What are the side ratios for special right triangles?

A 45-45-90 triangle has sides in the ratio 1 : 1 : square root of 2, where the two legs are equal and the hypotenuse is a leg times the square root of 2. A 30-60-90 triangle has sides in the ratio 1 : square root of 3 : 2, anchored on the short leg opposite the 30 degree angle.

How do you find the leg of a 45-45-90 triangle from the hypotenuse?

Divide the hypotenuse by the square root of 2, then rationalize. For example, a hypotenuse of 10 gives a leg of 10 divided by the square root of 2, which simplifies to 5 times the square root of 2.

When can you use special right triangle ratios instead of trigonometry?

Use them whenever a right triangle has angles of exactly 45-45-90 or 30-60-90. In those cases the fixed side ratios give you every side instantly, saving the time you would spend on the Pythagorean theorem or a trig function.

Where do special right triangles show up if a problem never names them?

The diagonal of a square creates two 45-45-90 triangles, and the altitude of an equilateral triangle creates two 30-60-90 triangles. Recognizing these hidden triangles lets you find diagonals and heights instantly.

CramClub • Special Right Triangles — SAT® Guide & Practice

$\text{leg} : \text{leg} : \text{hyp} = 1 : 1 : \sqrt{2}$

$\text{short} : \text{long} : \text{hyp} = 1 : \sqrt{3} : 2$

$h = \dfrac{s\sqrt{3}}{2}$

Two Triangles Worth Memorizing

The SAT reaches for two right triangles over and over, and both have side ratios that never change: the 45-45-90 and the 30-60-90. Memorize these ratios and you can find every side from a single given length — usually without a calculator and far faster than the Pythagorean theorem or trigonometry. Both appear on the exam's reference sheet, but real fluency beats stopping to look them up, and it lets you recognize these triangles hidden inside squares, equilateral triangles, and coordinate figures.

The 45-45-90 Triangle — Ratio $1 : 1 : \sqrt{2}$

A 45-45-90 triangle is an isosceles right triangle: the two legs are equal, and each acute angle is $45^\circ$ . Its sides always follow

$\text{leg} : \text{leg} : \text{hypotenuse} = 1 : 1 : \sqrt{2}.$

So if a leg is $5$ , the other leg is $5$ and the hypotenuse is $5\sqrt{2}$ . To reverse the process — going from hypotenuse back to a leg — divide the hypotenuse by $\sqrt{2}$ . This triangle is exactly half of a square sliced along its diagonal, which is why the diagonal of a square with side $s$ is $s\sqrt{2}$ .

The 30-60-90 Triangle — Ratio $1 : \sqrt{3} : 2$

A 30-60-90 triangle has sides in the ratio

$\text{short leg} : \text{long leg} : \text{hypotenuse} = 1 : \sqrt{3} : 2.$

The anchor is the short leg, which sits opposite the $30^\circ$ angle. From it:

the hypotenuse is always twice the short leg;
the long leg (opposite the $60^\circ$ angle) is the short leg times $\sqrt{3}$ .

This triangle is exactly half of an equilateral triangle cut down its altitude, which is why the height of an equilateral triangle with side $s$ is $\dfrac{s\sqrt{3}}{2}$ .

Triangle	Angles	Side ratio	Anchor side
45-45-90	$45^\circ, 45^\circ, 90^\circ$	$1 : 1 : \sqrt{2}$	either leg
30-60-90	$30^\circ, 60^\circ, 90^\circ$	$1 : \sqrt{3} : 2$	short leg (opposite $30^\circ$ )

How to Actually Use the Ratios

Think of the ratio as a scale factor. In a 30-60-90 triangle, whatever number multiplies the " $1$ " in the ratio multiplies every entry. If the short leg is $6$ , scale by $6$ : the long leg is $6\sqrt{3}$ and the hypotenuse is $12$ . If instead you are given the hypotenuse, first divide by $2$ to recover the short leg, then scale up. The order matters: always translate back to the anchor side (short leg for 30-60-90, either leg for 45-45-90) before finding the others.

A Fully Worked 30-60-90 Example

The hypotenuse of a 30-60-90 triangle measures $14$ . Find the exact lengths of both legs.

\text{short leg} &= \frac{14}{2} = 7 \quad(\text{opposite the } 30^\circ \text{ angle})\\ \text{long leg} &= 7\sqrt{3} \quad(\text{opposite the } 60^\circ \text{ angle}) \end{aligned}$$ Sanity-check with the Pythagorean theorem: $7^2 + (7\sqrt{3})^2 = 49 + 147 = 196 = 14^2.$ ✓ ## Where They Hide You will rarely be told "this is a special right triangle." Instead you will meet a **square** (its diagonal makes two 45-45-90 triangles), an **equilateral triangle** (its altitude makes two 30-60-90 triangles), or a coordinate figure whose slope creates a $30^\circ$ or $45^\circ$ angle. Recognizing the hidden triangle is what turns a slow computation into a five-second answer. These ratios are simply a shortcut for the same relationships you would otherwise extract from [right triangle trigonometry](/test-prep/concepts/right-triangle-trigonometry) or the [Pythagorean theorem](/test-prep/concepts/pythagorean-theorem) — they save the computation whenever the angles are $30^\circ$, $45^\circ$, or $60^\circ$. ## Common Mistakes to Avoid - Multiplying the short leg by $2$ to get the long leg. The factor $2$ gives the **hypotenuse**; the long leg uses $\sqrt{3}$. - Forgetting to rationalize when dividing by $\sqrt{2}$: $\dfrac{s}{\sqrt{2}} = \dfrac{s\sqrt{2}}{2}$. - Mixing up which side faces $30^\circ$ versus $60^\circ$ — the shortest side always faces the smallest angle. - Assuming *any* right triangle is "special." The ratios apply only to 45-45-90 and 30-60-90 angle sets.

A 45-45-90 triangle has legs of length $5$ . Find the exact length of the hypotenuse.

A 45-45-90 triangle has a hypotenuse of length $10$ . Find the exact length of each leg.

A 30-60-90 triangle has a short leg of length $6$ . Find the long leg and the hypotenuse.

The hypotenuse of a 30-60-90 triangle measures $14$ . Find the exact lengths of both legs.

An equilateral triangle has side length $12$ . Find its exact area.

A 30-60-90 triangle has a long leg of length $5\sqrt{3}$ . Find its perimeter.

Doubling the short leg to get the long leg of a 30-60-90 triangle.

The factor $2$ produces the hypotenuse. The long leg is the short leg times $\sqrt{3}$ , not $2$ .

Leaving an answer as $\dfrac{s}{\sqrt{2}}$ with a radical in the denominator.

Rationalize: $\dfrac{s}{\sqrt{2}} = \dfrac{s}{\sqrt{2}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{s\sqrt{2}}{2}$ . SAT answer choices are always in simplified radical form.

Assigning the $\sqrt{3}$ side to the $30^\circ$ angle instead of the $60^\circ$ angle.

The shortest side faces the smallest angle. The short leg ( $1$ ) faces $30^\circ$ ; the long leg ( $\sqrt{3}$ ) faces $60^\circ$ ; the hypotenuse ( $2$ ) faces $90^\circ$ .

Applying a special-triangle ratio to a right triangle whose angles are not 45-45-90 or 30-60-90.

Confirm the angle set first. If the acute angles are anything other than two $45^\circ$ or a $30^\circ/60^\circ$ pair, use the Pythagorean theorem or trigonometry instead.

A 45-45-90 right triangle has legs of length 6. What is the length of its hypotenuse?

In a 30-60-90 triangle, the short leg (opposite the 30 degree angle) has length 5. What is the length of the hypotenuse?

A 45-45-90 triangle has a hypotenuse of length 10. What is the length of each leg?

In a 30-60-90 triangle, the side opposite the 60 degree angle has length 9 times the square root of 3. What is the length of the hypotenuse?

An equilateral triangle has a side length of 10. What is the length of its altitude (height)?

A square has a diagonal of length 14. What is the length of each side of the square?

$\text{leg} : \text{leg} : \text{hyp} = 1 : 1 : \sqrt{2}$

$\text{short} : \text{long} : \text{hyp} = 1 : \sqrt{3} : 2$

$h = \dfrac{s\sqrt{3}}{2}$

Two Triangles Worth Memorizing

The 45-45-90 Triangle — Ratio $1 : 1 : \sqrt{2}$

A 45-45-90 triangle is an isosceles right triangle: the two legs are equal, and each acute angle is $45^\circ$ . Its sides always follow

$\text{leg} : \text{leg} : \text{hypotenuse} = 1 : 1 : \sqrt{2}.$

The 30-60-90 Triangle — Ratio $1 : \sqrt{3} : 2$

A 30-60-90 triangle has sides in the ratio

$\text{short leg} : \text{long leg} : \text{hypotenuse} = 1 : \sqrt{3} : 2.$

The anchor is the short leg, which sits opposite the $30^\circ$ angle. From it:

the hypotenuse is always twice the short leg;
the long leg (opposite the $60^\circ$ angle) is the short leg times $\sqrt{3}$ .

This triangle is exactly half of an equilateral triangle cut down its altitude, which is why the height of an equilateral triangle with side $s$ is $\dfrac{s\sqrt{3}}{2}$ .

Triangle	Angles	Side ratio	Anchor side
45-45-90	$45^\circ, 45^\circ, 90^\circ$	$1 : 1 : \sqrt{2}$	either leg
30-60-90	$30^\circ, 60^\circ, 90^\circ$	$1 : \sqrt{3} : 2$	short leg (opposite $30^\circ$ )

How to Actually Use the Ratios

A Fully Worked 30-60-90 Example

The hypotenuse of a 30-60-90 triangle measures $14$ . Find the exact lengths of both legs.

\text{short leg} &= \frac{14}{2} = 7 \quad(\text{opposite the } 30^\circ \text{ angle})\\ \text{long leg} &= 7\sqrt{3} \quad(\text{opposite the } 60^\circ \text{ angle}) \end{aligned}$$ Sanity-check with the Pythagorean theorem: $7^2 + (7\sqrt{3})^2 = 49 + 147 = 196 = 14^2.$ ✓ ## Where They Hide You will rarely be told "this is a special right triangle." Instead you will meet a **square** (its diagonal makes two 45-45-90 triangles), an **equilateral triangle** (its altitude makes two 30-60-90 triangles), or a coordinate figure whose slope creates a $30^\circ$ or $45^\circ$ angle. Recognizing the hidden triangle is what turns a slow computation into a five-second answer. These ratios are simply a shortcut for the same relationships you would otherwise extract from [right triangle trigonometry](/test-prep/concepts/right-triangle-trigonometry) or the [Pythagorean theorem](/test-prep/concepts/pythagorean-theorem) — they save the computation whenever the angles are $30^\circ$, $45^\circ$, or $60^\circ$. ## Common Mistakes to Avoid - Multiplying the short leg by $2$ to get the long leg. The factor $2$ gives the **hypotenuse**; the long leg uses $\sqrt{3}$. - Forgetting to rationalize when dividing by $\sqrt{2}$: $\dfrac{s}{\sqrt{2}} = \dfrac{s\sqrt{2}}{2}$. - Mixing up which side faces $30^\circ$ versus $60^\circ$ — the shortest side always faces the smallest angle. - Assuming *any* right triangle is "special." The ratios apply only to 45-45-90 and 30-60-90 angle sets.