Circles, triangles, squares and rectangles

Geometry · level 1, lesson 1 of 5

A question to start

Here is a riddle. I am a shape. I have 3 straight sides and 3 corners. You can turn me upside down, stretch me tall, or squash me flat, and I keep my name. What am I?

You may have said "a triangle" right away. Good! But here is the surprising part. A triangle can look very different. It can be tall and thin like a party hat. It can be low and wide like a slice of pizza lying down. It can stand on its point. And it is still a triangle. Why? Because a shape's name does not come from how big it is or which way it points. It comes from counting. In this lesson you will learn to count two things, sides and corners, and then you will be able to name shapes like a real mathematician.

The idea

Four shapes to know

Look at these four shapes. You see them every day.

• A circle is round, like a coin, a clock or the top of a cup.
• A triangle has 3 sides, like a slice of sandwich cut from corner to corner.
• A square has 4 sides that are all the same length, like a cracker or a sticky note.
• A rectangle has 4 sides too, like a door, a book or a phone.

Geometry is the part of math about shapes. Mathematicians do not just look at a shape and guess its name. They look at its parts.

Sides and corners

A side is a straight edge of a shape. Run your finger along it. It does not bend.

A corner is where two sides meet. It is a pointy place. If you were an ant walking along the edge of the shape, a corner is where you would have to turn.

Let us count the triangle. Put your finger on one corner (the dots). Walk along a side to the next corner. That is side 1. Walk to the next corner. That is side 2. Walk back to where you started. That is side 3. So a triangle has 3 sides and 3 corners.

Now count the square: 4 sides and 4 corners.

Here is a good question. Why does a triangle have the same number of sides and corners? Think of the ant again. The ant walks a side, then turns at a corner, then walks a side, then turns at a corner. Walk, turn, walk, turn. Every side ends at a corner. So when the ant gets back home, it has walked the same number of sides as it has turned corners. That is true for every shape with straight sides: the number of sides and the number of corners always match.

The circle is different

A circle has no straight sides and no corners. It is one smooth curve that goes round and round. The ant walking on a circle never has to make a sharp turn. It just keeps gently turning a little bit all the time.

There is one more special thing about a circle. Find its middle point. Every point on the circle is the same distance from the middle. That is why a wheel is a circle: the middle of the wheel (where the axle goes) stays the same height as the wheel rolls. A square wheel would go bump, bump, bump!

Triangles come in many looks

All of these are triangles. Count the sides and corners of each one: 3 and 3, every time.

So big or small, tall or flat, pointing up or pointing down, if it has 3 straight sides and 3 corners, it is a triangle.

But be careful. These are not triangles:

• The first one has a gap. A shape must be closed, with no gap, like a fence all the way round a field.
• The second one has bendy sides. The sides of a triangle must be straight.
• The third one has 4 sides. Count them!

Squares and rectangles

A square has 4 straight sides and 4 corners. Its sides are all the same length. Its corners are all square corners.

What is a square corner? It is the corner of a sheet of paper, of a book, of a door. It is not sharp and pointy, and it is not wide and flat. You can test a corner with the corner of a piece of paper: if they match exactly, it is a square corner.

The shape on the left has 4 square corners. The little squares drawn inside its corners show it. The shape on the right has 4 sides too, but its corners are not square. Two are sharp and two are wide. It is not a rectangle and not a square.

A rectangle has 4 straight sides and 4 square corners. But its sides do not have to be all the same length. Usually two sides are long and two sides are short.

Here is a fun fact that surprises many grown-ups. A square follows all the rules of a rectangle: 4 sides, 4 square corners. So a square is a special kind of rectangle, the kind with all sides the same. Every square is a rectangle. But not every rectangle is a square, because most rectangles have long sides and short sides.

Turning a shape does not change its name

Look at this square. Now turn it so it stands on one corner.

Some people call the turned one a "diamond". But nothing about it changed! It still has 4 sides, all the same length, and 4 square corners. It is the same square, just turned. Turning a shape, or moving it, or flipping it over, never changes its name.

Worked examples

Example 1. Name the shape. A shape has 4 straight sides. Two sides are long and two are short. All 4 corners are square corners. What is it?

1. Count the sides: 4. So it is not a triangle, and it is not a circle.
2. Check the corners: all square corners. So it is a rectangle or a square.
3. Check the sides: they are not all the same length (two long, two short).
4. So it is a rectangle, not a square. A door is a good example.

Example 2. Count the shapes in a picture. How many of each shape can you find in this house?

1. The big part of the house: 4 sides, all the same, square corners. 1 square.
2. The roof: 3 sides, 3 corners. 1 triangle.
3. The door: 4 sides, two long and two short, square corners. 1 rectangle.
4. The windows: 4 equal sides, square corners. 2 squares.
5. The sun: round, no corners. 1 circle.

So the picture has 3 squares, 1 triangle, 1 rectangle and 1 circle. That is 6 shapes in all. (Did you count the big square of the house as well as the windows? It is easy to miss the biggest shape!)

Example 3. A shape with 6 corners. Sam draws a shape with straight sides. He counts 6 corners. How many sides does it have?

1. Remember the ant: walk a side, turn at a corner, walk a side, turn at a corner.
2. Every side ends at a corner, and every corner starts a new side.
3. So the number of sides is the same as the number of corners: 6 sides.

A shape with 6 sides has a name too: a hexagon. You can see hexagons in a honeycomb. You will meet many more shapes like this in level 2.

Try it

1. How many sides does a triangle have? How many corners?
2. I have no corners and no straight sides. What am I?
3. Is a slice of pizza (cut from the middle) a triangle? Look closely at the side at the crust.
4. A shape has 4 sides that are all the same length and 4 square corners. What is it? Can you give it a second name too?
5. Find 3 rectangles in your room. Check their corners with the corner of a piece of paper.
6. Mia says: "This triangle is upside down, so it is not a triangle any more." Is she right?
7. A shape has 5 corners. How many sides does it have?
8. Draw a shape with 4 sides that is not a rectangle. What did you change?

Answers

1. 3 sides and 3 corners. Count with your finger: walk along a side, stop at a corner, three times, and you are back where you started.
2. A circle. It is one smooth round curve, with no straight parts and no sharp turns.
3. Almost, but not quite. Two sides are straight, but the crust is a curved side. A true triangle has 3 straight sides. So a pizza slice is "nearly a triangle". A sandwich cut from corner to corner is a real triangle.
4. A square. Its second name is rectangle, because it has 4 sides and 4 square corners, which is all a rectangle needs. A square is a special rectangle.
5. Some ideas: a door, a book, a window, a picture frame, the screen of a tablet, a drawer. Their corners match the corner of the paper exactly.
6. No. Turning a shape does not change its name. It still has 3 straight sides and 3 corners, so it is still a triangle.
7. 5 sides. The number of sides always matches the number of corners, because every side ends at a corner. (A shape with 5 sides is called a pentagon.)
8. One way: draw 4 sides but make the corners not square, like a shape that leans over, or like the right-hand shape in the "square corners" picture. Another way: make one side much shorter so the top is narrower than the bottom. You changed the corners. A rectangle must have 4 square corners.

Common mistake

Thinking a turned shape is a different shape. Many children (and some grown-ups) say a square standing on its corner is "a diamond, not a square", or that an upside-down triangle "is not a triangle". But a shape's name comes from counting its sides and corners and checking them, not from which way it points. Turn the page and look again: nothing about the shape changed.

Thinking a square is not a rectangle. A rectangle needs 4 sides and 4 square corners. A square has both. So a square is a rectangle, just a special one with all its sides the same length.

Where it leads

• Next lesson: Flat and solid, where shapes jump off the page and become things you can hold, like balls and boxes.
• This level: Level 1: First shapes. The whole topic: Geometry.
• Counting sides and corners is real counting. Practise counting in Numbers.
• Shapes in a row that repeat, like circle, square, circle, square, make a pattern: Patterns.
• Later, in level 2, you will name shapes with 5, 6, 7 and 8 sides.