In this chapter we learn to measure, compare, and classify geometric shapes more precisely using angles, triangles, quadrilaterals, and 3D shapes.
Measuring Line Segments
Line segments can be compared by placing them side by side, or by using a ruler or divider. The length is measured in millimetres (mm), centimetres (cm), or metres (m).
Angles and the Protractor
A protractor is used to measure angles in degrees. Place the centre of the protractor at the vertex of the angle, align the baseline with one arm, and read where the other arm points.
- Types of Angles (revisited)
- Acute: 0° to 90° (exclusive)
- Right: exactly 90°
- Obtuse: 90° to 180° (exclusive)
- Straight: 180°
- Reflex: 180° to 360° (exclusive)
- Complete: 360°
- Types of Triangles
- By sides:
- Equilateral: all three sides equal, all angles 60°.
- Isosceles: two sides equal, base angles equal.
- Scalene: all sides different.
- By angles:
- Acute triangle: all angles acute.
- Right triangle: one angle is 90°.
- Obtuse triangle: one angle is obtuse.
- Types of Quadrilaterals
- Square: all sides equal, all angles 90°.
- Rectangle: opposite sides equal, all angles 90°.
- Rhombus: all sides equal, opposite angles equal.
- Parallelogram: opposite sides parallel and equal.
- Trapezium: one pair of parallel sides.
- Kite: two pairs of consecutive equal sides.
- 3D Shapes
- Three-dimensional shapes have length, breadth, and height:
- Cube: 6 square faces, 12 edges, 8 vertices.
- Cuboid: 6 rectangular faces, 12 edges, 8 vertices.
- Cylinder: 2 circular faces, 1 curved surface.
- Cone: 1 circular base, 1 curved surface, 1 vertex.
- Sphere: 1 curved surface, no edges, no vertices.
A triangle has angles 45°, 55°, and 80°. Classify it.
All angles are less than 90°, so it is an acute triangle.
A triangle has sides 5 cm, 5 cm, and 8 cm. What type is it?
Two sides are equal (5 cm each), so it is an isosceles triangle.
Classify an angle of 270°.
270° is between 180° and 360°, so it is a reflex angle.
How many edges does a cube have?
A cube has 12 edges.
A quadrilateral has all sides equal but angles are not all 90°. What is it?
A rhombus.
Name the 3D shape with one vertex, one circular base, and a curved surface.
A cone.
How many right angles does a rectangle have?
A rectangle has 4 right angles, one at each corner.
Euler's Formula for 3D Shapes
F + V - E = 2, where F = Faces, V = Vertices, E = Edges.
Check for cube: 6 + 8 - 12 = 2. ✓
Common mistakes
- A square is a special type of rectangle and rhombus.
- All equilateral triangles are also isosceles, but not vice versa.
- A sphere has no flat faces, no edges, and no vertices.
Summary
Angles are measured with a protractor. Triangles are classified by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Quadrilaterals have varied properties. 3D shapes satisfy Euler's formula: F + V - E = 2.