What are the angles of an isosceles right triangle?

The angles are always 45°, 45°, and 90°. The right angle sits between the two equal legs, and each base angle is 45° because the remaining 90° of angle measure splits equally between two congruent sides.

What is the ratio of sides of a 45-45-90 triangle?

The side ratio is $1 : 1 : \sqrt{2}$. Both legs share the same length, and the hypotenuse is $\sqrt{2}$ times that length. For example, if each leg is 5, the hypotenuse is $5\sqrt{2}$.

How do you find the hypotenuse of an isosceles right triangle?

Multiply the leg length by $\sqrt{2}$. This comes directly from the Pythagorean theorem: $a^2 + a^2 = c^2$ simplifies to $c = a\sqrt{2}$. There is no need to compute each squared term separately if you memorize this shortcut.

Isosceles Right Triangle — Definition, Formula & Examples

An isosceles right triangle is a triangle with one 90° angle and two equal-length sides (legs). Its three angles always measure 45°, 45°, and 90°, making it the classic 45-45-90 triangle.

An isosceles right triangle is a triangle in which exactly one interior angle is a right angle and the two sides forming that right angle are congruent. By the angle sum property, the two remaining angles each measure 45°. The ratio of the legs to the hypotenuse is $1 : 1 : \sqrt{2}$ , a consequence of the Pythagorean theorem applied to two equal legs.

Key Formula

\text{hypotenuse} = a\sqrt{2} \qquad \text{Area} = \frac{a^2}{2}

Where:

$a$ = Length of each equal leg (the two sides forming the right angle)
$a\sqrt{2}$ = Length of the hypotenuse (the side opposite the 90° angle)

How It Works

Start by identifying the two equal legs — these are the sides that form the right angle. The third side, opposite the 90° angle, is the hypotenuse. If each leg has length

a

, you find the hypotenuse by applying

a\sqrt{2}

. Conversely, if you know the hypotenuse

c

, each leg equals

\frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}

. The area is simply

\frac{a^2}{2}

because the two legs serve as the base and height. Recognizing this triangle lets you skip the full Pythagorean theorem and work directly with the

1:1:\sqrt{2}

ratio.

Worked Example

Problem: An isosceles right triangle has legs of length 6 cm. Find the hypotenuse, area, and perimeter.

Step 1: Identify the known values. Both legs equal 6 cm, and the angles are 45°-45°-90°.

a = 6

Step 2: Find the hypotenuse using the ratio formula.

c = a\sqrt{2} = 6\sqrt{2} \approx 8.49 \text{ cm}

Step 3: Calculate the area. The two legs act as base and height.

A = \frac{a^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18 \text{ cm}^2

Step 4: Calculate the perimeter by adding all three sides.

P = 6 + 6 + 6\sqrt{2} = 12 + 6\sqrt{2} \approx 20.49 \text{ cm}

Answer: Hypotenuse =

6\sqrt{2} \approx 8.49

cm, Area = 18 cm², Perimeter =

12 + 6\sqrt{2} \approx 20.49

cm.

Another Example

This example reverses the direction: the hypotenuse is given instead of the legs, requiring students to solve for the leg length first.

Problem: The hypotenuse of an isosceles right triangle is 10 cm. Find the length of each leg and the area.

Step 1: You know the hypotenuse, so work backward to find the legs. Set up the relationship.

c = a\sqrt{2} \implies a = \frac{c}{\sqrt{2}}

Step 2: Substitute the hypotenuse and rationalize the denominator.

a = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} \approx 7.07 \text{ cm}

Step 3: Compute the area using the leg length.

A = \frac{a^2}{2} = \frac{(5\sqrt{2})^2}{2} = \frac{50}{2} = 25 \text{ cm}^2

Answer: Each leg =

5\sqrt{2} \approx 7.07

cm, Area = 25 cm².

Why It Matters

Isosceles right triangles appear throughout geometry, trigonometry, and standardized tests like the SAT and ACT, where the 45-45-90 ratio is expected knowledge. Architects and engineers encounter this triangle whenever a square is divided by its diagonal, such as in roof framing, tile cutting, and structural bracing. Mastering it also builds the foundation for understanding exact values of

\sin 45°

and

\cos 45°

in trigonometry courses.

Common Mistakes

Mistake: Using the 30-60-90 ratio (

1 : \sqrt{3} : 2

) on a 45-45-90 triangle.

Correction: Always check the angles first. A 45-45-90 triangle uses the ratio

1 : 1 : \sqrt{2}

. The two special right triangles have completely different side relationships.

Mistake: Multiplying the hypotenuse by

\sqrt{2}

to find a leg (instead of dividing).

Correction: The hypotenuse is already the longest side. To go from hypotenuse to leg, divide by

\sqrt{2}

(or equivalently multiply by

\frac{\sqrt{2}}{2}

Mistake: Forgetting to rationalize the denominator when writing exact answers.

Correction: Many teachers require

\frac{10\sqrt{2}}{2} = 5\sqrt{2}

rather than

\frac{10}{\sqrt{2}}

. Practice rationalizing so your final answers are in standard form.

Check Your Understanding

Each leg of an isosceles right triangle is 8 cm. What is the hypotenuse?

Hint: Use hypotenuse = leg × √2.

Answer:

8\sqrt{2} \approx 11.31

The hypotenuse of a 45-45-90 triangle is 14 cm. What is the area?

Hint: First find the leg:

a = \frac{14}{\sqrt{2}} = 7\sqrt{2}

. Then use

A = \frac{a^2}{2}

Answer: 49 cm²

A square has a diagonal of 12 cm. What is the side length of the square?

Hint: The diagonal splits a square into two isosceles right triangles. The diagonal is the hypotenuse.

Answer:

6\sqrt{2} \approx 8.49

Related Terms

Angle Sum Property of a Triangle — Proves the two base angles each equal 45°
Acute Triangle — A different triangle classification by angles
Base of an Isosceles Triangle — The hypotenuse serves as the base here
Altitude of a Triangle — Each leg doubles as an altitude in this triangle
Base of a Triangle — Either leg can be chosen as the base for area
Centroid — Located at one-third the leg length from the right angle
AA Similarity — All 45-45-90 triangles are similar by AA
ASA Congruence — Can prove two isosceles right triangles congruent
AAS Congruence — Alternative congruence test for these triangles
Centers of a Triangle — Symmetry axis aligns several triangle centers