Angles in shapes

Every corner of a flat shape is an angle. The two sides that meet at the corner are its arms. So the corners can be named the same way as any angle: acute, right or obtuse, by holding them against a square corner.

The rule

Look only at the marked corner, on the inside of the shape. If it is narrower than a square corner, it is acute. If it is exactly square, it is a right angle. If it is wider, it is obtuse.

How to name a corner

Find the marked corner. It has a small curve drawn in it, or a small square if it is a right angle. Ignore the other corners for now.
Look at the two sides that meet at that corner. They are the arms of the angle. Look at the gap between them inside the shape, not outside it.
Compare it with a square corner, like the corner of a page. Narrower is acute, the same is a right angle, and wider is obtuse.
If the shape is lying on its side, turn the page or tilt your head. Turning a shape does not change its corners.

Worked examples

What kind of angle is marked on this square?

A right angle, and the small square says so. Every corner of a square is a right angle, 90 degrees, and so is every corner of a rectangle.

What kind of angle is marked on this triangle?

An acute angle. This is an equilateral triangle, with three equal sides, and each of its corners is 60 degrees, narrower than a square corner. All three corners are acute.

What kind of angle is marked on this hexagon?

An obtuse angle. In a regular hexagon, with six equal sides and six equal corners, every corner is 120 degrees, wider than a square corner.

What kind of angle is marked on this rhombus?

An obtuse angle. The marked corner at the top is wider than a square corner. The corners at the left and right are narrower, so they are acute. A shape can have both kinds, and this rhombus has two of each.

The corners of regular shapes

In a regular shape every side is the same length and every corner is the same size. The more sides it has, the wider its corners are.

Triangle: 60 degrees, acute. Square: 90 degrees, right. Pentagon: 108 degrees, obtuse. Hexagon: 120 degrees, obtuse. Octagon: 135 degrees, obtuse.

So a regular triangle has acute corners, a square has right angles, and every regular shape with five sides or more has obtuse corners. Regular and irregular shapes explains what makes a shape regular.

Why a triangle's angles add up to 180 degrees

Draw any triangle on paper and cut it out. Tear off its three corners and put them side by side with their points touching. They always make a straight line, and a straight line is 180 degrees. So the three angles of any triangle add up to 180.

That is why no triangle can have two right angles: 90 + 90 is already 180, leaving nothing for the third corner. And a triangle can have at most one obtuse angle.

Any straight-sided shape can be cut into triangles by drawing lines from one corner to the others. A square makes 2 triangles, so its corners add up to 2 × 180 = 360, and each of the four is 360 ÷ 4 = 90. A hexagon makes 4 triangles: 4 × 180 = 720, and 720 ÷ 6 = 120 for each corner of a regular one. That is why more sides means wider corners.

Where mistakes happen

Naming the shape instead of the corner. Not every triangle has acute corners. A right-angled triangle has one right angle and two acute ones, and some triangles have an obtuse corner.
Thinking a turned shape has different corners. A square stood on one corner, like a diamond, still has four right angles.
Expecting every corner of a shape to be the same. A rhombus, a parallelogram and many other shapes have acute and obtuse corners together.
Looking at the outside of the corner. The angle that counts is the gap inside the shape. The angle outside, the long way round, is bigger than a straight line.
Thinking more sides make sharper corners. It is the other way round, and a regular octagon's corners are wider than a square's.

Questions people ask

Can a triangle have a right angle?

Yes, one. It is called a right-angled triangle, and its other two corners are always acute, because together they make the other 90 degrees.

Are all the corners of a rectangle right angles?

Yes, all four. That is what makes it a rectangle and not just any shape with four sides.

What if a corner is very nearly a right angle?

Then you need a protractor to tell, and even a careful eye can be wrong. The app only marks corners that are clearly one kind or another, so a corner it asks about is either a right angle or at least 12 degrees away from one.

Do the angles of a four-sided shape always add up to 360 degrees?

Yes, for any shape with four straight sides, because it can be cut into two triangles of 180 degrees each.