Printable pre-algebra, algebra and geometry worksheets

Special Right Triangles Worksheet Generator

Make printable 45-45-90 and 30-60-90 triangle practice – each problem drawn with its given side, answers in simplest radical form on a separate key.

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This free special right triangles worksheet generator makes printable practice sheets for 45-45-90 and 30-60-90 triangles, with an answer key. It runs in your web browser, needs no sign-up, and nothing is uploaded, because the problems are made on your device. Pick the problem types you want, set the difficulty and the number of problems, and print the sheet or save it as a PDF. Each sheet can include finding missing sides, side ratios, word problems and area and perimeter questions. Teachers can make several versions for one class, a single type gives a 30 60 90 triangle worksheet or a 45 45 90 triangle worksheet, and everything you make is free to use in class or in products you sell, such as Teachers Pay Teachers resources.
Special right triangles worksheet example

How to use the Special Right Triangles Worksheet Generator

  1. Tick one or more boxes under Problem types. The 45-45-90 and 30-60-90 side-finding types are ticked when the page opens. If you tick several, the problems are shared evenly between them and grouped by type.
  2. Choose a Difficulty: Easy, Medium or Hard.
  3. Choose the Number of problems: 6, 8, 10, 12, 15, 20, 24 or 30.
  4. Open More options to set the Columns (Automatic, 1, 2 or 3), the Paper size (Letter or A4), a Worksheet title, and whether to add a Name and date line. Leave Answer key on a separate page ticked to keep the key, or untick it.
  5. Check the preview on the right. It shows the sheet as it will print, with every side-finding problem drawn as a labelled triangle.
  6. Click Show answers to see each answer under its problem in the preview. Click it again to hide the answers.
  7. Click Print / Save as PDF. In the print dialog, choose Save as PDF if you want a file instead of paper.

Click New worksheet to get a fresh random set with the same settings. Click Copy link to copy a link that opens exactly the same worksheet again, since the problem set is stored in the link as a number.

What are special right triangles?

Special right triangles are right triangles whose side lengths follow fixed ratios, so you can find every side from just one side. There are two types: the 45-45-90 triangle and the 30-60-90 triangle. Using these ratios lets you give exact answers with radicals instead of decimals.

In a 45-45-90 triangle, the two legs are equal, and the hypotenuse is a leg times √2. The sides are in the ratio x : x : x√2. If you know the hypotenuse h, each leg is h/√2, which is the same as h√2/2. For example, if the hypotenuse is 6, each leg is 6√2/2 = 3√2.

In a 30-60-90 triangle, the sides are x (the short leg, opposite 30°), x√3 (the long leg, opposite 60°) and 2x (the hypotenuse). The hypotenuse is always the longest side, and the short leg is always the shortest.

How do you find the sides of a 30-60-90 triangle?

Find x first, then use the ratio to get the other two sides. Start by checking which side you are given.

  1. If you have the hypotenuse, divide it by 2 to get the short leg x. Then multiply x by √3 to get the long leg.
  2. If you have the short leg, multiply it by √3 to get the long leg, and multiply it by 2 to get the hypotenuse.
  3. If you have the long leg, divide it by √3 to get x. Then the short leg is x and the hypotenuse is 2x.
  4. Rationalize any denominator. For example, if the long leg is 10, then x = 10/√3 = (10√3)/3. The hypotenuse is 2x = (20√3)/3.

If the hypotenuse is 14, the short leg is 7 and the long leg is 7√3. You can check your work by confirming that the short leg, the long leg and the hypotenuse follow the ratio x : x√3 : 2x.

Sample problems from this generator

  1. x1445°45° Answer: x = 7√2
  2. x1245°45° Answer: x = 6√2
  3. yx660°30° Answer: x = 3√3, y = 3
  4. yx630°60° Answer: x = 3, y = 3√3

Frequently asked questions

What is simplest radical form?

A radical is a root such as √2 or √3. An expression is in simplest radical form when the number under the root has no square factors, so √12 becomes 2√3, and no radical stays in the denominator. The answers on this tool follow that rule, so 10/√3 is written as (10√3)/3.

Can I sell these special right triangles worksheets as a printable on Etsy or TpT?

Yes. Everything you make here is royalty-free for personal, classroom and commercial use, including Teachers Pay Teachers, Etsy printables, KDP workbooks and print-on-demand products. No credit is required. The only thing you may not do is copy the generator or the site itself to run a competing service. See the terms.

What do the word problems ask for?

Word problems give the exact answer in simplest radical form. If the answer contains a radical, they also give a decimal rounded to the nearest tenth. For example, a square with sides of 6 feet has a diagonal of 6√2 ≈ 8.5 feet.

Does the tool work on a phone or tablet?

Yes, it works on phones and tablets as well as computers. Printing needs a printer, or you can use the Save as PDF option in the print dialog.

Are my settings saved?

Yes. Your settings are remembered in your browser for your next visit. If you open a shared link, the worksheet uses that link’s settings instead.