Free Isosceles Right Triangle Calculator

What is an Isosceles Right Triangle?

An isosceles right triangle is a right-angled triangle where the two legs (the sides that form the right angle) have identical lengths. Consequently, the two acute angles are equal, each measuring $45^{\circ}$. This specific configuration is why it is commonly called a 45‑45‑90 triangle. The relationship between the sides follows from the Pythagorean theorem: the hypotenuse $h$ equals the leg length $a$ multiplied by $\sqrt{2}$.

Using the Calculator

The Isosceles Right Triangle Calculator accepts any known side (leg or hypotenuse) as input. Once you provide that value, the tool instantly computes:

• The other leg (always equal to the given leg if a leg is entered)
• The hypotenuse
• The perimeter
• The area

If you need a missing value, select whether the known side is a leg or the hypotenuse, enter its measurement, and the calculator does the rest.

Perimeter Calculation

For an isosceles right triangle, the perimeter $P$ is the sum of the two equal legs and the hypotenuse:

$$P = a + a + h = 2a + h$$

Using the relationship $h = a\sqrt{2}$, the perimeter can also be expressed as $P = a(2 + \sqrt{2})$.

Example: If one leg $a = 5\ \text{cm}$,

$$h = 5\sqrt{2} \approx 7.07\ \text{cm},\quad P = 2 \times 5 + 7.07 = 17.07\ \text{cm}.$$

The Isosceles Right Triangle Perimeter Calculator functionality performs this calculation automatically.

Area Calculation

The area of any triangle is $\dfrac{1}{2} \times \text{base} \times \text{height}$. In an isosceles right triangle, because both legs are perpendicular, one leg can serve as the base and the other as the height. Therefore:

$$\text{Area} = \dfrac{1}{2} a^{2}$$

Example: With $a = 4\ \text{cm}$,

$$\text{Area} = \dfrac{1}{2} \times 4^{2} = \dfrac{1}{2} \times 16 = 8\ \text{cm}^{2}.$$

The built-in Isosceles Right Triangle Area Calculator saves you from manual multiplication.

Finding the Leg Length from the Hypotenuse

If you know the hypotenuse $h$ but not the leg length, rearrange the Pythagorean relation:

$$a = \dfrac{h}{\sqrt{2}}$$

Example: For a hypotenuse of $5\ \text{cm}$,

$$a = \dfrac{5}{\sqrt{2}} \approx 3.54\ \text{cm}.$$

This inverse operation is handled directly by the Isosceles Right Triangle Hypotenuse Calculator — just select “hypotenuse known” and you get the leg lengths.

Constructing an Isosceles Right Triangle with Straightedge and Compass

Creating a precise isosceles right triangle on paper does not require a protractor. Follow these steps:

1. Draw a base line — Use a ruler to draw a horizontal segment. Mark the endpoints $A$ and $B$.
2. Construct the perpendicular bisector — Place the compass at $A$ with a radius greater than half $AB$, and draw an arc above and below the line. Repeat from $B$ so the arcs intersect. Connect the two intersection points with a straight line – this line is the perpendicular bisector of $AB$ and crosses $AB$ at point $O$.
3. Mark the apex — With the compass still set to the same radius (or at least to $OA$), draw an arc centered at $O$ that cuts the perpendicular bisector above $AB$. Label that intersection $C$.
4. Complete the triangle — Draw straight segments from $A$ to $C$ and from $B$ to $C$. Triangle $ABC$ is the required isosceles right triangle, with the right angle at $C$ and legs $AC = BC$.

This classical construction leverages the fact that the circumcenter lies at the midpoint of the hypotenuse; here the apex is placed at the midpoint of the hypotenuse’s perpendicular bisector.

Key Properties At a Glance

Property	Value
Leg length	$a$
Hypotenuse	$a\sqrt{2}$
Acute angles	$45^{\circ}$ each
Area	$\dfrac{1}{2}a^{2}$
Perimeter	$a(2 + \sqrt{2})$

The consistent $\sqrt{2}$ ratio makes the isosceles right triangle easy to scale and analyze. Whether you need to compute missing dimensions or verify designs, the 45 45 90 Triangle Calculator (another name for the same tool) provides quick, reliable outputs.