Acute Angle
GeometryAn 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. It measures between $0^\circ$ and $90^\circ$. Acute angles look "sharp" or narrow, like the tip of an arrowhead or a slice of pizza.
Example
A $45^\circ$ angle is acute. The angles at the tip of a star shape are acute. When clock hands show 1 o'clock, the smaller angle between them is acute.
Key Insight
The word "acute" means sharp. Acute angles are sharp-looking. Acute triangles have all three angles less than $90^\circ$ - all three corners look sharp.
Definition
An acute angle has a measure strictly between $0^\circ$ and $90^\circ$ ($0 < \theta < 90$). It is smaller than a right angle. In a right triangle, the two non-right angles are always acute and they are complementary (they sum to $90^\circ$).
Example
Angles of $30^\circ$, $45^\circ$, $60^\circ$, and $89^\circ$ are all acute. In a $30$-$60$-$90$ triangle, both the $30^\circ$ and $60^\circ$ angles are acute. The complement of a $30^\circ$ angle is $60^\circ$.
Key Insight
The trigonometric ratios (sine, cosine, tangent) are most naturally defined for acute angles in a right triangle. Extending trig to all angles requires the unit circle definition, but the acute case is the gateway to understanding all of trigonometry.
Definition
An acute angle satisfies $0 < \theta < \pi/2$ radians. In a triangle, the angle at vertex $A$ is acute iff the dot product of the two adjacent side vectors is positive. A triangle is acute iff $a^2 + b^2 > c^2$ for all three combinations of sides.
Example
In triangle with sides $5, 6, 7$: check all three: $25+36>49$ (yes), $25+49>36$ (yes), $36+49>25$ (yes). All conditions hold, so the triangle is acute. If any failed, that angle would be obtuse.
Key Insight
The condition for an acute triangle ($a^2 + b^2 > c^2$ for all sides) is the strict converse of the Pythagorean theorem. This connects angle classification directly to the algebraic relationship between side lengths, a bridge between geometric and algebraic reasoning.