Isosceles Right Triangle Calculator

What is an Isosceles Right Triangle?

An isosceles right triangle (also known as a 45-45-90 triangle) is a unique geometric figure that combines the properties of both an isosceles triangle and a right triangle. It features one $90^\circ$ right angle and two equal acute angles of $45^\circ$ ($\frac{\pi}{4}\text{ radians}$). Because the two acute angles are equal, the two legs opposite them must also have equal lengths ($a = b$).

The hypotenuse ($c$) lies opposite the $90^\circ$ angle. By applying the Pythagorean theorem, the relationship between the legs and the hypotenuse is $c = a\sqrt{2}$. This makes solving any 45-45-90 triangle straightforward when any single side, area, perimeter, or altitude is known.

Key Formulas of an Isosceles Right Triangle

Given a leg length $a$ (where $a = b$) and hypotenuse $c$:

• Hypotenuse: $c = a\sqrt{2} \approx 1.4142 \times a$
• Leg length from hypotenuse: $a = \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2} \approx 0.7071 \times c$
• Area: $A = \frac{1}{2} a^2 = \frac{c^2}{4}$
• Perimeter: $P = 2a + c = a(2 + \sqrt{2}) \approx 3.4142 \times a$
• Altitude to Hypotenuse ($h$): $h = \frac{a}{\sqrt{2}} = \frac{c}{2}$
• Inradius ($r$): $r = \frac{a}{2}(2 - \sqrt{2}) = \frac{c}{2}(\sqrt{2} - 1)$
• Circumradius ($R$): $R = \frac{c}{2} = \frac{a\sqrt{2}}{2}$

How to Calculate Missing Values (Step-by-Step)

Depending on which metric you already have, you can determine all other measurements using these concise procedures:

1. When the leg ($a$) is known: Multiply $a$ by $\sqrt{2}$ to obtain hypotenuse $c$. The area is $\frac{1}{2} a^2$, and the perimeter is $2a + c$.
2. When the hypotenuse ($c$) is known: Divide $c$ by $\sqrt{2}$ to find each leg length $a$. Square $c$ and divide by 4 to get the total area.
3. When the area ($A$) is known: The leg length is $a = \sqrt{2A}$, and the hypotenuse is $c = 2\sqrt{A}$.
4. When the perimeter ($P$) is known: The leg length is $a = \frac{P}{2 + \sqrt{2}} = P(1 - \frac{\sqrt{2}}{2})$, and the hypotenuse is $c = P(\sqrt{2} - 1)$.
5. When the altitude to hypotenuse ($h$) is known: The hypotenuse is simply $c = 2h$, the leg length is $a = h\sqrt{2}$, and the area is $A = h^2$.

Related Geometry Calculators

Explore other helpful triangle and polygon tools on OnlineMiniTools:

• Right Triangle Calculator: Calculate any general right triangle with Pythagorean theorem and trigonometry.
• Hypotenuse Calculator: Find the longest side of any right triangle from known legs or angles.
• Isosceles Triangle Calculator: Calculate general isosceles triangles with acute, right, or obtuse apex angles.
• Area of a Trapezoid Calculator: Compute the area of trapezoids and quadrilaterals.