Geometry — Class 5 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 5 Mathematics, Unit 1. The 2D perspective of 3D objects, and an introduction to angles. Taught in June.
1. Properties of 3D shapes
A 3D shape is a solid object with three dimensions: length, breadth and height.
To describe one, count its faces, edges and vertices.
| Shape | Properties | Examples |
|---|---|---|
| Cube | 6 faces, all sides equal; 8 vertices; 12 edges | Dice, ice cube |
| Cuboid | 6 faces, opposite faces equal; 8 vertices; 12 edges | Matchbox, brick |
| Cylinder | Two flat surfaces and one curved surface; the distance between the flat surfaces is its height | Tin, drum |
| Sphere | One curved surface; every point on it is the same distance from the centre; no edges, no vertices | Ball, laddu |
| Cone | One flat circular surface and one curved surface; one vertex | Ice cream cone |
Watch the sphere's definition — it is the same idea as a circle, carried into three dimensions. Every point of the surface is equidistant from the centre.
2. Rotation: turning a shape
Turn a shape about a fixed point and you have rotated it. A complete turn is one full circle, and we describe smaller turns as fractions of it.
A quarter turn is 1/4 of a full circle, a half turn is 1/2, and there are also 1/3 and 1/6 turns.
Some shapes look exactly the same after being turned. A square looks unchanged after a quarter turn. The letter O looks the same after any turn at all. The letters X, A, O, I and S have their own rotation behaviour, and the numbers 1, 0 and 8 are worth testing for yourself.
Try it with a ventilator fan: after a quarter turn the blades sit where the next blade was, so the fan looks unchanged.
3. Reflection: the mirror image
Write the letter D and hold it to a mirror. What comes back is its reflection — the same shape, but flipped left to right.
Stand in front of a mirror and step backwards. Your image does not get smaller; it stays the same size, and it appears to move back by exactly as much as you did.
Here is a way to make reflections yourself. Fold a sheet of paper in two, dip a thread in ink, lay it inside the fold and pull it out. Open the paper: the design on one side is the mirror image of the design on the other.
4. Lines of symmetry
When a line divides a figure into two parts and both parts match exactly, the figure is symmetrical about that line. The line is called the line of symmetry, or the axis of symmetry.
| Figure | Lines of symmetry |
|---|---|
| Square | 4 |
| Rectangle | 2 |
| Equilateral triangle | 3 |
| Circle | Countless |
A square can be folded in four different ways so the halves match: down the middle both ways, and along each diagonal. A rectangle only manages two — its diagonals do not give matching halves, which surprises many students.
Some capital letters are symmetrical (A, H, M, T, O) and some are not (F, G, J, P, R).
5. Nets: flattening a solid
A net is a two-dimensional shape which can be folded to make a three-dimensional solid.
Open out a small box and lay it flat. The shape left on the table is the net of a cuboid.
- Net of a cube: six equal squares, folded along the dotted lines.
- Net of a cuboid: six rectangles, with opposite ones equal.
- Net of a cylinder: a rectangle and two equal circles. Join the rectangle's two edges along its breadth so that its length becomes the circumference of the circles.
- Net of a cone: a circle and a part of a larger circle.
6. Angles
An angle is the shape formed by two line segments or rays starting from a common point.
That common point is the vertex, and the two line segments are the arms.
If the two arms are BA and BC and they meet at B, then B is the vertex and the angle is written ∠ABC — with the vertex letter always in the middle.
Angles are all around you: in bridges, buildings, cell phone towers, the wings of planes, bicycles, windows and doors. Look at the angle between a ladder and the ground, between the branches of a tree, between a staircase and the floor, or between the two hands of a clock.
7. Types of angle
Angles are classified by their measure in degrees.
| Type | Measure | Examples |
|---|---|---|
| Acute | Greater than 0° and less than 90° | 5°, 50°, 80° |
| Right | Exactly 90° | Corner of a book |
| Obtuse | Greater than 90° and less than 180° | 100°, 120°, 170° |
| Straight | Exactly 180° | Two right angles together |
A straight angle is formed by combining two consecutive right angles: 90° + 90° = 180°.
Finding right angles. A set square is the tool for this. There are two set squares in a geometry box, and both contain a 90° corner. Ram uses one to cut a rectangle from a wooden board, because a rectangle needs four right angles.
8. Complementary and supplementary angles
Two angles whose measures add to 90° are called complementary angles. If angle AOB is 30° and angle BOC is 60°, then 30° + 60° = 90°, so the pair is complementary. The two angles need not touch: an angle of 70° in one figure and an angle of 20° in another are still complementary.
Two angles whose measures add to 180° are called supplementary angles, 180° being the measure of a straight angle.
| Pair | Sum | Name |
|---|---|---|
| 30° and 60° | 90° | Complementary |
| 45° and 45° | 90° | Complementary |
| 110° and 70° | 180° | Supplementary |
| 90° and 90° | 180° | Supplementary |
To find the complement of an angle, subtract it from 90. To find its supplement, subtract it from 180. So the complement of 25° is 65°, and its supplement is 155°.
An angle of 120° has a supplement but no complement, because it is already larger than 90°.
9. Perimeter
The perimeter of a closed shape is the distance all the way round its edge. Walk once round the boundary of a field and the distance you cover is its perimeter.
You could add the four sides of a rectangle one at a time, but opposite sides of a rectangle are equal, so it is quicker to add two and double them.
Perimeter of a rectangle = (2 × length) + (2 × breadth)
Take rectangle PQRS with length 5 cm and breadth 2 cm. Its perimeter is (2 × 5) + (2 × 2) = 10 + 4 = 14 cm.
All four sides of a square are equal, which shortens the rule further.
Perimeter of a square = 4 × side
A square of side 7 cm has perimeter 4 × 7 = 28 cm. Perimeter is a length, so its unit is a plain length unit such as cm or m, never a square unit.
10. Fractals
A fractal is a never-ending pattern that repeats itself at different scales. Zoom in on any part of it and the same shape appears again, smaller. This property is called self-similarity.
Fractals look extremely complicated, yet each one is made by repeating a single simple step over and over. Begin with a triangle, mark the midpoint of each side, and take away the middle triangle. Do the same to each of the three triangles left, then to the nine after that. The figure grows intricate although the rule never changed.
Nature is full of them. A fern frond is built from smaller fronds of the same shape, and so are the florets of a cauliflower, the arms of a snowflake and the branching of a river into its tributaries.
11. Worked examples
Example 1. True or false: a cuboid has 7 vertices.
Solution: False. A cuboid has 8 vertices, like a cube.
Example 2. True or false: a cylinder has two flat surfaces.
Solution: True, plus one curved surface.
Example 3. How many lines of symmetry does a rectangle have?
Solution: 2 — one horizontal and one vertical through the middle. The diagonals are not lines of symmetry.
Example 4. What shapes make up the net of a cylinder?
Solution: A rectangle and two equal circles.
Example 5. Angles below 90° and above 0° are called what?
Solution: Acute angles.
Example 6. Which angle is formed by combining two consecutive right angles?
Solution: A straight angle, of 180°.
Example 7. Name the vertex and arms of the angle ∠ABC.
Solution: The vertex is B, and the arms are BA and BC.
12. Practice
- How many edges and vertices does a cube have?
- Which 3D shape has one curved surface and no vertices?
- What is the distance between the two flat surfaces of a cylinder called?
- How many lines of symmetry does a square have?
- What is a net?
- How many equal squares form the net of a cube?
- Define an angle.
- Angles between 90° and 180° are called what?
- What is the measure of a straight angle?
- Which tool in the geometry box is used to draw right angles?
- Find the complement of 38°.
- Find the supplement of 96°.
- A rectangle is 9 cm long and 4 cm broad. Find its perimeter.
- A square has a perimeter of 36 cm. Find the length of one side.
13. Answers
- 12 edges and 8 vertices.
- The sphere.
- Its height.
- Four.
- A two-dimensional shape that folds to make a three-dimensional solid.
- Six.
- The shape formed by two line segments or rays starting from a common point.
- Obtuse angles.
- 180°.
- The set square, whose corner measures 90°.
- 90 - 38 = 52°.
- 180 - 96 = 84°.
- (2 × 9) + (2 × 4) = 18 + 8 = 26 cm.
- 36 ÷ 4 = 9 cm.
14. Summary
- A 3D shape has length, breadth and height; describe it by faces, edges and vertices.
- Cube and cuboid both have 6 faces, 12 edges and 8 vertices; a cube's faces are all equal.
- A sphere has one curved surface with every point equidistant from the centre.
- Rotation turns a shape about a point; some shapes look unchanged after a quarter or half turn.
- A mirror image is the same size as the object and flipped left to right.
- A line of symmetry divides a figure into two exactly matching parts; a square has 4 and a rectangle 2.
- A net is a flat shape that folds into a solid: six squares make a cube, a rectangle and two circles make a cylinder.
- An angle is formed by two rays from a common vertex, written with the vertex in the middle.
- Acute is under 90°, right is 90°, obtuse is between 90° and 180°, and straight is 180°.
