In a 30-60-90 triangle the longer leg is 15. Find the shorter leg and the hypotenuse.

shorter leg 5sqrt(3), hypotenuse 10sqrt(3)

In a 30-60-90 triangle the hypotenuse is 20. Find the shorter leg and the longer leg.

shorter leg 10, longer leg 10sqrt(3)

In a 30-60-90 triangle the hypotenuse is 22. Find the shorter leg and the longer leg.

shorter leg 11, longer leg 11sqrt(3)

Special right triangles worksheet

Two triangles whose side ratios are worth memorizing: a 45-45-90 triangle has sides x, x and x√2, and a 30-60-90 triangle has sides x, x√3 and 2x. They turn up constantly, and with the ratios in hand no trigonometry is needed.

Not quite the one you need? Same words, different skill:

Right Triangle Trigonometry (Geometry). Ratios from the sides, and sides from an angle.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Special Right Triangles Date Period Find each missing length. Find sides of special right triangles Find the missing side. Find the missing side. Find the missing side. Find the missing side. Find the legs of a special triangle In a 30-60-90 triangle the longer leg is 15. Find the shorter leg and the hypotenuse. In a 45-45-90 triangle the hypotenuse is 18. Find each leg. In a 30-60-90 triangle the hypotenuse is 20. Find the shorter leg and the longer leg. In a 30-60-90 triangle the hypotenuse is 22. Find the shorter leg and the longer leg. Find $x$ and $y$ Find $x$ and $y$ . Find $x$ and $y$ . Find $x$ and $y$ . Solve triangles split by an altitude An altitude of 4 splits a triangle into a 45-45-90 triangle and a 30-60-90 triangle whose $30^{\circ}$ angle is at the base. Find the length of the whole base. The altitude to the base of a triangle is 2, and the base angles are $30^{\circ}$ and $45^{\circ}$ . Find the length of the side that meets the base at the $30^{\circ}$ angle. In triangle $A B C$ , $A = 45^{\circ}$ and $C = 30^{\circ}$ , and the altitude $\overline{B D}$ to $\overline{A C}$ is 6. Find $A B$ . Find $x$ with a radical side Find $x$ . Find $x$ . Find $x$ . Solve chained special right triangles Triangle $A B C$ is a 45-45-90 triangle with legs $8$ . Its hypotenuse $\overline{A C}$ is a leg of 45-45-90 triangle $A C D$ . Find the hypotenuse $A D$ . Triangle $A B C$ is a 30-60-90 triangle with short leg $8$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the hypotenuse $A D$ . Triangle $A B C$ is a 45-45-90 triangle with legs $7$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the long leg of $A C D$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Special Right Triangles

Date Period

Find each missing length.

Find sides of special right triangles
Find the missing side.
Find the missing side.
Find the missing side.
Find the missing side.

Find the legs of a special triangle
In a 30-60-90 triangle the longer leg is 15. Find the shorter leg and the hypotenuse.
In a 45-45-90 triangle the hypotenuse is 18. Find each leg.
In a 30-60-90 triangle the hypotenuse is 20. Find the shorter leg and the longer leg.
In a 30-60-90 triangle the hypotenuse is 22. Find the shorter leg and the longer leg.

Find $x$ and $y$
Find $x$ and $y$ .
Find $x$ and $y$ .
Find $x$ and $y$ .

Solve triangles split by an altitude
An altitude of 4 splits a triangle into a 45-45-90 triangle and a 30-60-90 triangle whose $30^{\circ}$ angle is at the base. Find the length of the whole base.
The altitude to the base of a triangle is 2, and the base angles are $30^{\circ}$ and $45^{\circ}$ . Find the length of the side that meets the base at the $30^{\circ}$ angle.
In triangle $A B C$ , $A = 45^{\circ}$ and $C = 30^{\circ}$ , and the altitude $\overline{B D}$ to $\overline{A C}$ is 6. Find $A B$ .

Find $x$ with a radical side
Find $x$ .
Find $x$ .
Find $x$ .

Solve chained special right triangles
Triangle $A B C$ is a 45-45-90 triangle with legs $8$ . Its hypotenuse $\overline{A C}$ is a leg of 45-45-90 triangle $A C D$ . Find the hypotenuse $A D$ .
Triangle $A B C$ is a 30-60-90 triangle with short leg $8$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the hypotenuse $A D$ .
Triangle $A B C$ is a 45-45-90 triangle with legs $7$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the long leg of $A C D$ .



Sigma Prep · sigmaprep.io/worksheets Name Special Right TrianglesAnswer keyVersion 1 Date Period Find each missing length. Find sides of special right triangles Find the missing side. $7$ Find the missing side. $3 \sqrt{2}$ Find the missing side. $7 \sqrt{3}$ Find the missing side. $7 \sqrt{2}$ Find the legs of a special triangle In a 30-60-90 triangle the longer leg is 15. Find the shorter leg and the hypotenuse.shorter leg $5 \sqrt{3}$ , hypotenuse $10 \sqrt{3}$ In a 45-45-90 triangle the hypotenuse is 18. Find each leg.each leg $9 \sqrt{2}$ In a 30-60-90 triangle the hypotenuse is 20. Find the shorter leg and the longer leg.shorter leg $10$ , longer leg $10 \sqrt{3}$ In a 30-60-90 triangle the hypotenuse is 22. Find the shorter leg and the longer leg.shorter leg $11$ , longer leg $11 \sqrt{3}$ Find $x$ and $y$ Find $x$ and $y$ . $x = 6$ , $y = 6$ Find $x$ and $y$ . $x = 12 \sqrt{3}$ , $y = 24$ Find $x$ and $y$ . $x = 22$ , $y = 11$ Solve triangles split by an altitude An altitude of 4 splits a triangle into a 45-45-90 triangle and a 30-60-90 triangle whose $30^{\circ}$ angle is at the base. Find the length of the whole base. $4 + 4 \sqrt{3}$ The altitude to the base of a triangle is 2, and the base angles are $30^{\circ}$ and $45^{\circ}$ . Find the length of the side that meets the base at the $30^{\circ}$ angle. $4$ In triangle $A B C$ , $A = 45^{\circ}$ and $C = 30^{\circ}$ , and the altitude $\overline{B D}$ to $\overline{A C}$ is 6. Find $A B$ . $6 \sqrt{2}$ Find $x$ with a radical side Find $x$ . $\frac{5 \sqrt{6}}{2}$ Find $x$ . $24$ Find $x$ . $\frac{4 \sqrt{6}}{3}$ Solve chained special right triangles Triangle $A B C$ is a 45-45-90 triangle with legs $8$ . Its hypotenuse $\overline{A C}$ is a leg of 45-45-90 triangle $A C D$ . Find the hypotenuse $A D$ . $16$ Triangle $A B C$ is a 30-60-90 triangle with short leg $8$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the hypotenuse $A D$ . $32$ Triangle $A B C$ is a 45-45-90 triangle with legs $7$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the long leg of $A C D$ . $7 \sqrt{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Special Right TrianglesAnswer keyVersion 1

Date Period

Find each missing length.

Find sides of special right triangles
Find the missing side. $7$
Find the missing side. $3 \sqrt{2}$
Find the missing side. $7 \sqrt{3}$
Find the missing side. $7 \sqrt{2}$

Find the legs of a special triangle
In a 30-60-90 triangle the longer leg is 15. Find the shorter leg and the hypotenuse.shorter leg $5 \sqrt{3}$ , hypotenuse $10 \sqrt{3}$
In a 45-45-90 triangle the hypotenuse is 18. Find each leg.each leg $9 \sqrt{2}$
In a 30-60-90 triangle the hypotenuse is 20. Find the shorter leg and the longer leg.shorter leg $10$ , longer leg $10 \sqrt{3}$
In a 30-60-90 triangle the hypotenuse is 22. Find the shorter leg and the longer leg.shorter leg $11$ , longer leg $11 \sqrt{3}$

Find $x$ and $y$
Find $x$ and $y$ . $x = 6$ , $y = 6$
Find $x$ and $y$ . $x = 12 \sqrt{3}$ , $y = 24$
Find $x$ and $y$ . $x = 22$ , $y = 11$

Solve triangles split by an altitude
An altitude of 4 splits a triangle into a 45-45-90 triangle and a 30-60-90 triangle whose $30^{\circ}$ angle is at the base. Find the length of the whole base. $4 + 4 \sqrt{3}$
The altitude to the base of a triangle is 2, and the base angles are $30^{\circ}$ and $45^{\circ}$ . Find the length of the side that meets the base at the $30^{\circ}$ angle. $4$
In triangle $A B C$ , $A = 45^{\circ}$ and $C = 30^{\circ}$ , and the altitude $\overline{B D}$ to $\overline{A C}$ is 6. Find $A B$ . $6 \sqrt{2}$

Find $x$ with a radical side
Find $x$ . $\frac{5 \sqrt{6}}{2}$
Find $x$ . $24$
Find $x$ . $\frac{4 \sqrt{6}}{3}$

Solve chained special right triangles
Triangle $A B C$ is a 45-45-90 triangle with legs $8$ . Its hypotenuse $\overline{A C}$ is a leg of 45-45-90 triangle $A C D$ . Find the hypotenuse $A D$ . $16$
Triangle $A B C$ is a 30-60-90 triangle with short leg $8$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the hypotenuse $A D$ . $32$
Triangle $A B C$ is a 45-45-90 triangle with legs $7$ . Its hypotenuse $\overline{A C}$ is the short leg of 30-60-90 triangle $A C D$ . Find the long leg of $A C D$ . $7 \sqrt{6}$

How to do these

A 30-60-90 triangle has its short leg $5$ . Find the hypotenuse

In a 30-60-90 the hypotenuse is twice the short leg.
2 times 5.

The answer is $10$ .

Where students go wrong

Doubling the wrong side. It is the short leg that doubles to give the hypotenuse, not the long one. The long leg is the short leg times root 3.

Also called 45 45 90 triangle, 30 60 90 triangle or special triangles.

What you can put on this worksheet

45-45-90 and 30-60-90: Find the missing side. — a right triangle with base unknown, height 6 and hypotenuse x → $12$
From the long leg or the hypotenuse: In a 45-45-90 triangle the hypotenuse is 16. Find each leg. → each leg $8 \sqrt{2}$
Both missing sides at once: Find $x$ and $y$ . — a right triangle with base 8√3, height x and hypotenuse y → $x = 8$ , $y = 16$
Two special triangles sharing an altitude: An altitude of 9 splits a triangle into a 45-45-90 triangle and a 30-60-90 triangle whose $30^{\circ}$ angle is at the base. Find the hypotenuse of the 45-45-90 triangle. → $9 \sqrt{2}$
Special right triangles with a radical side given: Find $x$ . — a right triangle, labeled 6√2, x, 45° → $6$
Two special right triangles chained: Triangle $A B C$ is a 45-45-90 triangle with legs $6$ . Its hypotenuse $\overline{A C}$ is a leg of 45-45-90 triangle $A C D$ . Find the hypotenuse $A D$ . → $12$

Questions about these worksheets

Yes. Every special right triangles sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the special right triangles questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: 45-45-90 and 30-60-90, from the long leg or the hypotenuse, both missing sides at once, two special triangles sharing an altitude, special right triangles with a radical side given and two special right triangles chained. Tick as many as you want and set how many of each, or let it spread them evenly.

Doubling the wrong side. It is the short leg that doubles to give the hypotenuse, not the long one. The long leg is the short leg times root 3.

Special right triangles worksheet

Find sides of special right triangles

Find the legs of a special triangle

Find xxx and yyy

Solve triangles split by an altitude

Find xxx with a radical side

Solve chained special right triangles

Find sides of special right triangles

Find the legs of a special triangle

Find xxx and yyy

Solve triangles split by an altitude

Find xxx with a radical side

Solve chained special right triangles

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these special right triangles worksheets free?+

Do the special right triangles worksheets come with an answer key?+

Are these special right triangles worksheets printable?+

What grade are these special right triangles worksheets for?+

Can I get a different version of the same special right triangles worksheet?+

Is this a free alternative to Kuta Software for special right triangles?+

Can I choose how hard the special right triangles questions are?+

What kinds of question are on a special right triangles worksheet?+

What do students get wrong with special right triangles?+

Find $x$ and $y$

Find $x$ with a radical side

Find $x$ and $y$

Find $x$ with a radical side