3-D Volume - ISEE Middle Level Mathematics Achievement

1. State the formula for the volume of any pyramid in terms of base area $B$ and height $h$.

   $V = \frac{1}{3}Bh$. The volume is one-third of the base area multiplied by the height to the apex.

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

   $V = \frac{1}{3}\pi r^2 h$. The volume is one-third of the base area $\pi r^2$ multiplied by the height.

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

   $45\pi$. Compute the base area $\pi r^2$ and multiply by height.

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

   $8$. Solve for height by dividing volume by $\pi r^2$.

5. A rectangular prism has volume $120$, length $10$, and width $3$. What is the height $h$?

   $4$. Solve for height by dividing volume by length times width.

6. A cone has volume $36\pi$, radius $r = 3$, and height $h$ unknown. What is $h$?

   $12$. Solve for height by rearranging the cone volume formula.

7. State the formula for the volume of any prism in terms of base area $B$ and height $h$.

   $V = Bh$. The volume is the base area multiplied by the height, which is the perpendicular distance between bases.

8. State the formula for the volume of a cube with side length $s$.

   $V = s^3$. The volume is the side length raised to the third power, as all dimensions are equal.

9. State the formula for the volume of a right rectangular pyramid with base area $B$ and height $h$.

   $V = \frac{1}{3}Bh$. The volume is one-third of the product of the base area and the height perpendicular to the base.

10. A cone and cylinder share the same base radius $r$ and height $h$. What is $\frac{V_{\text{cone}}}{V_{\text{cyl}}}$?

    $\frac{1}{3}$. The cone's volume is one-third that of the cylinder with identical base and height.

11. What is the volume of a cube with side length $s = 6$?

    $216$. Cube the side length to compute the volume.

12. A pyramid has volume $30$, height $5$, and base area $B$ unknown. What is $B$?

    $18$. Solve for base area by rearranging the pyramid volume formula.

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

    $V = \frac{4}{3}\pi r^3$. The formula derives from integrating the cross-sectional area of a sphere.

14. State the formula for the volume of a rectangular prism with length $l$, width $w$, and height $h$.

    $V = lwh$. The volume is calculated as the product of the three dimensions: length, width, and height.

15. What is the volume of a rectangular prism with $l = 5$, $w = 3$, and $h = 4$?

    $60$. Multiply the length, width, and height to find the volume.

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

    $36\pi$. Apply the sphere volume formula with the given radius.

17. A cylinder has diameter $10$ and height $4$. What is its volume in terms of $\pi$?

    $100\pi$. Use radius half of diameter in $\pi r^2 h$.

18. What is the volume of a prism with base area $B = 12$ and height $h = 7$?

    $84$. Multiply the base area by the height to obtain the volume.

19. A sphere has diameter $12$. What is its volume in terms of $\pi$?

    $288\pi$. Use radius half of diameter in $\frac{4}{3}\pi r^3$.

20. What is the volume of a cone with radius $r = 6$ and height $h = 3$ in terms of $\pi$?

    $36\pi$. Calculate one-third of $\pi r^2$ times height.

21. What is the volume of a pyramid with base area $B = 27$ and height $h = 6$?

    $54$. Take one-third of the product of base area and height.

22. What is the volume of a cylinder with radius $r = 2$ and height $h = 9$ in terms of $\pi$?

    $36\pi$. Compute $\pi r^2 h$ with the given values.

23. What is the volume of a triangular prism with base area $B = 15$ and prism height $h = 8$?

    $120$. Multiply the triangular base area by the prism height.

24. A cone has diameter $8$ and height $9$. What is its volume in terms of $\pi$?

    $48\pi$. Use radius half of diameter in $\frac{1}{3}\pi r^2 h$.

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

    $V = \pi r^2 h$. The volume is the base area $\pi r^2$ multiplied by the height.

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