Obtuse angle

An obtuse angle is an angle that measures more than 90° and less than 180°. It is wider than a right angle, the square corner of a sheet of paper, but not yet a straight line. Angles of 100°, 135° and 179° are all obtuse. A triangle with one obtuse angle is called an obtuse triangle.

What is an obtuse angle?

Angles measure how far one ray turns to reach another. A quarter turn is 90°, a right angle, and a half turn is 180°, a straight line. Anything between those two is obtuse.

Exactly 90° is a right angle, not obtuse, and exactly 180° is a straight angle. The word obtuse means blunt: an obtuse angle opens wide, while an acute angle is sharp and narrow.

A quick way to check

Hold the corner of a sheet of paper against the angle, with one edge along one ray. If the other ray falls outside the paper's corner, the angle is obtuse. If it lines up with the edge, it is a right angle, and if it falls inside, it is acute.

Obtuse angle examples

In everyday life you see obtuse angles in a reclining chair leaned back, an open laptop pushed far back, the blades of a ceiling fan seen from below, the corners of a stop sign (each 135°) and the hands of a clock at 4:00.

Obtuse angles on a clock

The hour hand moves 30° each hour and the minute hand 6° each minute. At 4:00 the hands are 4 × 30° = 120° apart, and at 5:00 they are 150° apart: both obtuse. At 10:10 the hour hand is at 305° and the minute hand at 60°, measured clockwise from 12, so the smaller angle between them is 360° − 245° = 115°, also obtuse.

Types of angles

Obtuse vs reflex angles

Every pair of rays makes two angles: one on the inside and one going the long way round. They add up to 360°. If the inside angle is 125°, the outside angle is 360° − 125° = 235°, which is reflex. Unless a question says otherwise, an angle means the smaller one.

The cosine of an angle shows the split between acute and obtuse neatly.

Here x is the angle in degrees and y is its cosine. Every obtuse angle has a negative cosine, which is why the law of cosines can tell you when a triangle is obtuse.

How to measure an obtuse angle

1. Put the center point of the protractor on the vertex.
2. Line up the zero line with one ray.
3. Read where the other ray crosses the scale that starts at 0 on your first ray.
4. Check the reading is more than 90. If you read 55 for an angle that looks wide, you used the wrong scale: the angle is 180° − 55° = 125°.

Under the Common Core standards, US students learn to draw obtuse angles and measure angles with a protractor in grade 4.

Obtuse triangles

An obtuse triangle has one angle greater than 90°. It can only have one, because two obtuse angles would already add up to more than 180°, the total for every triangle. Its other two angles are always acute.

Example 1: is it obtuse?

Two angles of a triangle are 20° and 30°. The third is 180° − 20° − 30° = 130°, which is more than 90°, so this is an obtuse triangle.

Example 2: from the side lengths

A triangle has sides 5, 6 and 9. Compare the square of the longest side with the sum of the squares of the other two:

81 is more than 61, so the angle opposite the longest side is obtuse. The law of cosines gives its size: cos C = (25 + 36 − 81) ÷ (2 × 5 × 6) = −1/3, so C ≈ 109.5°. The Trigonometry calculator solves any triangle like this.

Check that you can classify angles and triangles.

Obtuse angles in shapes

• Regular polygons with 5 or more sides have obtuse interior angles: a regular pentagon has 108° corners, a hexagon 120° and an octagon 135°.
• A parallelogram that is not a rectangle has two obtuse and two acute angles.
• A trapezoid often has obtuse angles at its shorter base.

Obtuse angles in radians

In radians a right angle is π/2 (about 1.571) and a straight angle is π (about 3.142), so an obtuse angle is between π/2 and π. For example, 2 radians, about 114.6°, is obtuse.

Common mistakes

• Calling 90° or 180° obtuse. Obtuse angles are strictly between them.
• Reading the wrong protractor scale, which turns 125° into 55°. Always check whether the angle looks wider than a square corner.
• Confusing obtuse and reflex. An angle over 180° is reflex, not obtuse.
• Thinking a triangle can have two obtuse angles. It can only have one.
• Judging by eye. Diagrams are not always drawn to scale. Use the measurements given.

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.

Read next

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