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Acute Angle

Acute Angle

An angle that has measure less than 90°.

 

Two rays forming an acute angle (less than 90°), labeled "acute angle" near the vertex.

 

 

See also

Obtuse angle, degrees

Key Formula

0°<θ<90°0° < \theta < 90°
Where:
  • θ\theta = The measure of the angle in degrees

Worked Example

Problem: In triangle ABC, angle A measures 50° and angle B measures 70°. Determine whether each angle in the triangle is acute.
Step 1: Check angle A. Since 0° < 50° < 90°, angle A is acute.
∠A=50°⇒acute\angle A = 50° \quad \Rightarrow \quad \text{acute}
Step 2: Check angle B. Since 0° < 70° < 90°, angle B is acute.
∠B=70°⇒acute\angle B = 70° \quad \Rightarrow \quad \text{acute}
Step 3: Find angle C using the triangle angle sum property: the three interior angles must add to 180°.
∠C=180°−50°−70°=60°\angle C = 180° - 50° - 70° = 60°
Step 4: Check angle C. Since 0° < 60° < 90°, angle C is also acute.
∠C=60°⇒acute\angle C = 60° \quad \Rightarrow \quad \text{acute}
Answer: All three angles (50°, 70°, and 60°) are acute. Because every angle in the triangle is acute, this is called an acute triangle.

Another Example

Problem: Classify each angle as acute or not acute: 15°, 89°, 90°, 120°.
Step 1: 15° is between 0° and 90°, so it is acute.
0°<15°<90°✓0° < 15° < 90° \quad \checkmark
Step 2: 89° is between 0° and 90°, so it is acute. Note that it is just barely less than a right angle.
0°<89°<90°✓0° < 89° < 90° \quad \checkmark
Step 3: 90° is not less than 90°, so it is not acute — it is a right angle.
90°≮90°×90° \not< 90° \quad \times
Step 4: 120° is greater than 90°, so it is not acute — it is an obtuse angle.
120°>90°×120° > 90° \quad \times
Answer: 15° and 89° are acute angles. 90° is a right angle, and 120° is an obtuse angle.

Frequently Asked Questions

Can an angle of exactly 90° be called acute?
No. An acute angle must measure strictly less than 90°. An angle of exactly 90° is called a right angle. Even 89.999° is technically acute, but 90° is not.
What is an acute triangle?
An acute triangle is a triangle in which all three interior angles are acute, meaning every angle measures less than 90°. If even one angle is 90° or more, the triangle is not acute — it is a right triangle or an obtuse triangle, respectively.

Acute Angle vs. Obtuse Angle

An acute angle measures between 0° and 90° (exclusive), while an obtuse angle measures between 90° and 180° (exclusive). Acute angles look sharp and narrow; obtuse angles look wide and blunt. The boundary between them is the right angle at exactly 90°, which is neither acute nor obtuse.

Why It Matters

Acute angles are everywhere in geometry and real life — the pitch of a roof, the tilt of a ramp, or the corner of a slice of pizza. Classifying angles as acute, right, or obtuse is one of the first steps in analyzing triangles, which in turn underpins trigonometry, engineering, and architecture. In trigonometry, the sine, cosine, and tangent of acute angles are all positive, which simplifies many calculations.

Common Mistakes

Mistake: Including 90° as an acute angle.
Correction: An acute angle must be strictly less than 90°. An angle of exactly 90° is a right angle, not an acute angle.
Mistake: Forgetting that 0° is not an acute angle either.
Correction: An acute angle must be greater than 0°. A 0° angle (where both rays overlap) is sometimes called a zero angle and is not classified as acute.

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