How to count sides and corners

A side is one straight edge of a flat shape. A corner is a point where two sides meet. Counting them sounds easy, but on a shape with 6 or 8 sides it is easy to count one twice or to miss one. A simple trick stops that.

The rule

A flat shape with straight sides has as many corners as it has sides. To count either, mark where you start, go round the shape once in one direction, and stop when you get back to the start.

How to count them

Pick a starting corner, or a starting side. Put a finger on it, or make a small pencil mark next to it.
Count it as 1. Then go round the shape in one direction, counting each new corner or side as you reach it.
Stop as soon as you are back at your mark. Do not count the first one again.
Check: the number of corners is always the same as the number of sides. If your two counts are different, count again.

Worked examples

How many corners does this shape have?

Start at the ringed corner and count round: 1, 2, 3, 4, 5, 6, 7, 8. The next corner is the ringed one again, so stop. The shape has 8 corners, so it is an octagon.

How many sides does this shape have?

Start with the coloured side at the top and count round: 1, 2, 3, 4, 5. The coloured side comes next, so stop. 5 sides. The short side counts just as much as the long ones, so this is a pentagon.

How many sides and how many corners does this shape have?

It has 4 sides, numbered 1 to 4, and 4 corners, the dots. Each side ends at a corner and each corner joins two sides, so the two counts match.

How many corners do a circle and an oval have?

None. Each one is a single smooth curve with no straight sides, so there is nowhere for two sides to meet. They are not polygons, and their sides are not usually counted.

Corners or vertices?

Corner is the everyday word, and the one younger children use. In maths the corner of a shape is also called a vertex, and more than one are vertices. A triangle has 3 vertices, which means the same as 3 corners. Children meet the word again when they count the faces, edges and vertices of solid shapes.

Why corners and sides always match

Walk round a shape with your finger. You go along a side and reach a corner. You go along the next side and reach the next corner. Side, corner, side, corner: they take turns all the way round until you are back at the start. There is one corner at the end of every side, so the two counts come out the same.

That gives you a free check. Count the sides, then count the corners. If the numbers are different, you have slipped somewhere.

Marking the start matters because going round a shape is a loop. Without a mark, nothing tells you when the loop is finished, so it is easy to carry on and count the first side or corner twice.

Where mistakes happen

Counting the starting corner again at the end. On an octagon that gives 9. Stop when you get back to your mark.
Missing a short side. A very short side is still a side, and its two ends are still corners.
Counting here and there instead of going round in order. That is how a side gets counted twice and another one missed.
Thinking a bigger shape has more sides. Size makes no difference. A tiny triangle and a huge one both have 3 sides and 3 corners.

Questions people ask

How many sides does a circle have?

Books disagree. Some say one curved side, some say none, and some say infinitely many. They all agree that a circle has no straight sides and no corners, so it is not a polygon. Math Basics never asks how many sides a circle has, for that reason.

Are corners and angles the same thing?

Nearly. A corner is the point where two sides meet. The angle is how far the shape opens at that corner. Every corner has one angle inside the shape, so a triangle has 3 corners and 3 angles. Types of angles shows how to name them.

What about a shape with corners that point inwards?

It works the same way. An arrow or a star has some corners that point into the shape, but each one still counts, and the corners still match the sides. The outline of a five-pointed star has 10 sides and 10 corners.