55. What is the volume of one cube with edge length $\frac{1}{3}$?

    $\left(\frac{1}{3}\right)^3=\frac{1}{27}$. Cube volume is edge length cubed: $(\frac{1}{3})^3 = \frac{1}{27}$.

56. Find the volume: $l=1\frac{1}{2}$, $w=\frac{2}{3}$, $h=\frac{3}{4}$.

    $\frac{3}{4}$ cubic units. Convert: $1\frac{1}{2} = \frac{3}{2}$, then $\frac{3}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} = \frac{18}{24} = \frac{3}{4}$.

57. Identify the missing width $w$ if $V=3$, $l=2$, and $h=\frac{3}{4}$ for a right rectangular prism.

    $w=2$. Solve $V = lwh$: $3 = 2 \times w \times \frac{3}{4}$, so $w = 2$.

58. State the volume formula for a right rectangular prism using base area and height.

    $V=Bh$. Volume equals base area times height for any prism.

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