Geometry — Class 4 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 4 Mathematics, Unit 1. Properties of 2D shapes, creating 2D shapes, and properties of 3D objects. Taught in June.
1. Drawing a circle, and naming its parts
Place a bangle on a sheet of paper and trace around it. What you get is a circle. But tracing does not show you what a circle really is, so the textbook asks you to draw one with a thread instead.
Mark a point on the paper and call it O. Tie one end of a thread at O and a pencil at the other end. Keep the thread stretched tight and swing the pencil all the way round. The curve you draw is a circle, and O is its centre.
Notice what the thread did. It never changed length. So every point on that circle is exactly the same distance from O. That distance is the radius.
Now take any two points on the circle and join them with a straight line. That line segment is a chord. If the points are L and M, the chord is LM.
One chord is special. A chord that passes through the centre O is the diameter. Because it runs from the circle, through the centre, and out to the circle again, it is made of two radii laid end to end:
Diameter = 2 × radius, and the radius is half the diameter.
The diameter is also the longest chord you can draw in a circle. Any chord that misses the centre is shorter.
Worked from the book. The radius of a circle is 5 cm. Find the diameter. Diameter = 2 × radius = 2 × 5 = 10 cm.
2. Quadrilaterals
Any closed figure with four sides is a quadrilateral. Every quadrilateral has 4 sides, 4 vertices (corners), and 2 diagonals — the lines joining opposite corners.
| Quadrilateral | What makes it special | Everyday examples |
|---|---|---|
| Square | 4 equal sides; diagonals equal | Carrom board, chess board |
| Rectangle | Opposite sides equal; diagonals equal | Mat, blackboard, post card |
| Rhombus | 4 equal sides; diagonals not equal | Kite, tile, sign board |
| Parallelogram | Opposite sides parallel and equal | Slanted signboard |
| Trapezium | Only one pair of parallel sides | Handbag flap |
The rhombus is the one to watch. It has four equal sides just like a square, so children often call it a square. The difference is in the diagonals: a square's two diagonals are equal in length, a rhombus's are not.
In a parallelogram EFGH, opposite sides are parallel and equal, so EF = GH and FG = HE.
3. Perimeter of a quadrilateral
The perimeter of a closed figure is the sum of the lengths of all its sides. Think of walking right around the edge of a field — the distance you walk is the perimeter.
For quadrilateral ABCD, perimeter = AB + BC + CD + DA.
Worked from the book. A quadrilateral has sides 3 cm, 6 cm, 8 cm and 2 cm. Perimeter = 3 + 6 + 8 + 2 = 19 cm.
Worked from the book. A trapezium IJKL has sides 7 cm, 5 cm, 3 cm and 4 cm. Perimeter = IJ + JK + KL + LI = 7 + 5 + 3 + 4 = 20 cm.
There is no shortcut here — you add the actual side lengths. Do not multiply unless the sides really are equal.
4. Building shapes with a tangram
A tangram is a Chinese puzzle about a thousand years old. It is made of five or seven flat geometrical pieces called tans. The rule is that you use every piece, and the pieces may touch but must not overlap.
From the same handful of triangles and quadrilaterals you can build a boat, a bird, a running figure or a house. That is the lesson: shapes are not fixed things — bigger shapes are made out of smaller ones.
5. Tessellations
A tessellation is what you get when you cover a surface with tiles, leaving no gaps and no overlaps. Look at the floor of your classroom or the wall of a bathroom.
Square tiles tessellate. So do rectangles, triangles and hexagons — which is why honeycombs are hexagonal.
But you cannot fill a circle with square tiles. A circle is bounded by a curve, and a square is bounded by straight lines, so the straight edges can never follow the curve. Around the rim you would always be left with gaps.
6. Properties of 3D objects
A 3D object takes up space. To describe one, count three things: faces (the flat or curved surfaces), edges (where two faces meet), and vertices (the corners).
| 3D object | Faces | Edges | Vertices | Examples |
|---|---|---|---|---|
| Cube | 6 (all equal squares) | 12 | 8 | Dice, ice cube, Rubik's cube |
| Cuboid | 6 (opposite faces equal) | 12 | 8 | Matchbox, brick, eraser, book |
| Sphere | 1 curved surface | 0 | 0 | Ball, laddu, marble |
| Cylinder | 2 flat + 1 curved | 2 | 0 | Tin, drum, candle |
| Cone | 1 flat + 1 curved | 1 | 1 | Ice cream cone, birthday cap |
A cube and a cuboid have exactly the same counts: 6 faces, 12 edges, 8 vertices. Counting alone will not separate them. The difference is that all six faces of a cube are equal squares, while a cuboid only has its opposite faces equal.
7. Nets
A net is a flat figure that you can cut out and fold up into a solid. Unfold a matchbox carefully and flatten it — the cross-shaped outline left on the table is the net of a cuboid.
Nets are genuinely useful. Floor maps of houses, layout plans of buildings and bridges all start as flat drawings that stand for solid things.
Two quick ones you can make yourself: fold a semicircle into a cone, and roll a rectangle into a cylinder.
8. Iterative patterns
Look at a tree, a river, a shell, a cloud or a leaf and you will find a pattern in it. Many of these are built the same way, by doing one thing over and over.
Iteration means the repeated application of a process. You carry out a step, then carry out the very same step on the result, and keep going.
Start with the shapes you can already draw, a circle, a spiral and an oval, and iteration turns them into patterns:
- Draw a small circle. Around it draw a slightly larger circle, then a larger one again. The rule never changes, only the size.
- Draw a curve that turns about a centre and creeps outward a little on every turn, and you have a spiral, exactly the shape of a snail shell.
- Draw one oval, turn the paper a little and draw another, and keep turning by the same amount. The overlapping ovals make a rosette.
A spirograph draws these for you. Tape a bottle cap with two or three holes in it to a sheet of paper, put a pencil point through one hole and drag it round. The pencil repeats the same looping motion and a complicated figure appears from a very simple rule.
9. Worked examples
Example 1. The diameter of a circle is 14 cm. Find its radius.
Solution: Radius is half the diameter. Radius = 14 ÷ 2 = 7 cm.
Example 2. Find the perimeter of a rectangle with sides 8 cm, 3 cm, 8 cm and 3 cm.
Solution: Perimeter = 8 + 3 + 8 + 3 = 22 cm.
Example 3. A shape has four equal sides but its diagonals are of different lengths. Name it.
Solution: A rhombus. A square also has four equal sides, but a square's diagonals are equal.
Example 4. How many edges does a cuboid have, and how many vertices?
Solution: A cuboid has 12 edges and 8 vertices — the same as a cube.
Example 5. Which 3D object has one vertex and one flat surface?
Solution: A cone. Its flat surface is a circle and its single vertex is the tip.
Example 6. Why can a circle not be tiled with squares?
Solution: A circle has a curved boundary and a square has straight edges. Straight edges cannot follow a curve, so gaps are always left around the rim.
10. Practice
- The radius of a circle is 9 cm. Find its diameter.
- Which is the longest chord of a circle?
- Fill in the blank: all the radii of a circle are ______.
- Find the perimeter of a quadrilateral with sides 10 cm, 5 cm, 2 cm and 7 cm.
- Find the perimeter of a rectangle whose sides are 7 cm, 4 cm, 7 cm and 4 cm.
- A cube has ______ faces, ______ edges and ______ vertices.
- Name a 3D object with no vertices and no edges.
- How many pieces does a tangram have?
- Name the two diagonals of quadrilateral PQRS.
- Name the shape you get by drawing a curve that turns about a centre and moves outward a little on each turn.
- What does the word iteration mean?
11. Answers
- Diameter = 2 × 9 = 18 cm.
- The diameter.
- Equal.
- 10 + 5 + 2 + 7 = 24 cm.
- 7 + 4 + 7 + 4 = 22 cm.
- 6 faces, 12 edges, 8 vertices.
- A sphere.
- Five or seven pieces, called tans.
- PR and QS.
- A spiral.
- Doing the same process over and over, each time on the result of the step before.
12. Summary
- A circle is the set of points at a fixed distance (the radius) from a centre.
- A chord joins two points on a circle; a chord through the centre is the diameter, and diameter = 2 × radius. The diameter is the longest chord.
- All four-sided closed figures are quadrilaterals: 4 sides, 4 vertices, 2 diagonals.
- A rhombus has four equal sides but unequal diagonals; a square has four equal sides and equal diagonals.
- Perimeter is the sum of the side lengths — add, do not multiply.
- Tangrams build big shapes from small ones; tessellations tile a surface with no gaps or overlaps.
- Cube and cuboid both have 6 faces, 12 edges, 8 vertices; a cube's faces are all equal squares.
- A sphere has one curved surface and no edges or vertices; a cone has one vertex.
- A net is a flat figure that folds into a solid.
