3-D Volume - ISEE Upper Level: Mathematics Achievement

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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