Lesson 1: Side Lengths and Areas of Squares

Property

The area ($A$) of a square is found by squaring its side length ($s$):

To find the side length of a square given its area, you take the square root of the area:

Examples

Property

This geometric method finds the length of a tilted line segment by constructing a square. The area of the tilted square is found by subtracting the areas of the four corner triangles from the area of a larger, surrounding square. The side length of the tilted square is the square root of its calculated area. For example, a tilted square with an area of 2 square units has a side length of $\sqrt{2}$.

Examples

• A tilted square is inside a $4 \times 4$ square, with its corners at the midpoints of the outer sides. The large area is 16. The four corner triangles each have area $\frac{1}{2} \times 2 \times 2 = 2$. The tilted square's area is $16 - 4(2) = 8$. Its side length is $\sqrt{8}$.
• A tilted square is inside a $5 \times 5$ square. Its corners are 1 unit from each corner of the larger square. The large area is 25. The four corner triangles are $1 \times 4$ right triangles, each with area $\frac{1}{2}(1)(4) = 2$. The tilted square's area is $25 - 4(2) = 17$. Its side length is $\sqrt{17}$.
• A large square has side length 6. A tilted square inside has corners that divide the sides into segments of length 2 and 4. The large area is 36. The four corner triangles have area $\frac{1}{2}(2)(4) = 4$. The tilted square's area is $36 - 4(4) = 20$. Its side length is $\sqrt{20}$.

Explanation

This is a clever geometric trick! By drawing a tilted square inside a bigger, straight one, we can find its area by subtraction. The side length of this tilted square is then simply the square root of that area.

Property

If $n^2 = m$, then $m$ is the square of $n$.
If $n^2 = m$, then $n$ is a square root of $m$.

Square Root Notation

$\sqrt{m}$ is read "the square root of $m$".
If $n^2 = m$, then $n = \sqrt{m}$, for $n \geq 0$.
The symbol $\sqrt{\phantom{m}}$ is called a radical sign. The expression under the radical sign is called the radicand. The positive square root is also called the principal square root.