Associated Topics || Dr. Math Home || Search Dr. Math

### Defining a Square Centimeter

```Date: 04/09/2002 at 12:44:34
From: Young Kim
Subject: Geometry - Definition of 2-D Area

Dear Dr.Math,

I know that area is defined by the number of squares that cover a
given section of the x-y plane. And we use unit square to cover the
section to be measured. And books I've looked in say that the area of
a unit square is 1 square centimeter because the two sides of the unit
square are each 1 cm and we just multiply the two sides. But I cannot
understand!

It seems to me that the whole point of measuring area comes down to
measuring the area of a square of l cm length and width of 1 cm. And
if we just multiply length and width to get the area of the square
without knowing why, then we are back to the starting point!

My question is, Why is the area of a unit square the product of the
two sides? By unit square, I mean a square of 1 cm length and 1 cm
width.

Kim
```

```
Date: 04/09/2002 at 12:58:59
From: Doctor Peterson
Subject: Re: Geometry - Definition of 2-D Area

Hi, Young Kim.

Before we can measure anything, we have to define the unit with which
we will measure it. In this case, we define something called a square
centimeter as the area of a one-centimeter square. That doesn't need
proof, since the concept doesn't exist until we make this definition.

Given that definition, we can find the area of any rectangle (with
integral sides, to start with) by laying out H rows of W unit squares.
We can count them by multiplying, and find that the area is then W*H.

If we then apply this to the unit square, we of course get 1*1 = 1,
but this just shows that our calculation is consistent with the
definition. It also motivates our notation, where we write a square
centimeter as cm^2 (with an exponent) because

1 cm * 1 cm = (1*1)(cm*cm) = 1 cm^2

The fact that we are multiplying two centimeter measures makes it
reasonable to call this area 1 cm^2. But again, that is not the
definition of a square centimeter, only a conclusion from the
definition.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/13/2002 at 14:53:23
From: Young Kim
Subject: Geometry - Definition of 2-D Area

Dear Dr.Peterson,

explanation resolved my confusion once and for all. I really

Best Regards,
Young Kim
```
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
Middle School Definitions
Middle School Two-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search