Acute angle

An acute angle is an angle that measures more than 0° and less than 90°. It is smaller than a right angle, the square corner of a sheet of paper. Angles of 30°, 45° and 89° are all acute. A triangle whose three angles are all acute is called an acute triangle.

What is an acute angle?

An angle is formed where two rays meet at a point called the vertex, and its size is how far one ray turns to reach the other, measured in degrees. A full turn is 360°, so a quarter turn, a right angle, is 90°. Any angle smaller than that quarter turn, but bigger than nothing at all, is acute.

Exactly 90° is not acute: that is a right angle. And 0° is not acute either, because the two rays point the same way, so there is no opening at all.

A quick way to check

Hold the corner of a sheet of paper, an index card or a book against the angle, with one edge along one ray. If the other ray is inside the corner, the angle is acute. If it lines up with the edge, it is a right angle. If it is outside, it is obtuse.

Acute angle examples

In everyday life, you see acute angles in a slice of pizza, the point of a pencil, the tip of a star, the letters V and A, a partly open laptop lid or door, and the hands of a clock at 2:00, which make a 60° angle.

Acute angles on a clock

The hour hand moves 30° every hour and the minute hand moves 6° every minute, so you can work out the angle between them. At 2:00 the hands are 2 × 30° = 60° apart: acute. At 4:00 they are 120° apart: obtuse. At 12:10, the minute hand has moved 60° and the hour hand 5°, so they are 55° apart: acute.

Types of angles

Angles are named by their size. An acute angle is the smallest kind.

A good way to see the line between acute and obtuse is the cosine of the angle, which you meet in trigonometry.

Here x is the angle in degrees and y is its cosine. For every acute angle the cosine is positive, at exactly 90° it is zero, and for every obtuse angle it is negative. That is the reason behind the triangle test further down this page.

How to measure an acute angle

1. Put the center point of the protractor on the vertex of the angle.
2. Line up the protractor's zero line with one ray.
3. Read the scale where the other ray crosses it. Use the scale that starts at 0 on the ray you lined up.
4. Check that your reading makes sense. For an acute angle it must be less than 90. If you read 140, you used the wrong scale, and the angle is really 40°.

Most protractors have two scales running in opposite directions. Picking the wrong one is the most common measuring mistake, and the acute or obtuse check catches it every time. In the US, students learn to measure angles in whole degrees with a protractor, and to draw acute, right and obtuse angles, in grade 4.

Acute triangles

An acute triangle is a triangle in which all three angles are acute. Every triangle has at least two acute angles, because its angles add up to 180°: if one angle is 90° or more, the other two must share 90° or less. So you name a triangle by its largest angle.

An equilateral triangle is always acute, with three 60° angles. In fact it is the only regular polygon with acute angles: a square has 90° corners and a regular pentagon has 108° corners.

Example 1: is the triangle acute?

Two angles of a triangle are 50° and 60°. The third is 180° − 50° − 60° = 70°. Every angle is under 90°, which makes it an acute triangle.

Example 2: using the side lengths

A triangle has sides of 6, 7 and 8. Is it acute? Compare the square of the longest side with the sum of the squares of the other two:

64 is less than 85, so the largest angle is less than 90° and the triangle is acute. If the two numbers were equal the triangle would be right, and if the longest side's square were bigger it would be obtuse. This works because of the law of cosines and the cosine graph above.

Check that you can sort angles and triangles.

Acute angles in radians

In radians, a right angle is π/2 (about 1.571), so an acute angle is between 0 and π/2. For example, π/6 (30°) and π/4 (45°) are acute, but 2 radians (about 114.6°) is obtuse.

Common mistakes

• Calling 90° acute. Acute means strictly less than 90°. Exactly 90° is a right angle.
• Reading the wrong protractor scale, which turns 40° into 140°. Always ask whether the angle looks smaller or bigger than a square corner.
• Calling a triangle acute because it has acute angles. Every triangle has at least two. It is acute only if all three are.
• Judging by the drawing. Diagrams are not always to scale. Use the measurements or angle marks you are given.
• Thinking a longer ray means a bigger angle. The size of an angle depends only on the turn between the rays, not how long they are drawn.

Practice with a short timed paper

Angle facts come up in every middle and high school geometry course. The daily Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math papers are short timed practice tests of 8 questions, with a new one every day. The Trigonometry course shows how acute angles in right triangles lead to sine, cosine and tangent.

Read next

Maths · GeometryObtuse angle: definition, examples and obtuse trianglesWhat an obtuse angle is, how it compares with acute, right and reflex angles, how to measure one, and how to tell if a triangle is obtuse, with examples.