Special right triangles: 45-45-90 and 30-60-90

Q: Why is the hypotenuse of a 45-45-90 triangle a leg times the square root of 2?

Both legs are x, so the Pythagorean theorem gives x^2 + x^2 = 2x^2 for the hypotenuse squared. The square root of 2x^2 is x\sqrt{2}. That is also why the diagonal of any square is its side times \sqrt{2}.

Q: Do I have to rationalize the denominator?

The value is the same either way: \frac{12}{\sqrt{2}} and 6\sqrt{2} are equal. Many teachers want the rationalized form, and answer choices are usually written that way, so convert before you compare. Multiply the top and bottom by the root on the bottom.

The key idea

45-45-90: half of a square

Cut a square along its diagonal and you get two 45-45-90 triangles. The legs are equal. The Pythagorean theorem gives the hypotenuse: $x^2 + x^2 = 2x^2$ , so the hypotenuse is $x\sqrt{2}$ .

\text{leg} : \text{leg} : \text{hypotenuse} = x : x : x\sqrt{2}

30-60-90: half of an equilateral triangle

Cut an equilateral triangle with side 2x straight down the middle. Each half has a hypotenuse of 2x, a short leg of x (half the base) and a long leg of $x\sqrt{3}$ (the height).

\text{short leg} : \text{long leg} : \text{hypotenuse} = x : x\sqrt{3} : 2x

The short leg is always across from the 30° angle, and the long leg is across from the 60° angle. Smallest angle, smallest side.

Going from the side you know to the side you want
Triangle	You know	Do this
45-45-90	A leg	Hypotenuse = leg $\times \sqrt{2}$
45-45-90	The hypotenuse	Leg = hypotenuse $\div \sqrt{2}$
30-60-90	The short leg	Long leg = short leg $\times \sqrt{3}$ , hypotenuse = short leg $\times 2$
30-60-90	The hypotenuse	Short leg = hypotenuse $\div 2$ , then use the row above
30-60-90	The long leg	Short leg = long leg $\div \sqrt{3}$ , then use the rows above

Dividing by a square root leaves a root on the bottom of a fraction. To rationalize, multiply the top and bottom by that root. The value does not change, because you are multiplying by 1.

\frac{12}{\sqrt{2}} = \frac{12}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}

Worked examples

Example 1: 45-45-90, leg to hypotenuse

Problem A 45-45-90 triangle has legs of 7. Find the hypotenuse.

In a 45-45-90 triangle, the hypotenuse is a leg times $\sqrt{2}$ .
$c = 7\sqrt{2}$
Check with the Pythagorean theorem: both sides come to 98.
$7^2 + 7^2 = 98 = (7\sqrt{2})^2$

Answer $7\sqrt{2}$ , about 9.90.

Example 2: 45-45-90, hypotenuse to leg

Problem The hypotenuse of a 45-45-90 triangle is 12. Find the length of each leg.

Call each leg x. The hypotenuse is a leg times $\sqrt{2}$ .
$x \sqrt{2} = 12$
Divide both sides by $\sqrt{2}$ .
$x = \frac{12}{\sqrt{2}}$
Rationalize: multiply the top and bottom by $\sqrt{2}$ , then simplify.
$x = \frac{12\sqrt{2}}{2} = 6\sqrt{2}$

Answer Each leg is $6\sqrt{2}$ , about 8.49.

Example 3: 30-60-90, hypotenuse to legs

Problem The hypotenuse of a 30-60-90 triangle is 18. Find both legs.

The short leg is half the hypotenuse.
$\frac{18}{2} = 9$
The long leg is the short leg times $\sqrt{3}$ .
$9 \cdot \sqrt{3} = 9\sqrt{3}$
Size check: $9\sqrt{3}$ is about 15.6, which sits between 9 and 18, as the middle side should.

Answer Short leg 9, long leg $9\sqrt{3}$ (about 15.59).

Example 4 (test-hard): an equilateral triangle from its height

Problem An equilateral triangle has a height of 9. Find its side length and its area.

The height splits the triangle into two 30-60-90 triangles. The height is the long leg, across from the 60° angle.
Short leg = long leg divided by $\sqrt{3}$ . Rationalize.
$\frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}$
A side of the equilateral triangle is the hypotenuse, which is twice the short leg.
$2 \cdot 3\sqrt{3} = 6\sqrt{3}$
Area is half of base times height.
$\frac{1}{2} \cdot 6\sqrt{3} \cdot 9 = 27\sqrt{3}$

Answer Side $6\sqrt{3}$ (about 10.39) and area $27\sqrt{3}$ (about 46.77).

Common mistakes

Putting $\sqrt{3}$ on the hypotenuse. In a 30-60-90 triangle the hypotenuse is 2x and the long leg is $x\sqrt{3}$ . Fix: the hypotenuse is the longest side, and 2 is bigger than $\sqrt{3} \approx 1.73$ .
Mixing up the legs. The short leg is across from 30°, not next to it. Fix: find the 30° angle and look straight across.
Multiplying when you should divide. Going from the hypotenuse to a leg makes the side shorter, so you divide by $\sqrt{2}$ (or by 2). Fix: check that legs come out shorter than the hypotenuse.
Using the 45-45-90 ratio on a 30-60-90 triangle. Only a triangle with two equal legs gets the $\sqrt{2}$ rule.
Rounding too early. Keep $\sqrt{2}$ and $\sqrt{3}$ exact until the last step, then round once.

Quick methods

Practice

5 practice questions

A 45-45-90 triangle has legs of 9. What is the hypotenuse?
1. $9\sqrt{2}$
2. $9\sqrt{3}$
3. $18$
4. $\frac{9\sqrt{2}}{2}$
Show answer
Answer: $9\sqrt{2}$
The hypotenuse is a leg times $\sqrt{2}$ . $9\sqrt{3}$ uses the 30-60-90 ratio, 18 doubles like a 30-60-90 hypotenuse, and $\frac{9\sqrt{2}}{2}$ divides by $\sqrt{2}$ when you should multiply.
The short leg of a 30-60-90 triangle is 5. What is the long leg?
1. $10$
2. $5\sqrt{2}$
3. $5\sqrt{3}$
4. $10\sqrt{3}$
Show answer
Answer: $5\sqrt{3}$
The long leg is the short leg times $\sqrt{3}$ . 10 is the hypotenuse, $5\sqrt{2}$ uses the 45-45-90 ratio, and $10\sqrt{3}$ doubles before multiplying.
In a 30-60-90 triangle, the side across from the 60° angle is 12. What is the hypotenuse?
1. $24$
2. $8\sqrt{3}$
3. $4\sqrt{3}$
4. $12\sqrt{3}$
Show answer
Answer: $8\sqrt{3}$
The short leg is $\frac{12}{\sqrt{3}} = 4\sqrt{3}$ , and the hypotenuse is twice that, $8\sqrt{3}$ (about 13.86). 24 treats 12 as the short leg, $4\sqrt{3}$ is the short leg, and $12\sqrt{3}$ multiplies by $\sqrt{3}$ instead of dividing.
A square has a diagonal of 14. What is its area?
1. $49$
2. $98$
3. $196$
4. $98\sqrt{2}$
Show answer
Answer: $98$
The diagonal splits the square into two 45-45-90 triangles, so each side is $\frac{14}{\sqrt{2}} = 7\sqrt{2}$ . The area is $(7\sqrt{2})^2 = 98$ . 196 squares the diagonal, 49 squares half of it, and $98\sqrt{2}$ multiplies the side $7\sqrt{2}$ by the diagonal 14.
An equilateral triangle has sides of 8. What is its height? Give an exact answer.
Show answer
Answer: $4\sqrt{3}$
The height splits the triangle into two 30-60-90 triangles with hypotenuse 8 and short leg 4. The height is the long leg, $4\sqrt{3}$ , about 6.93.

Frequently asked questions

How do I remember which side gets the square root of 3?

Picture half of an equilateral triangle. The hypotenuse is a full side, 2x. The short leg is half a side, x. The only side left, the height, gets $x\sqrt{3}$ . Since $\sqrt{3} \approx 1.73$ is between 1 and 2, the long leg sits between the other two sides.

Why is the hypotenuse of a 45-45-90 triangle a leg times the square root of 2?

Both legs are x, so the Pythagorean theorem gives $x^2 + x^2 = 2x^2$ for the hypotenuse squared. The square root of $2x^2$ is $x\sqrt{2}$ . That is also why the diagonal of any square is its side times $\sqrt{2}$ .

Do I have to rationalize the denominator?

The value is the same either way: $\frac{12}{\sqrt{2}}$ and $6\sqrt{2}$ are equal. Many teachers want the rationalized form, and answer choices are usually written that way, so convert before you compare. Multiply the top and bottom by the root on the bottom.

How do I know a triangle is 30-60-90 if the angles are not labeled?

Look at the sides. If the hypotenuse is exactly twice one leg, it is a 30-60-90 triangle. If the two legs are equal, it is 45-45-90. The height of an equilateral triangle and the diagonal of a square also make these triangles, so watch for those shapes.