# 3-Dimensional Shapes

## Rules

The volume of a non-tapering 3-dimensional object is the area of the base multiplied by the height.

Irregular objects and those which “taper” - or are not the same from the bottom to the top - are typically not tested on the GRE. However, for block-shaped objects, the formula is simply length(width)(height). However, there are two key tapering objects which do have simple formulas and could be tested - cones, and spheres.

The volume of a cone is one third of the area of the base multiplied by the height.

The area of the base will utilize the same formula for a circle.

The volume of a sphere is ${\displaystyle {\frac {4}{3}}\pi r^{3}}$ . A “sphere” is a round three-dimensional object, such as a ball or a globe.

## Practice

1. A cereal box (left) has a volume of 225. If the dimensions of the face of the cereal box are 18 and 5, what is the depth of the cereal box?

2. A cylinder (right) has a diameter of 22 and a height of 45. What is the volume of the cylinder?

3. A soccer ball has a diameter of 12. A tennis ball has a diameter of 2. The volume of the tennis ball is what percentage of the volume of the soccer ball?

## Answers to Practice Questions

1. 2.5

The volume of a block-shaped object is equal to length(width)(height). The length and width are known to be 18 and 5, and the volume is 225. Thus, 18(5)(width) = 225. The depth equals 225 divided by 18(5) or 2.5.

2.

The volume of a cylinder equals the area of the base multiplied by the height. Because the diameter of the cylinder is 22, the radius is 11 and the area of the base isThis means that the volume isor

3.

The volume of a sphere is Thus, the volume of the soccer ball is or ; the volume of the tennis ball isorThe ratio of their volumes is or