Volume With Formulas - SSAT Middle Level Quantitative

1. Identify the volume of a rectangular prism with dimensions $2.5$, $4$, and $3$.

   $30$. The product of the three dimensions yields the volume of the rectangular prism.

2. State the formula for the volume of a triangular prism with base area $B$ and prism length (height) $h$.

   $V = Bh$. The volume is the triangular base area multiplied by the prism's length, or height.

3. What is the volume of a prism with base area $B = 15$ and height $h = 7$?

   $105$. The prism's volume is the product of its base area and height.

4. If all dimensions of a rectangular prism are multiplied by $k$, how does volume change?

   $V$ is multiplied by $k^3$. Scaling all linear dimensions by $k$ cubes the factor for volume due to three dimensions.

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

   $V = \frac{1}{3}\pi r^2 h$. The volume is one-third of the base area times height, derived from the pyramid-cone relationship.

6. What is the volume of a triangular prism with triangle area $B = 12$ and prism length $h = 5$?

   $60$. Multiply the triangular base area by the prism's length to obtain volume.

7. If the radius of a cylinder triples and height stays the same, how does volume scale?

   $V$ becomes $9$ times as large. Tripling the radius squares to nine times the base area, scaling volume by nine.

8. 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 perpendicular dimensions, representing the space enclosed.

9. State the formula for the volume of a prism (any base) with base area $B$ and height $h$.

   $V = Bh$. The volume is the base area extruded uniformly along the height of the prism.

10. What is the height $h$ of a cone with $V = 12\pi$ and $r = 2$?

    $h = 9$. Rearrange the cone formula to solve for height as $3V / (\pi r^2)$.

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

    $V = \frac{4}{3}\pi r^3$. This formula integrates the sphere's geometry, yielding four-thirds pi times radius cubed.

12. What is the volume of a rectangular prism with $l = 8$, $w = 3$, and $h = 5$?

    $120$. Multiply the given length, width, and height to find the enclosed volume.

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

    $216$. Raise the side length to the third power to compute the cube's volume.

14. What is the volume of a cylinder with $r = 3$ and $h = 10$? (Leave in terms of $\pi$.)

    $90\pi$. Compute base area as $\pi r^2$ and multiply by height for the cylinder's volume.

15. What is the volume of a pyramid with base area $B = 48$ and height $h = 9$?

    $144$. Multiply base area by height and divide by three for the pyramid's volume.

16. What is the side length $s$ of a cube with volume $V = 125$?

    $s = 5$. Take the cube root of the volume to determine the side length.

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

    $36\pi$. Apply the sphere volume formula with the given radius to find the enclosed space.

18. If a rectangular prism's length doubles, with $w$ and $h$ unchanged, how does $V$ change?

    $V$ doubles. Doubling one dimension multiplies the original volume by two, as volume is proportional to each dimension.

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

    $V = s^3$. Since all edges are equal, the volume is the side length cubed to account for three dimensions.

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

    $r = 4$. Divide volume by $\pi h$ and take the square root to find the radius.

21. What is the height $h$ of a rectangular prism with $V = 72$, $l = 6$, and $w = 4$?

    $h = 3$. Solve for height by dividing volume by the product of length and width.

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

    $V = \pi r^2 h$. The volume equals the area of the circular base multiplied by the height of the cylinder.

23. What is the volume of a cone with $r = 6$ and $h = 9$? (Leave in terms of $\pi$.)

    $108\pi$. Take one-third of $\pi r^2 h$ to account for the tapering shape of the cone.

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

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

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