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# Geometry Study Guide for McGraw-Hill's ASVAB (page 4)

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?