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Flat and solid: cubes, balls, cones and cylinders

Geometry · level 1, lesson 2 of 5

A question to start

Can you pick up a square?

You can pick up a cracker that is shaped like a square. You can pick up a sticky note. But the square itself, the shape drawn on the paper, is flat. It has no thickness at all. You cannot hold it, roll it or stack it. It lives only on the page.

Now pick up a dice, a ball or a can of soup. These are not flat. They take up space. You can hold them, turn them round, look at them from the back. They are solid shapes. In this lesson you will meet six solid shapes that are hiding all around your house, and you will find out a clever trick: every solid shape has flat shapes hiding on it.

The idea

Flat shapes and solid shapes

A flat shape can be drawn on paper. It has a length and a width, but no thickness. Circles, triangles, squares and rectangles are flat shapes. Grown-ups call them 2D shapes ("two-D"), because they go in two directions: across and up.

A solid shape takes up space. You can hold it in your hand. It has length, width and height (thickness). Grown-ups call them 3D shapes ("three-D"), because they go in three directions: across, up, and towards you.

flat: square solid: cube flat: circle solid: ball

A square is flat. A cube is like a square that has grown thick, as thick as it is wide. A circle is flat. A ball is round in every direction you look.

Six solid shapes

cube box ball (sphere) cylinder cone pyramid

Cube. A cube is like a dice or a building block. Every face (every flat side) is a square, and all the squares are the same size. It has sharp corners and straight edges.

Box. A box is like a cereal box, a brick or a shoebox. Its flat sides are rectangles. Grown-ups call it a cuboid or a rectangular prism. A cube is a special box whose sides are all squares, just like a square is a special rectangle.

Ball. A ball is perfectly round, like a marble, an orange or a globe. The math name for a ball shape is a sphere (say "sfeer"). Every point on its outside is the same distance from its middle. It has no flat parts, no edges and no corners.

Cylinder. A cylinder (say "SIL-in-der") is like a can of soup, a toilet paper roll, or a glue stick. It has a circle at the top and a circle at the bottom, and a curved side that wraps round.

Cone. A cone is like an ice cream cone or a party hat. It has a circle at the bottom, and its curved side gets narrower and narrower until it ends in a point at the top.

Pyramid. A pyramid has a flat bottom (here a square) and triangle sides that lean in and meet at a point at the top. The famous old pyramids in Egypt have this shape.

Some roll, some slide, some stack

Here is a great experiment. Find some solid shapes at home: a ball, a can, a box, a block. Try three things with each one.

  1. Roll it. Push it gently on the floor. Does it roll?
  2. Slide it. Push it on a flat side. Does it slide without rolling?
  3. Stack it. Can you put another one on top so it stays?

Here is what you will find, and why.

Shape Rolls? Slides? Stacks? Why
ball yes, any way no no it is curved everywhere, with no flat part to rest on
cylinder yes, on its side yes, on an end yes, on its ends curved side for rolling, two flat circles for standing
cone in a circle yes, on its base not easily one flat circle at the bottom, but a point at the top
cube no yes yes all flat faces, so it sits still
box no yes yes all flat faces
pyramid no yes no (a point on top) flat bottom, but a point at the top

The big idea: curved parts roll, flat parts slide and stack. That is why balls are round (so they roll) and why bricks are boxes (so they stack into walls with no gaps). A builder could not make a wall out of balls!

The cone is funny. If you lay it on its curved side and push it, it does not roll in a straight line. It rolls round and round in a circle, because its point stays in one place while the wide end goes round.

Flat shapes hide on solid shapes

Here is the clever trick. Dip the bottom of a solid shape in paint and press it on paper, like a stamp. What shape do you get?

a circle a square a rectangle a circle
  • A cylinder stands on a circle, so it prints a circle.
  • A cube prints a square. Every face of a cube is a square, so any face you choose prints a square.
  • A box prints a rectangle (or, on its ends, maybe a square).
  • A cone prints a circle from its bottom.
  • A pyramid with a square bottom prints a square. If you press one of its slanted sides instead, it prints a triangle.

So every flat side of a solid shape is a flat shape. That is how flat shapes and solid shapes are connected. The flat sides of a solid are called its faces. You will count faces carefully in level 2.

Our round world

Is the Earth flat or solid? It is a solid, and its shape is very nearly a sphere: a huge ball. Mars, the red planet, is nearly a ball too. Big planets end up round because gravity pulls all their rock towards the middle from every side, and the shape that has every outside point the same distance from the middle is a ball.

Worked examples

Example 1. Name the solid. It has 6 flat faces. All of them are the same size and all are squares. What is it?

  1. It has flat faces, so it is not a ball, a cylinder or a cone (those have curved parts).
  2. All faces are squares, all the same size.
  3. That is a cube, like a dice.

Example 2. Solid or flat? Sort these: a drawing of a circle, a coin, an orange, a sticker, a tin can.

  1. A drawing of a circle lives on the paper: flat.
  2. A sticker is very thin. In math we treat it as flat: flat (a circle or other shape).
  3. An orange is round in every direction: solid, a sphere.
  4. A tin can: solid, a cylinder.
  5. A coin is tricky! It is very thin, but it does have a little thickness. Really it is a very, very short cylinder. Its face is a circle. So we say: the coin is a solid (a flat cylinder), and its face is a circle.

Example 3. Which shape could it be? "I can roll, and I can also stand still on a flat end. If you paint my end and stamp it, you get a circle." What am I?

  1. "I can roll" means I have a curved part. So I am a ball, a cylinder or a cone.
  2. "I can stand still on a flat end" means I have a flat part. A ball has no flat part, so I am not a ball.
  3. Stamping my end gives a circle. That fits both a cylinder and a cone!
  4. Which is it? Both answers fit the clues. A cylinder rolls in a straight line and a cone rolls in a circle, so one more clue, "I roll in a straight line", would tell us it is a cylinder. Sometimes a riddle has more than one answer, and a good mathematician says so.

Try it

  1. Is a drawing of a triangle flat or solid?
  2. Name the solid shape: a can of beans.
  3. Name the solid shape: a tennis ball.
  4. Which of these can roll: a cube, a ball, a cylinder, a box?
  5. You stamp a cube in paint. What flat shape do you get?
  6. Why do people build walls out of bricks and not out of balls?
  7. I have one flat face that is a circle, a curved side, and a point. What am I?
  8. Go on a shape hunt at home. Find one cube, one box, one sphere, one cylinder and one cone (or something nearly that shape).
Answers
  1. Flat. It is drawn on paper and has no thickness. You could not pick it up off the page.
  2. A cylinder: a circle on top, a circle on the bottom, and a curved side.
  3. A sphere (a ball). It is round in every direction.
  4. The ball and the cylinder. They have curved parts. The cube and the box have only flat faces, so they slide but do not roll.
  5. A square. Every face of a cube is a square.
  6. Bricks are boxes with flat faces, so they sit still on top of each other and fit side by side with no gaps. Balls touch only at a tiny point, so they roll off each other and leave big gaps.
  7. A cone, like an ice cream cone or a party hat turned upside down.
  8. Some ideas. Cube: a dice, a toy block, a sugar cube. Box: a cereal box, a book, a brick. Sphere: a ball, an orange, a marble. Cylinder: a can, a candle, a crayon (nearly), a toilet paper roll. Cone: an ice cream cone, a party hat, a traffic cone.

Common mistake

Calling a ball a circle. A circle is flat, drawn on paper. A ball is solid, round in every direction. When you look at a ball, its outline looks like a circle, which is why people mix them up. But you can hold a ball and you cannot hold a circle. The ball's math name is sphere.

Calling a cube a square. In the same way, a square is flat, and a cube is solid. A cube has 6 squares on it, one on each face.

Where it leads