We describe a nice way to do it, unfortunately in words. It really needs a picture.
Put down your cardboard rectangle, one corner at the origin, the long side along the positive $x$-axis. So the corners of your cardboard rectangle are at $(0,0)$, $(0,45)$, $(45,20)$, and $(0,20)$.
Draw a $30\times 30$ square, with corners $(0,0)$, $(30,0)$, $(30,30)$, and $(0,30)$.
Draw the line that joins $(0,30)$ to $(45,0)$.
This line will meet the top side of your cardboard rectangle at $P=(15,20)$, and will meet the right side of the square at $Q=(30,10)$. Let $R=(30,0)$ and $S=(45,0)$.
All set up! Use a razor knife to cut along the line $PS$. That will slice a substantial triangle from the cardboard. Leave it in place for now.
Use the razor knife to cut straight down along $QR$. This slices off a smallish triangle from the cardboard.
Slide the big triangle upward until its top side agrees with the top line of the square. It will.
Slide the little triangle way up so that it fills in the top left corner of the square. It will.
Done, two cuts.
It is a very pretty construction, works uniformly for all rectangles that are not too skinny. If the rectangle is very skinny, a not too hard adjustment can be made.
You will have to prove that this works. Straight coordinate or similar triangle geometry.
Remark: This construction is one of the steps in the proof of the Bolyai-Gerwien Theorem (which, as is so often the case, was proved a number of years earlier by at least two other people). The result is that if $A$ and $B$ are any polygonal regions with the same area, then $A$ can be cut into a finite number of polygonal pieces that can be reassembled to maske $B$.
Free square root worksheets (PDF and html)
On this page, you'll find an unlimited supply of printable worksheets for square roots, including worksheets for square roots only (grade 7) or worksheets with square roots and other operations (grades 8-10). Options include the radicand range, limiting the square roots to perfect squares only, font size, workspace, PDF or html formats, and more.
If you want the answer to be a whole number, choose "perfect squares," which makes the radicand to be a perfect square (1, 4, 9, 16, 25, etc.). If you choose to allow non-perfect squares, the answer is typically an unending decimal that is rounded to a certain number of digits.
The option "Only simplify, no answers as decimals" forces the answer NOT to be given as a rounded decimal, but instead the answer is simplified if possible, and the square root is left in the answer if it cannot be simplified. For example, an answer of √28 will be given in simplified form as 2√7. This option is useful for algebra 1 and 2 courses.
You can also make worksheets that include one or two other operations, besides taking a square root.
Basic instructions for the worksheets
Each worksheet is randomly generated and thus unique. The answer key is automatically generated and is placed on the second page of the file.
You can generate the worksheets either in html or PDF format — both are easy to print. To get the PDF worksheet, simply push the button titled "Create PDF" or "Make PDF worksheet". To get the worksheet in html format, push the button "View in browser" or "Make html worksheet". This has the advantage that you can save the worksheet directly from your browser (choose File → Save) and then edit it in Word or other word processing program.
Sometimes the generated worksheet is not exactly what you want. Just try again! To get a different worksheet using the same options:
- PDF format: come back to this page and push the button again.
- Html format: simply refresh the worksheet page in your browser window.
Ready-made square root worksheets
Use the generator to make customized worksheets for square roots.