\[A=s^2\]

\[A=(s)(s)\]

## How to Use the Area of a Square Formula

- Identify what measurements you were given.
- Identify which measurement you were asked to find.
- Substitute the given measurements into the Area of a Square Formula and simplify or solve the equation until you find the measurement you are looking for.

## Examples

Find the side length of the square.

*What measurements were you given?*

The diagram tells me that the area of the square is \(9\, in^2\).

\[\blue A=9\]

*Which measurement were you asked to find?*

The measurement I want to find is the side length of the square.

\[\blue s=s\]

*Do you have all the variables required for the Area of a Square Formula?*

Yes, in Steps 1 and 2, I defined both variables:

\[\blue A=9\]

\[\blue s=s\]

When I substitute these values into the Area of a Square Formula, I get…

\[{\blue 9}={\blue s}^2\]

I can solve this equation by square rooting both sides.

\[3=s\]

The side length of the square is \(\blue 3\, in\).

Find the area of the square.

*What measurements were you given?*

The diagram tells me that the diagonal of the square is \(12 \, cm\).

The Area of a Square Formula does not have a variable for the diagonal of the square, but I can use the diagonal and the Pythagorean Theorem to find the side length of the square.

\[s^2+s^2=12^2\]

\[2s^2=144\]

\[s^2=72\]

\[\green s=\sqrt{72}\]

*Which measurement were you asked to find?*

The measurement I want to find is the area of the square.

\[\green A=A\]

*Do you have all the variables required for the Area of a Square Formula?*

Yes, in Steps 1 and 2, I defined both variables:

\[\green s=\sqrt{72}\]

\[\green A=A\]

When I substitute these values into the Area of a Square Formula, I get…

\[{\green A}={\green \sqrt{72}}^2\]

I can simplify the right hand side of the equation. The square root and exponent of 2 cancel each other out, which gives me…

\[{\green A}={\green 72}\]

The area of the square is \(\green 72\, cm^2\).

Is this shape a square?

*What measurements were you given?*

The diagram tells me that the side length of the shape is \(4 \, ft\) and the area is \(15 \, ft^2\).

\[\purple s=4\]

\[\purple A=15\]

*Which measurement were you asked to find?*

I was actually not asked to find a measurement.

Instead, I was asked to determine if the shape is a square or not.

The shape does have right angles, so if I can show that all the sides are the same length, then that will prove that it is a square.

*Do you have all the variables required for the Area of a Square Formula?*

Yes, in Steps 1 and 2, I defined both variables:

\[\purple s=4\]

\[\purple A=15\]

When I substitute these values into the Area of a Square Formula, I get…

\[{\purple 15}={\purple 4}^2\]

When I simplify the right hand side of the equation, I get…

\[{\purple 15}=16\]

This shape is not a square because the area and side length measurements do not satisfy the Area of a Square Formula.

## Area of a Square Formula Proof

Area is a two-dimensional measurement that measures how many unit cubes will fit into two-dimensional space.

The top or bottom side length determines how many unit squares are in one row.

In this example, there are 4 unit squares in one row.

The left or right side length of a square determines how many rows of unit squares are in the square.

In this example, there are 4 rows of unit squares.

If you multiply the number of unit squares in one row by the number of rows, that will give you the total number of unit squares in the shape.

In this example, that would be…

\[\yellow 4\times 4 = 16\]

All squares have equal side lengths, so the area of any square with side length \(s\) is…

\[\yellow A = (s)(s)\]

Or you can simplify the formula and write the repeated multiplication as an exponent…

\[\yellow A=s^2\]