45°- 45°- 90° Triangle
In the figure below, drag the orange dots on each
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:
With the '' being the hypotenuse (longest side).
The sides are always in the ratio 1 : 1 :
This can be derived from
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
S is the length of either short side
Other triangle topics
Perimeter / Area
Congruence and Similarity
Triangle quizzes and exercises
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