3-D Volume - ISEE Upper Level Mathematics Achievement

1. What is the volume of a cylinder with diameter $d=10$ and height $h=3$ in terms of $\pi$?

   $75\pi$. Uses radius as half the diameter in the cylinder volume formula.

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

   $V = \frac{1}{3}Bh$. Takes one-third of the product of base area and height for the pyramid's volume.

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

   $V = \frac{4}{3}\pi r^3$. Integrates the cross-sectional areas to derive the volume enclosed by the sphere.

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

   $V = \pi r^2 h$. Multiplies the circular base area by the height to determine the cylinder's volume.

5. What is the volume of a cube with side length $s=4$?

   $64$. Cubes the side length to find the volume of the cube.

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

   $90$. Divides the product of base area and height by three for the pyramid's volume.

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

   $V = \frac{1}{3}\pi r^2 h$. Uses one-third of the cylinder volume formula due to the tapering shape of the cone.

8. State the formula for the volume of a cylinder using diameter $d$ and height $h$.

   $V = \frac{1}{4}\pi d^2 h$. Substitutes radius as half the diameter into the standard cylinder volume formula.

9. Identify the volume of a square pyramid with base side $s=6$ and height $h=5$.

   $60$. Calculates one-third of the square base area times height for the pyramid.

10. What is the height $h$ of a cylinder with volume $V=100\pi$ and radius $r=5$?

    $4$. Solves for height by dividing volume by $\pi r^2$ in the cylinder formula.

11. State the formula for the volume of a hemisphere with radius $r$.

    $V = \frac{2}{3}\pi r^3$. Represents half the volume of a full sphere with the same radius.

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

    $2$. Identifies the radius from the given volume expression matching the sphere formula.

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

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

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

    $V = lwh$. Multiplies the three dimensions to determine the enclosed space in a rectangular prism.

15. What is the volume of a cone with $r=3$ and $h=4$ in terms of $\pi$?

    $12\pi$. Takes one-third of $\pi r^2 h$ to calculate the cone's volume.

16. State the formula for the volume of a triangular prism with base area $B$ and prism height $h$.

    $V = Bh$. Multiplies the triangular base area by the prism height to find the total volume.

17. What is the volume of a hemisphere with radius $r=6$ in terms of $\pi$?

    $144\pi$. Uses half of the full sphere's volume for the hemisphere with the given radius.

18. State the formula for the volume of a square pyramid with base side $s$ and height $h$.

    $V = \frac{1}{3}s^2h$. Applies one-third of the base area times height for a pyramid with square base.

19. What is the volume of a rectangular prism with $l=6$, $w=2$, and $h=5$?

    $60$. Multiplies length, width, and height to compute the rectangular prism's volume.

20. State the formula for the volume of a right circular cone using diameter $d$ and height $h$.

    $V = \frac{1}{12}\pi d^2 h$. Substitutes radius as half the diameter into the standard cone volume formula.

21. State the formula for the volume of any prism with base area $B$ and height $h$.

    $V = Bh$. Multiplies the base area by the height to calculate the volume for any prism.

22. What is the volume of a cone with diameter $d=12$ and height $h=8$ in terms of $\pi$?

    $96\pi$. Substitutes radius as half the diameter into the cone volume formula.

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

    $126$. Multiplies the base area by the height to find the prism's volume.

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

    $V = s^3$. Cubes the side length since all dimensions are equal in a cube.

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

    $45\pi$. Multiplies the base area $\pi r^2$ by height to obtain the cylinder's volume.

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