# Why is the area of a $3$-by-$3$ square $3\times 3$ and not $3+3$?

When we need to find the area of a square, we multiply the sides. For example, the area of a square with one side as $$3$$ cm will be $$3\times 3 \text{ cm}^2$$.

My question is: What is the logic behind this? Why can't we just add the sides and get the area? For example, $$3+3=6\text{ cm}^2$$.

Thanks!

• I suggest you read and contemplate on the Wikipedia page: en.wikipedia.org/wiki/Area – Matti P. Nov 6 '18 at 8:39
• The area of a square should be proportional to both sides. (If we duplicate both sides, the area is multiplied by $4$) – ajotatxe Nov 6 '18 at 8:39
• Have you tried drawing a $3\times3$ square and counting how many $1\times1$ squares can fit inside it? – Rahul Nov 6 '18 at 8:39
• Here is a link: skillsyouneed.com/num/area.html – Sujit Bhattacharyya Nov 6 '18 at 8:42
• Okay, so consider that $3$-by-$3$ square. Paste four of them together edge-to-edge in the right way, and you'll get a $6$-by-$6$ square. If there's any justice, the area of the new square should be four-times that of the old square. However, $$6+6=12 \neq 24 = 4\cdot (3+3)$$ So, whatever rule there might be for calculating the area of a square, "add two sides" isn't it. – Blue Nov 6 '18 at 8:43

## 2 Answers

It is good to ask why! That is how all the science was discovered. For simplicity, one could think of the "area" in (centimeters, for example) as:

the number of squares of size 1 cm required to occupy a given region.

To be more accurate the region has to have no depth, it has to be flat, i.e. 2-dimensional only.

In the example below we have an Orange square of area $$1\ \mathrm{cm}^2$$ ($$1\ \mathrm{cm}$$ squared) and we want to fill the big square with such Orange squares. The area represents the number of Orange squares required.

You can count the number manually one or notice that:

The required number is $$3+3+3=9$$

The number $$9$$ here tells us that we need $$9$$ such Orange tiles (squares).

There is no magic, its all about the definition you set for the area. People agree on one definition and then device means to calculate the value corresponding to the definition.

or use multiplication to speed things up to get the correct answer $$3 \times 3=9$$

I hope this helps a bit. Edit: In response to the comment, here is what a $$25m$$ by $$0.5m$$ would look like after filling it with square tiles each having side of $$1m$$. Each tile can be split in half. In the drawing, the rectangle uses 2 halves except at the last tile, it uses only 1 half a tile. You don't need further adjustments. • Thanks a lot for your initial words of encouragement! What does it mean if a country's area is ,let's suppose, 23433323 Km^2 ? Does it mean the country's land can contain that many(23433323) squares and the area of each square is 1 m^2 ? Kindly help! I think my concept of 'what area really is' is not strong enough. Please help. – Sunny Nov 7 '18 at 15:41
• You are welcome. You are correct. A country is considered an "irregular shape", we can't figure the area of an irregular shape via formula directly. However, the meaning of the area is the same. If the country has an area of X $Km^2$, then you could fit X tiles (squares) each of side length $1$Km. If you really try to fill an irregular shape with square tiles, you will have to chip off some, because of the irregular shape but that won't affect the area figure. See an example here:sierra.nmsu.edu/morandi/coursematerials/areasofleaves.html – NoChance Nov 7 '18 at 21:36
• Thanks for explaining! The link was very insightful. So Can I conclude to this definition - "Finding the area of an object, shape, or a piece of land is actually to find how many squares can be fit into it" ? If area of a circle is 154cm^2, this means 154 squares can fit into it each with 1 cm^2 of area ? Am I correct all the way here ? – Sunny Nov 8 '18 at 1:03
• Generally yes, but the unit squares are not all going to be fitted in as complete tiles/squares if the shape is irregular, some squares/tiles will need to be cut to cover the curves, etc. In case of a circle, since $pi$ is not a finite number the area of a circle for some radius values may be a number with so many decimals that you can not fill completely in practice, but it looks you get the point. – NoChance Nov 8 '18 at 1:15
• Things are getting clear to me. Could you explain to me 'how many squares can be fitted and how' in a rectangle of 25 m by 0.5 m ? ( Will this rectangle need to be adjusted to a proper shape where it can make room for a 1m^2 because it's one side is 0.5 m which is less than 1 meter ? – Sunny Nov 8 '18 at 1:23

You can fit $$9$$ one-by-one squares into one three-by-three square.

What we imagine as "area" has the property that if something has area $$x$$, and some other thing has area $$y$$, then the two things together have area $$x+y$$. This propety can only hold for squares if their area is the product, not sum of their side lengths.

• I can't figure out the reason for this answer. Maybe I'm missing an understand of one of the ways people sometimes think that you have. Maybe you're thinking some people might see that a 2 by 2 square can be divided into 2 2 by 1 rectangles and so has an area equal to the sum of the length of an edge and an adjacent edge and so they make the mistake that that's also the case for a 3 by 3 square. – Timothy Jun 27 at 2:31
• @Timothy (1) Do you agree that you can fit 9 1x1 squares into one 3x3 square? (2) do you agree that the area of two things together is the sum of both their areas? If you answer yes to both questions, then you MUST agree that a 3x3 square has area 9. – 5xum Jun 27 at 6:31