Solving Problems with Volume Formulas - Geometry

1. What is the volume formula for a cone in terms of base area $B$ and height $h$?

   $V=\frac{1}{3}Bh$. General form using base area $B$ instead of $\pi r^2$.

2. A cone's radius doubles and height doubles. By what factor does $V$ change?

   Factor of $8$. $V \propto r^2h$, so $(2)^2 \cdot 2 = 8$ times volume.

3. A sphere's radius doubles. By what factor does the volume change?

   Factor of $8$. $V \propto r^3$, so doubling $r$ gives $(2)^3 = 8$ times volume.

4. Identify the error: using $V=\pi r^2l$ for a cone, where $l$ is slant height. What should be used?

   Use $V=\frac{1}{3}\pi r^2h$ with perpendicular $h$. Cone volume formula uses perpendicular height, not slant height.

5. What is the volume of a cylinder with $r=6$ and $h=4$?

   $144\pi$. $V = \pi(6)^2(4) = 144\pi$.

6. What is the volume formula for a sphere with radius $r$?

   $V=\frac{4}{3}\pi r^3$. Four-thirds $\pi$ times radius cubed.

7. A cylinder's radius doubles and height stays the same. By what factor does $V$ change?

   Factor of $4$. $V \propto r^2$, so doubling $r$ gives $ $(2)^2$ = 4 $ times volume.

8. What is the formula for the base area of a circular base with radius $r$?

   $B=\pi r^2$. Area formula for a circle with radius $r$.

9. What is the volume of a cylinder with $r=2$ and $h=9$?

   $36\pi$. $V = \pi(2)^2(9) = 36\pi$.

10. Find the height $h$ of a cone with volume $V$ and radius $r$.

    $h=\frac{3V}{\pi r^2}$. Solve $V = \frac{1}{3}\pi r^2 h$ for $h$.

11. What is the volume formula for a right circular cone with radius $r$ and height $h$?

    $V=\frac{1}{3}\pi r^2h$. One-third of a cylinder's volume with same base and height.

12. Find the base area $B$ of a pyramid with volume $V$ and height $h$.

    $B=\frac{3V}{h}$. Solve $V = \frac{1}{3}Bh$ for $B$.

13. Find the radius of a sphere with volume $\frac{4}{3}\pi$.

    $r=1$. Solve $\frac{4}{3}\pi = \frac{4}{3}\pi r^3$ for $r$.

14. What is the volume of a pyramid with base area $B=30$ and height $h=9$?

    $90$. $V = \frac{1}{3} \cdot 30 \cdot 9 = \frac{270}{3}$.

15. What is the volume of a cone with base area $B=27\pi$ and height $h=6$?

    $54\pi$. $V = \frac{1}{3} \cdot 27\pi \cdot 6 = \frac{162\pi}{3}$.

16. What is the volume formula for a right circular cylinder with radius $r$ and height $h$?

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

17. A cone has volume $25\pi$, radius $r=5$. What is the height $h$?

    $h=3$. Solve $25\pi = \frac{1}{3}\pi(5)^2 h$ for $h$.

18. A cylinder's height triples and radius stays the same. By what factor does $V$ change?

    Factor of $3$. $V \propto h$, so tripling $h$ gives $3$ times volume.

19. What is the volume of a sphere with diameter $8$?

    $\frac{256}{3}\pi$. Diameter 8 means $r = 4$; $V = \frac{4}{3}\pi(4)^3$.

20. A sphere has radius $r=5$. What is its volume?

    $\frac{500}{3}\pi$. $V = \frac{4}{3}\pi(5)^3 = \frac{500\pi}{3}$.

21. What is the relationship between a pyramid's volume and a same-base, same-height prism's volume?

    $V_{\text{pyr}}=\frac{1}{3}V_{\text{prism}}$. A pyramid always has one-third the volume of its corresponding prism.

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

    $r=\sqrt{12}$. Solve $12\pi = \frac{1}{3}\pi r^2(3)$ for $r$.

23. Find the radius of a cylinder with volume $64\pi$ and height $h=4$.

    $r=4$. Solve $64\pi = \pi r^2(4)$ for $r$.

24. Find the radius $r$ of a cone with volume $V$ and height $h$.

    $r=\sqrt{\frac{3V}{\pi h}}$. Solve $V = \frac{1}{3}\pi r^2 h$ for $r$.

25. What is the volume of a cylinder with $r=1.5$ and $h=8$?

    $18\pi$. $V = \pi(1.5)^2(8) = \pi \cdot 2.25 \cdot 8$.

26. What is the relationship between a cone's volume and a same-base, same-height cylinder's volume?

    $V_{\text{cone}}=\frac{1}{3}V_{\text{cyl}}$. A cone always has one-third the volume of its corresponding cylinder.

27. What is the volume formula for a pyramid with base area $B$ and height $h$?

    $V=\frac{1}{3}Bh$. One-third of base area times height.

28. Find the radius of a sphere with volume $36\pi$.

    $r=3$. Solve $36\pi = \frac{4}{3}\pi r^3$ for $r$.

29. What is the volume of a cone with $r=3$ and $h=5$?

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

30. Identify the height used in $V=\frac{1}{3}Bh$ for a pyramid: the slant height or the perpendicular height?

    The perpendicular height $h$. Volume formulas use perpendicular height, not slant height.

31. Find the height of a cone with volume $48\pi$ and radius $r=4$.

    $h=9$. Solve $48\pi = \frac{1}{3}\pi(4)^2 h$ for $h$.

32. Find the height $h$ of a cylinder with volume $V$ and radius $r$.

    $h=\frac{V}{\pi r^2}$. Solve $V = \pi r^2 h$ for $h$.

33. What is the volume of a sphere with $r=2$?

    $\frac{32}{3}\pi$. $V = \frac{4}{3}\pi(2)^3 = \frac{32}{3}\pi$.

34. Identify the height used in $V=\frac{1}{3}\pi r^2h$ for a cone: slant height or perpendicular height?

    The perpendicular height $h$. Volume formulas use perpendicular height, not slant height.

35. What is the volume of a cylinder with $r=3$ and $h=5$?

    $45\pi$. $V = \pi(3)^2(5) = \pi \cdot 9 \cdot 5$.

36. Find the radius $r$ of a sphere with volume $V$.

    $r=\sqrt[3]{\frac{3V}{4\pi}}$. Solve $V = \frac{4}{3}\pi r^3$ for $r$.

37. What is the volume of a cone with diameter $10$ and height $12$?

    $100\pi$. Diameter 10 means $r = 5$; $V = \frac{1}{3}\pi(5)^2(12)$.

38. What is the radius of a sphere with diameter $d$ (in terms of $d$)?

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

39. What is the volume of a cone if a same-base, same-height cylinder has volume $72\pi$?

    $24\pi$. Cone volume is one-third of cylinder volume.

40. Find the radius $r$ of a cylinder with volume $V$ and height $h$.

    $r=\sqrt{\frac{V}{\pi h}}$. Solve $V = \pi r^2 h$ for $r$.

41. What is the volume formula for a cylinder in terms of base area $B$ and height $h$?

    $V=Bh$. General form using base area $B$ instead of $\pi r^2$.

42. What is the volume of a cylinder with base area $B=18\pi$ and height $h=7$?

    $126\pi$. $V = 18\pi \cdot 7$.

43. Identify the error: using $V=\frac{1}{2}Bh$ for a pyramid. What is the correct coefficient?

    Coefficient $\frac{1}{3}$, so $V=\frac{1}{3}Bh$. Pyramid volume uses coefficient $\frac{1}{3}$, not $\frac{1}{2}$.

44. A cylinder has volume $200\pi$ and height $h=8$. What is the radius $r$?

    $r=5$. Solve $200\pi = \pi r^2(8)$ for $r$.

45. What is the volume of a cone with $r=2$ and $h=9$?

    $12\pi$. $V = \frac{1}{3}\pi(2)^2(9) = 12\pi$.

46. Identify the error: using $V=4\pi r^2$ for a sphere. What is the correct volume formula?

    $V=\frac{4}{3}\pi r^3$. Confused with surface area formula; volume uses $\frac{4}{3}\pi r^3$.

47. Find the height of a cylinder with volume $81\pi$ and radius $r=3$.

    $h=9$. Solve $81\pi = \pi(3)^2 h$ for $h$.

48. What is the volume of a cylinder with diameter $10$ and height $4$?

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

49. What is the volume of a pyramid if a same-base, same-height prism has volume $150$?

    $50$. Pyramid volume is one-third of prism volume.

50. What is the volume of a sphere with $r=3$?

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

51. What is the volume of a cone with $r=6$ and $h=4$?

    $48\pi$. $V = \frac{1}{3}\pi(6)^2(4) = \frac{144\pi}{3}$.

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