Measuring Three-Dimensional Figures

How do you determine surface area of a figure?

To determine the surface area of a figure, first think about disassembling the figure into separate faces. For instance, think about a right rectangular parallelepiped with dimensions of 6 centimeters by 8 centimeters by 20 centimeters.

There are six faces.

The surface area of the figure is the sum of the areas of the six faces.

A = 2(6)(8) + 2(6)(20) + 2(8)(20) = 656
square centimeters

How do you determine volume?

There are a number of volume formulas. In this section, we will explore two basic categories of volume formulas: non-pointed and pointed figures.

First, let’s review some formulas for non-pointed solids:

• Volume of a rectangular solid: lwh, where lw is the area of the base, and h is the height of the solid
• Volume of a cylinder: , where  is the area of the base, and h is the height of the solid
• Volume of a solid with trapezoidal bases and rectangular sides: , where  is the area of the trapezoid, and h is the height of the solid.

These first three formulas refer to non-pointed figures. Notice that each figure in this category has a volume that can be calculated by multiplying the area of its base by its height.

A rectangular solid is also called a rectangular parallelepiped.

A solid with trapezoidal bases and rectangular sides is also called a trapezoidal parallelepiped.

Next, let’s consider solids that are pointed at the top.

• Volume of a square pyramid: , where s is the length of a side on the square base, s² is the area of the square base, and h is the height of the pyramid
• Volume of a triangular pyramid: , where  is the area of the triangular base, and h is the height of the pyramid
• Volume of a cone: , where  is the area of the circular base, and h is the height of the cone

These three formulas refer to pointed figures. Notice that the volume of each pointed figure is  of the area of the base multiplied by the height.

To find the volume of a non-pointed solid, multiply the area of the base by the height of the solid, measured at a right angle to the base.

To find the volume of a pointed solid, multiply the area of the base by the height of the solid, measured at a right angle to the base, and divide by 3.

What happens when the dimensions of a figure are changed?

• When a two-dimensional figure is changed in size by a factor of n, the area is changed by a factor of n². When a three-dimensional figure is changed in size by a factor of n, the area is changed by a factor of n³.

Take the figure below, for example. The orange rectangle is 4 by 6 and has an area of 24 square units.

The yellow rectangle has the dimensions of the orange rectangle multiplied by a factor of three, which results in dimensions of 12 by 18. The area of this rectangle is 216 square units, which is the area of the orange rectangle multiplied by the square of the factor of three, or nine.

The blue rectangle has the dimensions of the orange rectangle multiplied by a factor of one-half, which results in dimensions of 2 by 3. The area is 6 square units, which is the area of the orange rectangle multiplied by the square of the factor of one-half, or one-fourth.

Let’s say that these rectangles are actually solids. If the orange solid were 4 by 6 by 2, it would have a volume of 48 cubic units.

If the dimensions of the yellow rectangular solid differ from the dimensions of the orange solid by a factor of three, its dimensions would be 12 by 18 by 6. The volume of the yellow rectangular solid would be 1,296 cubic units, which is the volume of the orange rectangular solid multiplied by the cube of the factor of three.

If the dimensions of the blue rectangular solid differ from the dimensions of the orange solid by a factor of one half, the dimensions would be 2 by 3 by 1. The volume of the blue rectangular solid would be 6 cubic units, which is the volume of the orange rectangular solid multiplied by the cube of the factor of one-half.

Review of New Vocabulary and Concepts

• The volume of a solid is measured in cubic units.
• The surface area of a solid is measured in square units.
• The volume of a non-pointed solid is found by multiplying the area of the base by the height of the solid.
• The volume of a pointed solid is found by multiplying the area of the base by the height of the solid and dividing by three.
• The surface area of a solid is found by finding the sum of the areas of all of the surfaces of the solid.
• When the dimensions of a figure are increased by a factor of n, any area is increased by a factor of n², and any volume is increased by a factor of n³.