# ACT Math : How to find the area of a 45/45/90 right isosceles triangle

## Example Questions

### Example Question #1 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

What is the area of an isosceles right triangle with a hypotenuse of ?

Explanation:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees, because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

### Example Question #2 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

What is the area of an isosceles right triangle with a hypotenuse of ?

Explanation:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

### Example Question #5 : Isosceles Triangles

What is the area of an isosceles right triangle with a hypotenuse of ?

Explanation:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

### Example Question #6 : Isosceles Triangles

is a right isosceles triangle with hypotenuse . What is the area of ?

Explanation:

Right isosceles triangles (also called "45-45-90 right triangles") are special shapes. In a plane, they are exactly half of a square, and their sides can therefore be expressed as a ratio equal to the sides of a square and the square's diagonal:

, where  is the hypotenuse.

In this case,  maps to , so to find the length of a side (so we can use the triangle area formula), just divide the hypotenuse by :

So, each side of the triangle is  long. Now, just follow your formula for area of a triangle:

Thus, the triangle has an area of .