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# Geometry Study Guide for McGraw-Hill's ASVAB (page 5)

By McGraw-Hill Professional
Updated on Jun 26, 2011

### Three-Dimensional (Solid) Figures

A figure is two-dimensional if all the points on the figure are in the same plane. A square and a triangle are two-dimensional figures. A figure is three-dimensional (solid) if some points of the figure are in a different plane from other points in the figure.

On solid figures, the flat surfaces are called faces. Edges are line segments where two faces meet. A point where three or more edges intersect is called a vertex.

Types of Solid Figures On the ASVAB, you may see problems related to these solid figures: rectangular solid (prism), cube, cylinder, and sphere.

Rectangular Solid (Prism) On a rectangular solid (also called a prism), all of the faces are rectangular. The top and bottom faces are called bases. All opposite faces on a rectangular solid are parallel and congruent.

Cube A cube is a rectangular solid on which every face is a square.

Cylinder A cylinder is a solid figure with two parallel congruent circular bases and a curved surface connecting the boundaries of the two faces.

Sphere A sphere is a solid figure that is the set of all points that are the same distance from a given point, called the center. The distance from the center is the radius (r) of the sphere.

Finding the Volume of Solid Figures Volume is the amount of space within a three-dimensional figure. Volume is measured in cubic units, such as cubic inches (in3) or cubic centimeters (cm3). A cubic inch is the volume of a cube with edges 1 inch long.

Volume of a Rectangular Solid To find the volume (V) of a rectangular solid, multiply the length (l ) times the width (w) times the height (h). The formula is V = lwh.

Example

If a rectangular solid has a length of 3 yards, a height of 1.5 yards, and a width of 1.5 yards, what is its volume?

V = lwh

V = (3)(1.5)(1.5)

V = 6.75 cubic yards (6.75 yd3)

Volume of a Cube On a cube, the length, width, and height are all the same: Each one equals 1 side (s). To find the volume (V) of a cube, multiply the length × width × height. This is the same as multiplying side × side × side. The formula is V = s × s × s = s3.

Example

If each side of a cube measures 9 feet, what is its volume?

V = s3

V = (9)3

V = 729 cubic feet (729 ft3)

Volume of a Cylinder To find the volume (V) of a cylinder, first find the area of the circular base by using the formula A = πr2. Then multiply the result times the height (h) of the cylinder. The formula is V = (πr2)h).

Example

If a cylinder has a height of 7 meters and a radius of 2 meters, what is its volume?

V = (πr2)h

V = 3.14(2)2(7)

V = 3.14(4)(7)

V = 87.92 cubic meters (87.92 m3)

Volume of a Sphere To find the volume of a sphere, multiply 4/3 times π times the radius cubed. The formula is

Example

If the radius of a sphere measures 12 inches, what is the volume?

### Geometry Word Problems

Word problems on the ASVAB may deal with geometry concepts such as perimeter, area, and volume. To solve these problems, you may also need to use information about different units of measure. To review units of measure, see Chapter 8.

Just as with other kinds of word problems, you can solve geometry problems by following a specific procedure. In the examples that follow, pay special attention to the procedure outlined in each solution. Follow this same procedure whenever you need to solve this kind of word problem.

Example

Sally is buying wood to make a rectangular picture frame measuring 11 in. × 14 in. The wood costs 25 cents per inch. How much will Sally have to pay for the wood?

Procedure

What must you find? Cost of the wood for the frame

What are the units? Dollars and cents

What do you know? Cost of the wood per inch; shape of the frame, measure of the frame

Create an equation and solve.

Length of wood needed for rectangular frame = 2l + 2w

Substitute values and solve:

Length of wood = 2(14) + 2(11)

Length of wood = 28 + 22 = 50 in.

If each inch costs 0.25, then

0.25 × 50 = \$12.50

Sergei is planting rosebushes in a rectangular garden measuring 12 ft × 20 ft. Each rosebush needs 8 ft2 of space. How many rosebushes can Sergei plant in the garden?

Procedure

What must you find? Number of rosebushes that can be planted in the garden

What are the units? Numbers

What do you know? Shape of the garden, garden length and width, amount of area needed for each rosebush

Create an equation and solve.

A = lw

Substitute values and solve.

A = 12 × 20 = 240 ft2

Each rosebush needs 8 ft2

240 ÷ 8 = 30

Sergei can plant 30 rosebushes in the garden.

Practice problems for this study guide can be found at:

Geometry Practice Problems for McGraw-Hill's ASVAB