Types of angles

An angle is the amount of turn between two lines that meet at a point. The two lines are called arms, and the point is called the vertex. An angle is named by how wide it opens, and the easiest way to judge that is to hold it against a square corner, like the corner of a page.

The rule

An acute angle is smaller than a right angle. A right angle is exactly a square corner, 90 degrees. An obtuse angle is bigger than a right angle but smaller than a straight line, so between 90 and 180 degrees.

How to tell which kind it is

Find the vertex, the point where the two arms meet. Look at the gap between the arms, not at how long the arms are.
Take something with a square corner, like a sheet of paper. Put its corner on the vertex and one of its edges along one arm.
If the other arm is inside the paper's corner, the angle is acute. If it lies exactly along the paper's other edge, it is a right angle. If it is outside, the angle is obtuse.
A small square drawn in the corner of an angle means it is a right angle. You do not need to check it.
To be exact, measure in degrees with a protractor. Less than 90 degrees is acute, 90 is right, and more than 90 but less than 180 is obtuse.

Worked examples

What kind of angle is this?

An acute angle. Put a square corner on the bottom arm and the other arm stays inside it. It opens 40 degrees, less than 90.

What kind of angle is this one, turned on a slant?

A right angle. The small square in the corner says so, and the corner of a page laid along one arm would line up with the other. Turning an angle does not change it, so it is still 90 degrees, even though neither arm is flat.

What kind of angle is this?

An obtuse angle. The arm leans back past where a square corner would be, so the angle is wider than a right angle. It opens 130 degrees, between 90 and 180.

What angle do the hands of a clock make at 4 o'clock?

360 ÷ 12 = 30
4 × 30 = 120

A full turn is 360 degrees and a clock has 12 numbers, so the hands turn 30 degrees from one number to the next. At 4 o'clock they are 4 × 30 = 120 degrees apart. 120 degrees is more than 90 and less than 180, so the hands make an obtuse angle. At 3 o'clock they make a right angle, 90 degrees, and at 2 o'clock an acute one, 60 degrees.

Straight and reflex angles

A straight angle is exactly 180 degrees. The two arms point in opposite directions and make a straight line, like the hands of a clock at 6 o'clock.

A reflex angle is bigger than a straight line, between 180 and 360 degrees. Every angle has a reflex angle on its other side, the long way round. If the gap between two arms is 40 degrees, the way round outside is 360 − 40 = 320 degrees.

Children meet these later than acute, right and obtuse. A corner of a shape like a square or a triangle is never straight or reflex, which is why the app's practice leaves them out.

Measuring with a protractor

Put the small cross or hole in the middle of the protractor's straight edge on the vertex. Turn it so the 0 line runs along one arm. Then follow the scale that starts at 0 on that arm round to the other arm, and read the number.

A protractor has two scales, one on the inside and one on the outside. Before you read it, decide whether the angle is acute or obtuse. If it is acute and you read 140, you have used the wrong scale, and the answer is 40.

Why a right angle is 90 degrees

Degrees measure turn. A full turn, all the way round, is split into 360 equal parts, and each part is one degree. A right angle is a quarter of a turn: 360 ÷ 4 = 90. A straight line is half a turn: 360 ÷ 2 = 180.

That is why the right angle is the one to compare against. Acute means less than a quarter turn, and obtuse means more than a quarter turn but less than a half.

360 was chosen thousands of years ago, probably because it is close to the number of days in a year, and it divides exactly by 2, 3, 4, 5, 6, 8, 9, 10 and 12. That makes halves, thirds, quarters and many other parts of a turn come out as whole numbers.

Where mistakes happen

Judging an angle by the length of its arms. Long arms do not make a big angle. Only the gap between them counts, and a small angle drawn with long arms is still small.
Thinking a right angle has to stand upright, with one arm flat and one straight up. A right angle turned on a slant is still a right angle.
Mixing up acute and obtuse. Acute is the small one. Some children remember it as "a cute little angle".
Reading the wrong scale on a protractor, and writing 140 degrees for an angle that is plainly smaller than a right angle. Decide acute or obtuse first, then measure.
Thinking the size of the arc drawn in the corner shows the size of the angle. A small arc and a big arc on the same angle mark the same angle.

Questions people ask

Why is it called a right angle?

The word comes from the Latin for upright. It has nothing to do with right and left, or with right and wrong.

Is 90 degrees acute or obtuse?

Neither. 90 degrees is a right angle. Acute is anything less than 90, and obtuse is anything more than 90, up to 180.

What does the little square in a corner mean?

It means the angle is a right angle, exactly 90 degrees. Books and worksheets draw it so that you do not have to measure.

How can my child make a right-angle checker?

Fold any piece of paper in half, then fold it in half again so that the folded edge lies exactly on top of itself. The corner where the folds meet is a right angle, and it can be held against angles to test them.