45°- 45°- 90° Triangle
A right triangle where the angles are 45°, 45°, and 90°.
Try this In the figure below, drag the orange dots on each vertex to reshape the triangle. Note how the angles remain the same, and it maintains the same proportions between its sides.
This is one of the 'standard' triangles you should be able recognize on sight. A fact you should commit to memory is:
The sides are always in the ratio 1 : 1 : .
With the '' being the hypotenuse (longest side).

This can be derived from Pythagoras' Theorem. This ratio will come in handy later in the study of trigonometry. In the figure above, as you drag the vertices of the triangle to resize it, the angles remain fixed and the sides remain in this ratio.

Because the base angles are the same (both 45°) the two legs are equal and so the triangle is also isosceles

Area of a 45-45-90 triangle

As you see from the figure on the right, two 45-45-90 triangles together make a square, so the area of one of them is half the area of the square. As a formula

where
S  is the length of either short side

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises

COMMON CORE

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