How do you find the smallest angle in a right angled triangle whose side lengths are 6cm, 13cm and 14 cm?

##25.4^@##, rounded to one decimal place.

Given is a right angled triangle with sides ##6, 12.65 and 14cm## (The lengths of sides have been modified appropriately for a right angled triangle. )

From the properties of such triangle we know that

1. smallest angle is opposite the smallest side.
2. largest side is the hypotenuse.

In the figure below We observe that the required angle ##BAC## is opposite the smallest side. From the definition of ##sin## function

##sin angle BAC="Perpendicular"/"Hypotenuse"## , we get ##sin angle BAC=6/14## ## angle BAC=sin^-1 (6/14)=25.4^@##, rounded to one decimal place. .-.-.-.-.-.-.-.-.-.-.--.-.

We can use definition of ##cos or tan ## functions as well.

Thanks to @dk_ch for pointing out that the set of lengths given in the problem can not form a right angled triangle, as these do not satisfy the Pythagorean equation

##h^2=b^2+p^2##.

Hence, it is not a right angled triangle for which general solutions posted by other contributors are valid.

The error appears to be due to socratic software glitch which is framing such questions and posting these.