Identifying the 45 – 45 – 90 Degree Triangle

By Mark Ryan

A 45 – 45 – 90 degree triangle (or isosceles right triangle) is a triangle with angles of 45°, 45°, and 90° and sides in the ratio of


Note that it’s the shape of half a square, cut along the square’s diagonal, and that it’s also an isosceles triangle (both legs have the same length). The following figure shows an example of a 45°- 45°- 90° triangle.


Try a couple of problems. Find the lengths of the unknown sides in triangles BAT and BOY shown in the following figure.


You can solve 45°- 45°- 90° triangle problems in two ways: the formal book method and the street-smart method. Try ’em both and take your pick.

Using the formal book method

The formal method uses the ratio of the sides from the first figure.


For triangle BAT, because one of the legs is 8, the x in the ratio is 8. Plugging 8 into the three x’s gives you


And for triangle BOY, the hypotenuse is 10, so you set the


from the ratio equal to 10 and solve for x:


That does it:


Using the street-smart method

Now for the street-smart method for working with the 45°- 45°- 90° triangle (this method is based on the same math as the formal method, but it involves fewer steps). Remember the 45°- 45°- 90° triangle as the “root two triangle.” Using that tidbit, do one of the following:


Look back at the lengths of the sides in triangle BAT and triangle BOY. In triangle BAT, the hypotenuse is the only side that contains a radical. In triangle BOY, the hypotenuse is the only side without a radical. These two cases are by far the most common ones, but in unusual cases, all three sides may contain a radical symbol.