Apply Volume Formulas - 8th Grade Math

1. A cylinder has height $h$. If radius doubles from $r$ to $2r$, how does volume change?

   Volume quadruples: $V \to 4V$. Volume is proportional to $r^2$, so doubling $r$ quadruples $V$.

2. State the formula for the volume of a sphere with radius $r$.

   $V=\frac{4}{3}\pi r^3$. Sphere volume uses radius cubed with coefficient $\frac{4}{3}\pi$.

3. A sphere has diameter $8$. What is $V$ in terms of $\pi$?

   $\frac{256}{3}\pi$. With $r = 4$, $V = \frac{4}{3}\pi(4)^3 = \frac{256}{3}\pi$.

4. A cylinder has $V = 72\pi$ and $r = 3$. What is the height $h$?

   $h = 8$. Solve $72\pi = \pi(3)^2h$ to get $h = \frac{72}{9} = 8$.

5. What is the volume of a cone with diameter $8$ and height $9$ in terms of $\pi$?

   $48\pi$. Diameter 8 gives $r=4$; $V=\frac{1}{3}\pi(4^2)(9)=48\pi$.

6. Find the height $h$ of a cone with volume $V=12\pi$ and radius $r=3$.

   $h=4$. From $12\pi=\frac{1}{3}\pi(3)^2h$, solve $12\pi=3\pi h$ for $h$.

7. What is the volume of a cone with $r = 3$ and $h = 5$? (Leave in terms of $\pi$.)

   $15\pi$. Apply $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(3)^2(5) = 15\pi$.

8. Find and correct the error: A student wrote cylinder volume as $V=2\pi rh$.

   Correct: $V=\pi r^2h$. Student confused cylinder volume with lateral surface area formula.

9. Identify the meaning of $h$ in the formulas $V=\pi r^2h$ and $V=\frac{1}{3}\pi r^2h$.

   $h$ is the perpendicular height. Vertical distance from base to top.

10. A sphere's radius doubles from $r$ to $2r$. By what factor does the volume change?

    Volume increases by $8$: $V \to 8V$. Volume is proportional to $r^3$, so doubling $r$ multiplies $V$ by $8$.

11. Find the radius $r$ of a cylinder with volume $V=64\pi$ and height $h=16$.

    $r=2$. From $64\pi=\pi r^2(16)$, divide by $16\pi$ to get $r^2=4$.

12. A sphere has $V = 288\pi$. What is the radius $r$?

    $r = 6$. Solve $288\pi = \frac{4}{3}\pi r^3$ to get $r^3 = 216$, so $r = 6$.

13. What is the volume of a sphere with $r = 3$? (Leave in terms of $\pi$.)

    $36\pi$. Apply $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(3)^3 = 36\pi$.

14. What is the volume of a cylinder with diameter $10$ and height $4$ in terms of $\pi$?

    $100\pi$. Diameter 10 gives $r=5$; $V=\pi(5^2)(4)=100\pi$.

15. What is the radius of a sphere with diameter $12$?

    $6$. Radius is half the diameter: $12÷2=6$.

16. Find the volume of a cylinder with diameter $10$ and height $3$ in terms of $\pi$.

    $75\pi$. Diameter $10$ gives $r=5$, so $V=\pi(5)^2(3)$.

17. What is the radius $r$ of a sphere with $V=\frac{4}{3}\pi(2^3)$?

    $2$. Given expression shows $r^3=8$, so $r=2$.

18. Identify the base area used in volume formulas for cylinders and cones when the radius is $r$.

    $B=\pi r^2$. Circular base area equals $\pi$ times radius squared.

19. What is the volume of a cylinder with $r=3$ and $h=5$ in terms of $\pi$?

    $45\pi$. Apply $V=\pi r^2h$ with $r=3$, $h=5$: $\pi(3^2)(5)=45\pi$.

20. What is the height $h$ of a cylinder with $V=64\pi$ and $r=4$?

    $4$. Solve $64\pi=\pi(4^2)h$ for $h$: $64\pi=16\pi h$, so $h=4$.

21. Which option is the correct volume formula for a cone: $\pi r^2h$ or $\frac{1}{3}\pi r^2h$?

    $V=\frac{1}{3}\pi r^2h$. Cone uses one-third of base times height.

22. Choose the correct volume formula for a sphere: $\pi r^2h$, $\frac{4}{3}\pi r^3$, or $\frac{1}{3}\pi r^2h$.

    $V=\frac{4}{3}\pi r^3$. Sphere formula has $r^3$, not $r^2h$.

23. Find the height $h$ of a cylinder with volume $V=50\pi$ and radius $r=5$.

    $h=2$. From $50\pi=\pi(5)^2h$, divide by $25\pi$ to get $h=2$.

24. State the formula for the volume of a cone with radius $r$ and height $h$.

    $V=\frac{1}{3}\pi r^2h$. Cone volume is one-third of cylinder volume with same base and height.

25. A cylinder and cone have the same $r$ and $h$. What is $\frac{V_{\text{cone}}}{V_{\text{cyl}}}$?

    $\frac{1}{3}$. Cone volume is $\frac{1}{3}$ of cylinder volume when dimensions match.

26. What is the volume of a sphere with diameter $12$ in terms of $\pi$?

    $288\pi$. Diameter 12 gives $r=6$; $V=\frac{4}{3}\pi(6^3)=288\pi$.

27. What is the volume of a cylinder with $r = 3$ and $h = 5$? (Leave in terms of $\pi$.)

    $45\pi$. Apply $V = \pi r^2 h = \pi(3)^2(5) = 45\pi$.

28. A sphere's radius is tripled. By what factor does the volume change?

    $27$. Volume scales with $r^3$: if $r$ triples, $V$ increases by $3^3=27$.

29. Find the volume of a cone with $r=3$ and $h=5$ in terms of $\pi$.

    $15\pi$. Apply $V=\frac{1}{3}\pi r^2h=\frac{1}{3}\pi(3)^2(5)=\frac{1}{3}\pi(45)$.

30. Identify the meaning of $r$ in the formulas $V=\pi r^2h$ and $V=\frac{4}{3}\pi r^3$.

    $r$ is the radius. Distance from center to edge of circular base.

31. A cylinder has $r=2$. If the height doubles, how does the volume change?

    Volume doubles. Volume $V=\pi r^2h$ is proportional to height.

32. Find the radius $r$ of a sphere with volume $V=\frac{4}{3}\pi\cdot 27$.

    $r=3$. Since $V=36\pi=\frac{4}{3}\pi(27)$, and $27=3^3$, so $r=3$.

33. Find the volume of a cone with diameter $12$ and height $4$ in terms of $\pi$.

    $48\pi$. Diameter $12$ gives $r=6$, so $V=\frac{1}{3}\pi(6)^2(4)$.

34. A cylinder has $r = 4$. If height doubles from $h$ to $2h$, how does volume change?

    Volume doubles: $V \to 2V$. Volume is proportional to height, so doubling $h$ doubles $V$.

35. What is the radius $r$ of a cylinder with $V=36\pi$ and $h=4$?

    $3$. Solve $36\pi=\pi r^2(4)$ for $r$: $r^2=9$, so $r=3$.

36. Find and correct the formula error: $V = \pi r^2 h$ is claimed to be the volume of a cone.

    Correct: $V = \frac{1}{3}\pi r^2 h$. Cone formula needs factor $\frac{1}{3}$, not just $\pi r^2 h$.

37. If a cone and cylinder have the same $r$ and $h$, what fraction of the cylinder's volume is the cone's volume?

    $\frac{1}{3}$. Cone volume formula has factor $\frac{1}{3}$ compared to cylinder.

38. A cylinder and cone have the same $r$ and $h$. What is $\frac{V_{\text{cyl}}}{V_{\text{cone}}}$?

    $3$. Cylinder volume is $3$ times cone volume when dimensions match.

39. A cone has diameter $12$ and height $9$. What is $V$ in terms of $\pi$?

    $108\pi$. With $r = 6$, $V = \frac{1}{3}\pi(6)^2(9) = 108\pi$.

40. A cylinder has diameter $10$ and height $4$. What is $V$ in terms of $\pi$?

    $100\pi$. With $r = 5$, $V = \pi(5)^2(4) = 100\pi$.

41. Find the volume of a sphere with $r=3$ in terms of $\pi$.

    $36\pi$. Apply $V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3)^3=\frac{4}{3}\pi(27)$.

42. A cone has $V = 12\pi$ and $r = 2$. What is the height $h$?

    $h = 9$. Solve $12\pi = \frac{1}{3}\pi(2)^2h$ to get $h = \frac{36}{4} = 9$.

43. What is the height $h$ of a cone with $V=48\pi$ and $r=6$?

    $4$. Solve $48\pi=\frac{1}{3}\pi(6^2)h$: $48\pi=12\pi h$, so $h=4$.

44. What is the volume of a cone with $r=3$ and $h=5$ in terms of $\pi$?

    $15\pi$. Apply $V=\frac{1}{3}\pi r^2h$ with $r=3$, $h=5$: $\frac{1}{3}\pi(9)(5)=15\pi$.

45. Find and correct the error: A student wrote the sphere volume as $V=4\pi r^3$.

    Correct: $V=\frac{4}{3}\pi r^3$. Missing the fraction $\frac{1}{3}$ in the coefficient.

46. What is the radius if the diameter of a circular base is $d$?

    $r = \frac{d}{2}$. Radius is half the diameter.

47. Identify the missing factor: A cone has volume $V=\_\pi r^2h$.

    $\frac{1}{3}$. Cone formula has factor $\frac{1}{3}$ before $\pi r^2h$.

48. What is the volume of a sphere with $r=3$ in terms of $\pi$?

    $36\pi$. Apply $V=\frac{4}{3}\pi r^3$ with $r=3$: $\frac{4}{3}\pi(27)=36\pi$.

49. State the formula for the volume of a cylinder with radius $r$ and height $h$.

    $V=\pi r^2h$. Base area $\pi r^2$ times height gives cylinder volume.

50. Which option is the correct volume formula for a cylinder: $\pi r^2h$ or $\frac{1}{3}\pi r^2h$?

    $V=\pi r^2h$. Cylinder uses full base times height, not one-third.

51. What is the relationship between a cylinder's volume and a cone's volume with the same $r$ and $h$?

    $V_{\text{cyl}}=3V_{\text{cone}}$. Cylinder holds three times cone's volume.

52. What is the relationship between a cone's volume and a cylinder's volume with the same $r$ and $h$?

    $V_{\text{cone}}=\frac{1}{3}V_{\text{cyl}}$. Cone holds exactly one-third of cylinder's volume.

53. Find the volume of a cylinder with $r=3$ and $h=5$ in terms of $\pi$.

    $45\pi$. Apply $V=\pi r^2h=\pi(3)^2(5)=\pi(9)(5)$.

54. Identify the correct base area formula for a circle with radius $r$ used in volume formulas.

    $B = \pi r^2$. Circle area formula used as base in cylinder and cone volumes.

Study these flashcards