Understanding Elementary Shapes
1. Triangles
A triangle is a three-sided polygon. Sum of its three angles = 180.
Classification by Sides
| Type | Definition | Example |
|---|---|---|
| Equilateral | All three sides equal | All angles = 60 |
| Isosceles | Two sides equal | Base angles equal |
| Scalene | No sides equal | All sides different |
Classification by Angles
| Type | Definition | Example |
|---|---|---|
| Acute-angled | All angles < 90 | 60, 70, 50 |
| Right-angled | One angle = 90 | 90, 45, 45 |
| Obtuse-angled | One angle > 90 | 110, 35, 35 |
Common Mistake: Thinking an equilateral triangle cannot be acute-angled. It is actually the most perfectly acute triangle (all angles 60).
Exam Focus (2 marks): 'Classify a triangle with sides 5 cm, 5 cm, 6 cm.'
Two sides equal: Isosceles triangle. Check angles: base angles are equal.
Worked Example: In a right triangle, if one acute angle is 35, find the other.
Sum of angles = 180. Right angle = 90. Other two sum to 90.
Other acute angle = 90 - 35 = 55.
2. Quadrilaterals
A quadrilateral is a four-sided polygon. Sum of its four angles = 360.
| Type | Properties |
|---|---|
| Square | All sides equal, all angles 90, diagonals equal and bisect at 90 |
| Rectangle | Opposite sides equal, all angles 90, diagonals equal |
| Rhombus | All sides equal, opposite angles equal, diagonals bisect at 90 |
| Parallelogram | Opposite sides parallel and equal, opposite angles equal |
| Trapezium | One pair of opposite sides parallel |
| Kite | Two pairs of adjacent equal sides |
Common Mistake: Thinking that all rectangles are squares. No: a square is a special rectangle where all sides are equal, but not all rectangles are squares.
Worked Example: In a parallelogram, one angle is 75. Find the other angles.
Opposite angle = 75. Adjacent angles are supplementary: 180 - 75 = 105.
Angles: 75, 105, 75, 105. Sum = 360.
3. Polygons
A polygon is a closed figure with three or more straight sides.
| Sides | Name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
Regular vs Irregular
- Regular polygon: All sides equal and all angles equal (e.g., equilateral triangle, square).
- Irregular polygon: Sides and angles are not all equal.
Exam Focus (2 marks): 'How many sides does a hexagon have? Is it regular if all its sides are 4 cm each and all angles are 120?'
6 sides. Yes, it is a regular hexagon because all sides and all angles are equal.
4. 3D Shapes
Key Terms
- Face: A flat surface of a 3D shape.
- Edge: Where two faces meet (a line segment).
- Vertex (vertices): Where three or more edges meet (a corner).
Common 3D Shapes
| Shape | Faces | Edges | Vertices | Description |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | All faces squares |
| Cuboid | 6 | 12 | 8 | Opposite faces equal rectangles |
| Sphere | 1 | 0 | 0 | Curved surface, no edges |
| Cylinder | 3 | 2 | 0 | 2 circular faces + curved surface |
| Cone | 2 | 1 | 1 | 1 circular face + curved surface |
| Pyramid | 5 | 8 | 5 | Square base + 4 triangular faces |
Common Mistake: Confusing the number of edges of a cube (12) with the number of faces (6). A cube has 6 faces, 12 edges, and 8 vertices.
Euler's Formula (for polyhedra)
F + V - E = 2
Worked Example: A polyhedron has 6 faces and 12 edges. Find the number of vertices.
Using F + V - E = 2: 6 + V - 12 = 2 => V - 6 = 2 => V = 8.
5. Nets of 3D Shapes
A net is a 2D pattern that can be folded to form a 3D shape.
- Cube: 11 different nets exist.
- Cuboid: 6 rectangular faces arranged in a cross pattern.
- Cylinder: Two circles and a rectangle.
- Cone: A sector of a circle and a base circle.
6. Self-Test
- Classify a triangle with angles 40, 70, 70 by sides and by angles.
- One angle of a parallelogram is 130. Find the others.
- How many diagonals does a pentagon have?
- A polyhedron has 5 faces and 5 vertices. Find its edges using Euler's formula.
- Name the 3D shape with: (a) 6 square faces (b) 2 circular faces (c) 1 vertex.
- True or false: A square is always a rectangle.
- Draw a rough sketch of a trapezium and mark its parallel sides.
7. Answers to Self-Test
- By angles: acute-angled (all < 90). By sides: isosceles (two equal angles => two equal sides).
- Opposite angle = 130. Adjacent = 180 - 130 = 50. Angles: 130, 50, 130, 50.
- 5 diagonals.
- F + V - E = 2 => 5 + 5 - E = 2 => E = 8.
- (a) Cube (b) Cylinder (c) Cone.
- True: a square satisfies all properties of a rectangle (opposite sides equal, all angles 90).
- Draw a quadrilateral with one pair of opposite sides parallel. Mark the parallel sides.
