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Isosceles Right Triangle

A right triangle is a triangle that has exactly one angle that measures $90$ degrees. Since the sum of measures of all angles in a triangle has to be 180 degrees, it is obvious that the sum of the remaining two angles would be another $90$ degrees. The two perpendicular sides are called the legs of the right triangle and the longest side that lies opposite the 90-degree angle is called the hypotenuse of the right triangle. A right triangle can be scalene (having all three sides of different length) or isosceles (having exactly two sides of equal length). It can never be an equilateral triangle. That is because in an equilateral triangle all the angles have to be equal. Thus each angle has to measure $60$ degrees. So we cannot have any angle measuring $90$ degrees in an equilateral triangle.

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Calculate Isosceles Triangle Area of a Right Triangle Calculator
 

Definition

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An isosceles right triangle is a right triangle that has its two legs equal in length. Since the two legs of the right triangle are equal in length, the corresponding angles would also be congruent. Thus, in an isosceles right triangle, two legs and the two acute angles are congruent. It would, therefore, look as follows:

Isosceles Right Triangle Definition
 
Since it is a right triangle, the angle between the two legs would be $90$ degrees and so the legs would obviously be perpendicular to each other.

Formula

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The most important formula associated with any right triangle is the Pythagorean theorem. According to this theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. Now, in an isosceles right triangle, the other two sides are congruent. Therefore they are of the same length l. Thus of the hypotenuse measures h, then the Pythagorean theorem for isosceles right triangle would be:
$(hypotenuse)^2$ = $(side)^2\ +\ (side)^2$

$h^2$ = $l^2\ +\ l^2$

Simplifying that we have:

$h^2$ = $2l^2$

This is the formula for the Pythagorean theorem for an isosceles right triangle.

Next. If the angles of a triangle measure $<\ A$, $<\ B$ and $<\ C$, then their sum:

$m\ <\ A\ +\ m\ <\ B\ +\ <\ C$

Should be equal to $180$ degrees.

Now, in a right triangle, one angle is $90$ degrees. So suppose $m\ <\ B$ = $90^{\circ}$. Then the sum of angles would be:

$m\ <\ A\ +\ 90\ +\ m\ <\ C$ = $180$

Then, in an isosceles right triangle, the other two angles are congruent. So the measures of angle $A$ and angle $C$ would be same. Let us say that they both are equal to $\theta$. Then the sum of angles would be:

$\theta\ +\ 90\ +\ \theta$ = $180$

Combining like terms we have:

$2 \theta\ +\ 90$ = $180$

Subtracting 90 from both the sides we have:

$2\ \theta\ +\ 90\ -\ 90$ = $180\ -\ 90$

$2\ \theta$ = $90$

Dividing both the sides by $2$ gives:

$\frac{2\ \theta}{2}$ = $\frac{90}{2}$

$\theta$ = $45^{\circ}$

Thus we see that the two congruent angles in an isosceles right triangle would measure $45$ degrees each. So the triangle would look as follows:
Isosceles Right Triangle Formula


Isosceles Right Triangle Area

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The area of any triangle is given by the formula:

$Area$ = $\frac{1}{2}$ $bh$

Where $b$ is the length of the base of the triangle and h is the altitude of the triangle. Altitude refers to the shortest distance between the opposite vertex and the base. It is same as the length of the perpendicular dropped from the opposite vertex on to the base. 

For a right triangle, if we consider one of the legs as the base, then the perpendicular from the opposite vertex to this base would simply be the other leg. 
 
Isosceles Right Triangle Area

So, if the length of the base of the right triangle is $b$ and that of the altitude is $a$, then the area would simply be:

$Area$ = $\frac{1}{2}$ $ba$

In an isosceles right triangle, the length of the two legs is equal. Let us say that they both measure $l$ then the area formula can be further modified to:

$Area$ = $\frac{1}{2}$ $l\ \times\ l$

$Area$ = $\frac{1}{2}$ $l^2$

$Area$ = $\frac{l^2}{2}$

Where $l$ is the length of the congruent sides of the isosceles right triangle.

Isosceles Right Triangle Perimeter

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The perimeter of any plane figure is defined as the sum of the lengths of the sides of the figure. So for a triangle, the perimeter would be the sum of all the sides of the triangle. Thus for a triangle with side lengths: $a,\ b$ and $c$. The perimeter would simply be:

$Perimeter$ = $a\ +\ b\ +\ c$

In an isosceles right triangle, we know that two sides are congruent. Suppose their lengths are equal to l and the hypotenuse measures h units. Thus the perimeter of an isosceles right triangle would be:

$Perimeter$ = $h\ +\ l\ +\ l$

$Perimeter$ = $h\ +\ 2l$

Examples

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Example 1:  The hypotenuse of an isosceles right triangle measures $10$ cm. Find the area and perimeter of the right triangle.

Solution:

First, we need to find the lengths of the congruent sides of the isosceles right triangle. For that we use the Pythagorean theorem for an isosceles right triangle:

$h^2$ = $2l^2$

Here, we know that hypotenuse, h=10. Substituting that 
we have:

$10^{2}$ = $2l^2$

$100$ = $2l^2$

$\frac{100}{2}$ = $l^2$

$l^2$ = $50$

$l$ = $\sqrt{50}$

$l$ = $\sqrt{25\ \times\ 2}$

$l$ = $5\sqrt{2}\ cm$

Next, we use this to find the area of the isosceles 
right triangle.

$Area$ = $\frac{l^2}{2}$ = $\frac{(5\ \sqrt{2})^2}{2}$ = $\frac{25\ \times\ 2}{2}$ = $\frac{50}{2}$ = $25\ cm^2$

And lastly we can now find the perimeter using the 
perimeter formula:

$Perimeter$ = $h\ +\ 2l$

$Perimeter$ = $10\ +\ 2\ (5\ \sqrt{2})$

$Perimeter$ = $10\ +\ 10\ \sqrt{2}$ $\leftarrow$ Exact

$Perimeter$ = $10\ +\ 14.14$ = $24.14$ $\leftarrow$ Approx
Example 2: The area of an isosceles right triangle is 32 square inches. Find the perimeter of this right triangle.

Solution:

The formula for the area of an isosceles right triangle is:

$Area$ = $\frac{l^2}{2}$

For our problem, we are given that $area$ = $32$ square inches. 
Substituting this value into the above formula we have:

$32$ = $\frac{l^2}{2}$

Now we solve that for $l$. Thus,

$32\ \times\ 2$ = $l^2$

$l^2$ = $64$

$l$ = $\sqrt{64}$

$l$ = $8$ inches

Next, we need to find the length of the hypotenuse. The Pythagorean 
theorem for an isosceles right triangle looks like this:

$h^2$ = $2l^2$

Substituting the value of $l$ above we have:

$h^2$ = $2(8)^2$

$h^2$ = $2(64)$

$h^2$ = $128$

$h$ = $\sqrt{128}$

$h$ = $\sqrt{64\ \times\ 2}$

$h$ = $8\ \sqrt{2}$

Now that we know the value of $h$ we can now find the perimeter.

$Perimeter$ = $h\ +\ 2l$

$Perimeter$ = $8\ \sqrt{2}\ +\ 2(8)$

$Perimeter$ = $8\ \sqrt{2}\ +\ 16$ $\leftarrow$ Exact

$Perimeter$ = $11.31\ +\ 15$ = $27.31$ in $\leftarrow$ Approx
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