GRE Exam: A Simple Guide To Solving Geometry Problems - Solids and Cylinders

Solids and Cylinders
So far, you have only looked at two-dimensional shapes. The GRE will occasionally test your knowledge of three-dimensional shapes. You will need to know how to calculate the surface area and volume of rectangular solids, cubes, and cylinders. A cube is a type of rectangular solid, so let’s consider cubes and rectangular solids together.

If your test date is very close, you should focus on memorizing the formulas in bold.

Surface Area of a Rectangular Solid and a Cube

The following figure is a rectangular solid:


The edge of a rectangular solid refers to any line that joins two vertices, and the face of a rectangular solid refers to any of the rectangles formed by four edges.

All rectangular solids have six faces, and opposite faces have the same dimensions.

The surface area of any three-dimensional shape refers to the total area covered by the shape’s surface.

It may seem tedious to calculate surface area of a rectangular solid, since you would presumably need to calculate the area of each face. But remember that opposite faces have the same dimensions. So instead of calculating the area of each face, just calculate the area of each unique face and double that.

The formula for calculating surface area of a rectangular solid is therefore:
SArectangular solid = 2lw + 2lh + 2wh

What is the surface area of a rectangular solid with dimensions 3 × 4 × 5?
SOLUTION: Use the preceding formula. Let l = 3, w = 4, and h = 5. Substitute: 2(3)(4) + 2(3)(5) + 2(4)(5) = 94.

A cube is a type of rectangular solid in which all edges are equal. Since all edges of a cube are equal, all faces will have an equal area.


Thus to calculate the surface area of a cube, just calculate the area of one face and multiply that by six (since the cube has six sides).

The formula for the surface area of a cube is
SAcube = 6e2, where e = the length of an edge

If the area of one face of a cube is 81, what is the surface area of the cube?
SOLUTION: Since all faces have the same area, the surface area of the cube = 81 × 6 = 486.

Volume of a Rectangular Solid and Cube
The volume of a rectangular solid refers to the amount of space within that shape. Volume is measured in cubic units, such as 2 cubic meters or 3.7 cubic inches.

To calculate the volume of a rectangular solid, just multiply all the dimensions:


The same rule applies to a cube. But since all the edges of a cube are equal, you just need one side of the cube to calculate a cube’s volume:


If the volume of a rectangular solid with a length of 4, a width of 2, and a height of x equals the volume of a cube with an edge of 4, what is x?
1
2
4
8
16

SOLUTION: Since the volumes are equal,

The correct answer is D.

Diagonal of a Rectangular Solid
The diagonal of a rectangular solid is the straight line distance from one vertex to the opposite vertex.


To determine the diagonal of a rectangular solid, use the following formula:


If a rectangular solid has dimensions of 5, 5, and 10, what is the greatest straight-line distance from one vertex to another vertex?
10
20
25



SOLUTION: Use the formula: Let l = 5, w = 5, and h = 10. The diagonal = .

The correct answer is E.
 

The diagonal of a cube = .The main diagonal of any cube can be found by multiplying the length of one side by the square root of 3. Where “x” is the cube side.

Cylinders
A cylinder is rectangular solid whose base is a circle.


To determine the volume of a cylinder, multiply the area of the base by the height.

In the case of a cylinder, the area of the base is πr2 since the base is a circle.

The formula for the volume of a cylinder is


For this question, write your answer in the box.

How many cubic feet of water are needed to fill a cylinder with a radius of 10 feet and a height of 6 feet?


SOLUTION: Use the volume formula: volume = πr2h = π102(6) = 600π

The formula for the surface area of a cylinder is:

This formula, however, is rarely tested.