Acute triangle

An acute triangle is a triangle with three acute angles, which are angles measuring less than 90°.

This yield sign is in the shape of an acute triangle.

Properties of acute triangles

Using the Pythagorean Theorem to find acute triangles

When the lengths of the sides of a triangle are known, the Pythagorean Theorem can be used to determine whether or not the triangle is an acute triangle.

For a right triangle with a hypotenuse of length c and leg lengths a and b, the Pythagorean Theorem states:

a2 + b2 = c2

On the other hand, in a triangle where a2 + b2 > c2, if side c is also the longest side, the triangle is an acute triangle.

Example:

Is triangle ABC an acute triangle?

Side AB above is the longest side of triangle ABC. To check if ABC is an acute triangle, let c = 13, a = 10 and b = 9:

a2 + b2 = 102 + 92 = 181

c2 = 132 = 169

181 > 169

Therefore triangle ABC is an acute triangle.