Shapes all around us

Look around the room you are sitting in. The door is a tall rectangle. The clock on the wall is a circle. The carrom board is a square. Your pencil box is a cuboid, the steel tumbler is nearly a cylinder, and the laddoo on the plate is almost a sphere.

Shapes are everywhere, and people have been naming them for thousands of years so that they can talk about them clearly: a carpenter asking for a plank, an architect drawing a temple, a baker cutting a cake, a bee building a honeycomb (though bees never learned the names!).

In this layer you will meet the big families of shapes: flat shapes you can draw on paper, and solid shapes you can hold in your hand. You will learn how to count their sides, corners, faces and edges, see how a flat pattern folds into a box, look at objects from above and from the side, and find the magic of symmetry in butterflies, rangoli and the Taj Mahal.

Draw a square on paper. You can measure how long it is and how wide it is, but you cannot pick it up off the page. It has no thickness. Shapes like this are called two-dimensional, or 2D for short, because they have two measurements: length and breadth.

Now pick up a matchbox. It has a length and a breadth, but it also has a height (or thickness). It takes up space. Shapes like this are called three-dimensional, or 3D: they have three measurements.

A rangoli on the floor is 2D: it is a pattern of colour lying flat. A laddoo is 3D: you can hold it, turn it round and take a bite. A photo of a laddoo is 2D again, because the photo is flat even though the laddoo in it is not.

Put your pencil on paper and draw without lifting it. Whatever you draw is called a curve in maths, even if it is made only of straight bits! (Yes, mathematicians call a straight line a kind of curve.)

Some curves end where they began, and fence in a region, like the outline of a pond or a cricket ground. These are closed curves. Others have two free ends, like a piece of string lying on the table or the letter S. These are open curves.

A closed curve that does not cross itself is called a simple closed curve. A figure of 8 is closed but not simple, because it crosses itself in the middle.

A simple closed curve splits the page into three parts: the interior (inside), the exterior (outside) and the boundary (the curve itself). A cricket ground is a perfect example: the grass inside the rope is the interior, the stands are the exterior, and the rope is the boundary. If the ball touches the rope, it is on the boundary, and that is four runs!

A polygon is a simple closed figure made only of straight line segments. The segments are its sides. The corners where two sides meet are its vertices (one corner is a vertex).

A triangle is a polygon. A square is a polygon. The outline of a star drawn with five straight strokes (without crossing lines) is a polygon. But a circle is not a polygon, because it has no straight sides, and the letter C is not a polygon because it is open.

Poly is an old Greek word meaning many, and gon comes from the word for angle or knee. So a polygon is a many-cornered shape.

Some polygons are extra neat: all their sides are the same length and all their angles are the same size. These are called regular polygons. A square is a regular quadrilateral. An equilateral triangle is a regular triangle. A honeycomb cell is (very nearly) a regular hexagon.

Polygons that are not like this are irregular. The side view of a house with a roof is an irregular pentagon: its sides are not all equal. A rectangle that is not a square is irregular too: its angles are equal, but its sides are not.

A triangle has 3 sides, 3 vertices and 3 angles. It is the polygon with the fewest possible sides: you cannot close a shape with only two straight sides.

Triangles are everywhere in building. Look at a railway bridge, an electricity tower or the frame of a roof: you will see triangles again and again. Why? Push on the corner of a square made of four sticks joined at the corners, and it squashes into a slanted shape. Push on a triangle made of three sticks and it will not change shape unless a stick bends or breaks. Three lengths fix a triangle completely. That makes triangles rigid, and engineers love them.

Any polygon with four sides is a quadrilateral (quadri means four, lateral means side). That covers a huge family: doors, windows, tiles, a cricket pitch, a carrom board, a diamond-shaped patang flying at Makar Sankranti, the slanted shape of a bench seen from the side.

Some family members are so common that they have their own names.

Tie a string to a peg in the ground, pull it tight and walk around the peg with a stick in the other hand, scratching the soil. The mark you make is a circle: every point on it is exactly the same distance from the peg. Gardeners in India still mark out round flower beds this way.

The peg is the centre of the circle. The length of the string is the radius. A line straight across the circle through the centre is a diameter, and it is always twice the radius. The distance all the way round the circle is called the circumference.

Why are wheels round? Because the axle at the centre is always the same distance (one radius) from the road. As a round wheel rolls, the axle glides along at a steady height and the cart does not bump. Try to imagine a square wheel: the axle would rise up as the wheel tipped onto a corner and crash down again as it landed on the next side. Every cart, cycle, bullock-cart and train in the world uses the circle's special property.

Now let us leave the page and pick things up. Here are the solid shapes you will meet most often, with something from an Indian home or street for each one.

To describe a solid shape exactly, we count three things:

• Faces: the flat surfaces. A dice has 6 faces, each with a different number of dots.
• Edges: the lines where two faces meet. Run your finger along the edge of a matchbox and you feel a sharp line.
• Vertices: the corners, where edges meet. A corner of a brick is a vertex.

Counting them carefully is harder than it sounds, because you cannot see them all at once. The trick is to count in an organised way.

Take an empty toothpaste box or a sweet box and carefully open it out along its glued flap until it lies flat. The flat shape you get is called a net of the box. Fold it back up along the creases and the box appears again.

A net is a flat pattern that folds up to make a solid. The net of a cube is made of 6 squares joined edge to edge. One famous cube net looks like a cross: a line of four squares with one extra square sticking out above and one below.

Now put a tumbler on the table. Look at it from the side: you see a rectangle. Look straight down from the top: you see a circle. The same solid looks like different flat shapes from different directions. These are called views.

Architects draw a top view (called a plan), a front view and a side view of every building, so that builders know exactly what to make. A map is a giant top view of a town.

Fold a paper butterfly down the middle and the two halves fit exactly on top of each other. The fold line is called a line of symmetry, or a mirror line, because if you stand a mirror on it, the reflection of one half looks just like the other half.

A shape with at least one line of symmetry is called symmetrical. You can find symmetry in a leaf, a human face (nearly), the letter A, a rangoli, a temple doorway and the most famous building in India: the Taj Mahal, whose front is almost perfectly symmetrical about a line down the middle of its great arch and dome.

A tangram is an old puzzle from China made by cutting one square into 7 pieces: 2 large triangles, 1 medium triangle, 2 small triangles, 1 square and 1 parallelogram. The challenge is to use all seven pieces, without overlapping, to make pictures: a cat, a boat, a running man, a house, even the letters of your name. Every picture has exactly the same area as the original square, because it uses the same pieces.

Puzzles like this train the eye to see how shapes fit together, turn and flip, which is exactly the skill architects, tailors and tile-layers use every day.