Acute Angle
An acute angle is an angle that measures greater than 0 degrees and less than 90 degrees.
Formula
0 < \text{angle} < 90^\circ
Definition
An acute angle is smaller than a right angle, measuring between $0^\circ$ and $90^\circ$ (strictly, $0 < \theta < \pi/2$ radians); acute angles look "sharp" or narrow, like the tip of an arrowhead. In a right triangle, the two non-right angles are always acute and complementary, summing to $90^\circ$, and in a triangle, the angle at vertex $A$ is acute if and only if the dot product of the two adjacent side vectors is positive: a triangle is acute if $a^2 + b^2 > c^2$ for all three combinations of its sides.
Example
A $45^\circ$ angle is acute, as are the sharp angles at the tips of a star shape; when clock hands show 1 o'clock, the smaller angle between them is acute. In a $30$-$60$-$90$ triangle, both the $30^\circ$ and $60^\circ$ angles are acute, and the complement of a $30^\circ$ angle is $60^\circ$. For a triangle with sides $5, 6, 7$: checking all three combinations, $25+36>49$, $25+49>36$, and $36+49>25$ all hold, confirming the triangle is acute.
Key Insight
The word "acute" means sharp, and acute triangles (all three angles less than $90^\circ$) look sharp at every corner. The trigonometric ratios (sine, cosine, tangent) are most naturally defined for acute angles in a right triangle, making the acute case the gateway to all of trigonometry, and the condition $a^2+b^2>c^2$ for an acute triangle is the strict converse of the Pythagorean theorem, a bridge between geometric and algebraic reasoning.