# Basic Formulas for Two- and Three-Dimensional Figures

Instructor: David Karsner
Geometric measurements can be taken for one-, two-, and three-dimensional shapes. This lesson explores the most common formulas one would use to find perimeter, area, surface area, and volume of several two- and three-dimensional figures.

## How Big is the Pool?

You have been given the responsibility of adding the right amount of chemicals to the pool to keep it safe. The only problem is that the directions on the chemical box are based on how large the pool is, and you don't know how large it is. The pool is in the shape of a box. You have a one cubic foot bucket and a 50 foot measuring tape. How would you determine the size of the pool?

One option is to dip the water out the pool with the one cubic foot bucket and keep count on how many buckets you dip out of the pool. This option is very time consuming and what would you do with all the water you are dipping out of the pool.

Another option is to use the formula for the volume of a prism. That just means making three measurements and doing a little multiplication. Which option sounds better?

Using a formula to find volume or area is a handy short-cut in a situation like this. This lesson will give you those formulas and explain how they are used.

## One-, Two-, and Three-Dimensional Measurements

One of the main tasks in geometry is measuring figures. You can make these measurements in one-, two-, and three-dimensional space. In one dimension, you are measuring distance (length). In two dimensions, you are measuring area (length and width). In three dimensions, you are measuring volume (length, width, and height).

Even when a figure is two- or three-dimensional, you can take measurements of the lower dimensions as well. Let's use a cylinder as an example. It is a three-dimensional figure. You can find the volume of the cylinder (a three-dimensional measurement), the area of the circular base (a two-dimensional measurement) and the height of the cylinder (a one-dimensional measurement).

## Finding Distance

Distance is a one-dimensional measurement of the amount of space between two points. If you start at point A and move to point B, how far have you moved? The units for distance (length) are given in single units, such as 14 miles, 200 feet, 32 inches, 55 meters, or 200 kilometers.

Even though distance is a one-dimensional measurement, the measure of distance can be used for two- and three-dimensional figures. The perimeter of a rectangle, the radius of a circle, the height of a pyramid are all examples of one-dimensional measurements for two- or three-dimensional figures. Distance is often used to describe perimeters, or the space around the outside of a shape. Circumference is the name given to the perimeter of a circle. Any shape that includes a circle will use Pi (3.14) as part of the formula.

## Finding Area

Area is the amount of space that a two-dimensional shape takes up. Since it is two-dimensional, the units will always be square units (for example, 12 sq. feet or 25 sq. centimeters). To find the area of something, you start by multiplying the two dimensions together.

To find the area of a rectangle, you multiply the length by the width. Although area is a two-dimensional measurement, it can also be used with three-dimensional objects. You can find the area of the base (two-dimensional) of a cone (three-dimensional).

## Finding Volume

Volume is the amount of space inside of a three-dimensional object. Since it is three-dimensional, volume measurements will always be in cubic units (such as cubic feet or cubic liters). To find the volume of many three-dimensional shapes, you begin by finding the area (two-dimensional) of the base and multiplying it by the height.

## Solving Problems

To find measurements of perimeter, area, and volume follow these steps:

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