What is an acute angle in math?
An acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.
Acute Angle in a picture
Formal definition and notation
An angle ∠ ABC is acute when its measure satisfies 0° < m∠ ABC < 90°. Both inequalities are strict: an angle of exactly 90° is a right angle, and a zero angle (both rays on top of each other) is not counted as acute. A triangle whose three angles are all acute is called an acute triangle.
| Symbol | Read it as | Meaning |
|---|---|---|
| 0° < θ < 90° | theta is between 0 and 90 degrees | the condition for an angle θ to be acute |
Key facts
- The word acute comes from Latin for sharp: an acute angle has a sharp, narrow point.
- The largest whole-number acute angle is 89°. There is no largest acute angle overall, since 89.9° and 89.99° are acute too.
- The two acute angles of a right triangle always add to 90°, so they are complementary.
- Every triangle has at least two acute angles, because the three angles add to 180° and only one can be 90° or more.
- Half of any angle less than 180° is acute: half of 150° is 75°.
Try it: make an acute angle
Drag the moving ray and watch the name change. Find the last whole degree that is still acute, then cross 90 and see what happens.
Acute Angle examples, step by step
Example 1: Sort the angles
Which of these angles are acute: 12°, 90°, 67°, 104°, 88.5°?
- An acute angle is greater than 0° and less than 90°.
- 12°, 67° and 88.5° are all under 90°.
- 90° is exactly a right angle, and 104° is larger than 90°, so neither is acute.
Answer: Three are acute: 12°, 67° and 88.5°.
Example 2: The other acute angle in a right triangle
A right triangle has one acute angle of 38°. Find the third angle.
- The angles of a triangle add to 180°.
- The right angle uses 90°, so the two acute angles share 180° − 90° = 90°.
- Third angle = 90° − 38° = 52°.
Answer: The third angle is 52°, which is also acute.
Example 3: Solve for x
An angle measures (4x + 6)°. For which whole-number values of x is the angle acute?
- Acute means 0 < 4x + 6 < 90.
- Subtract 6: −6 < 4x < 84.
- Divide by 4: −1.5 < x < 21.
- The whole numbers in this range are 0, 1, 2, …, 20.
Answer: Any whole number from 0 through 20 works: 21 values in all, from 6° up to 86°.
Common mistakes
The three names do not overlap. Less than 90 is acute, exactly 90 is right, between 90 and 180 is obtuse.
A protractor has two scales. Decide first whether the angle looks acute, then pick the reading under 90.
Every triangle has at least two acute angles, so the third angle decides the triangle type.
Not to be confused with
An obtuse angle is wider than a right angle, between 90° and 180°. 70° is acute; 110° is obtuse.
An acute angle is one angle. An acute triangle is a triangle whose three angles are all acute, such as 50°, 60°, 70°.
Quick check: 3 questions
Which angle is acute?
Answer: D. 81° is less than 90°, so it is acute. 90° is a right angle, 91° is obtuse, and 180° is a straight angle.Show answer and explanation
Two angles of a triangle are 25° and 70°. What kind of angle is the third one?
Answer: A. The third angle is 180° − 25° − 70° = 85°, which is under 90°, so it is acute. Choosing obtuse comes from adding only the two given angles (95°) and mistaking that sum for the third angle.Show answer and explanation
A right angle is cut into two angles by a ray. One part is 33°. What is the other part?
Answer: B. The parts add to 90°: 90° − 33° = 57°, also acute. 147° subtracts from 180°, 67° is an arithmetic slip, and 33° assumes the parts are equal.Show answer and explanation
Acute Angle: frequently asked questions
Is a 90 degree angle acute?
No. An angle of exactly 90° is a right angle. Acute angles are strictly less than 90°, so 89° and 89.9° are acute but 90° is not. The names acute, right and obtuse never overlap, which is why a test can ask you to classify any angle with exactly one of them.
Can a triangle have three acute angles?
Yes. A triangle with all three angles under 90° is an acute triangle, for example one with angles 55°, 60° and 65°. Every equilateral triangle is acute, since each of its angles is 60°. What a triangle cannot have is fewer than two acute angles, because its angles must add to 180°.
What are real-life examples of acute angles?
The tip of a pizza slice cut into 8 equal pieces makes a 45° angle. Clock hands at 1:00 form a 30° angle. A partly open laptop screen, the point of a pencil and the letter V all show acute angles. In each case the opening is narrower than the corner of a sheet of paper.
Practice and tools
Sources
- Common Core State Standards, Grade 4 Geometry
- Common Core State Standards, Grade 4 Measurement and Data
- OpenStax, Contemporary Mathematics, 10.2 Angles
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.G.A.1, CCSS 4.G.A.2, CCSS 4.MD.C.6
Cite this entry
Effortless Math Team. “Acute Angle.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/acute-angle/
Flashcard
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