Quadrilaterals and Mensuration: Class 8 Study Guide
Properties of shapes help us find unknown lengths and angles, and formulas tell us how much a solid holds.
A quadrilateral has four sides, and its interior angles add to 360°. A parallelogram has opposite sides parallel and equal, opposite angles equal and diagonals that bisect each other. A rectangle is a parallelogram with four right angles and equal diagonals. A rhombus has four equal sides and diagonals that cross at right angles. A square has all of these properties. A trapezium has one pair of parallel sides.
Mensuration measures lengths, areas and volumes. Surface area is the total area of the faces of a solid, and volume is the space it occupies.
Properties at a glance
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Parallelogram | opposite equal and parallel | opposite equal | bisect each other |
| Rectangle | opposite equal | all 90° | equal, bisect each other |
| Rhombus | all equal | opposite equal | perpendicular bisectors |
| Square | all equal | all 90° | equal and perpendicular |
| Kite | two pairs of adjacent equal sides | one pair opposite equal | perpendicular |
Area formulas
Trapezium: ½ × (sum of parallel sides) × height. Rhombus: ½ × d₁ × d₂. A trapezium with parallel sides 8 cm and 12 cm and height 5 cm has an area of ½ × 20 × 5 = 50 cm². A polygon can be split into triangles and trapezia to find its area.
Which sides are parallel?
With the correct letters.
Check the units.
In cm², m².
Solids
For a cuboid with length l, breadth b and height h: volume = l × b × h; total surface area = 2(lb + bh + hl). For a cube of side a: volume = a³; surface area = 6a². For a cylinder of radius r and height h (with π = 22/7): curved surface area = 2πrh, total surface area = 2πr(r + h), volume = πr²h. Units: 1 m³ = 1000 litres, and 1 cm³ = 1 mL.
| Solid | Volume | Total surface area |
|---|---|---|
| Cube | a³ | 6a² |
| Cuboid | l × b × h | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πr(r + h) |
Worked examples and common mistakes
Example: A cylindrical tank with r = 7 m and h = 10 m holds (22/7) × 49 × 10 = 1540 m³, or 1 540 000 litres. Common mistakes: using diameter in place of radius, adding the wrong faces in surface area and mixing units. Draw and label every solid.
Check your understanding: model answers
1. 40 square units. 2. 60 cm³. 3. (22/7) × 49 × 5 = 770 cubic units.
✨ Key Takeaways
- Quadrilateral angles sum to 360°.
- Know the diagonal properties.
- Trapezium area = ½ × (a + b) × h.
- Volume of a cylinder = πr²h.
💬 Questions for Discussion & Reflection
- Find the area of a rhombus with diagonals 10 and 8.
- Find the volume of a cuboid 5 × 4 × 3 cm.
- The radius of a cylinder is 7 and its height 5. Find its volume (π = 22/7).