Learning Objectives
- Calculate area of trapezium, quadrilateral, and other polygons
- Find surface area of cube, cuboid, and cylinder
- Calculate volume of cube, cuboid, and cylinder
- Understand the relationship between area and volume
Key Concepts
Area of Trapezium
Formula: Area = ½ × (sum of parallel sides) × height = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them.
Area of a General Quadrilateral
Divide the quadrilateral into two triangles by drawing a diagonal. Area = ½ × diagonal × (sum of the two perpendicular heights from the opposite vertices to the diagonal).
Formula: Area = ½ × d × (h₁ + h₂)
Area of a Rhombus
Area = ½ × d₁ × d₂, where d₁ and d₂ are the diagonals of the rhombus.
Surface Area of Cuboid
Total Surface Area (TSA): 2(lb + bh + hl), where l = length, b = breadth, h = height.
Lateral Surface Area (LSA): 2h(l + b) — area of four walls.
Surface Area of Cube
Total Surface Area: 6a², where a is the side length.
Lateral Surface Area: 4a².
Surface Area of Cylinder
Curved Surface Area (CSA): 2πrh, where r is the radius and h is the height.
Total Surface Area: 2πr(r + h).
Volume
Volume of Cuboid: l × b × h
Volume of Cube: a³
Volume of Cylinder: πr²h
Volume is measured in cubic units (cm³, m³). 1 litre = 1000 cm³. 1 m³ = 1000 litres.
Summary
Mensuration deals with measurement of area and volume. Areas of trapezium, quadrilaterals, and rhombus use diagonals and heights. Surface area and volume formulas for cubes, cuboids, and cylinders are essential for solving practical problems involving space and capacity.
Important Terms
- Perimeter: Total length of the boundary of a shape
- Area: The measure of the surface enclosed by a shape (in square units)
- Volume: The amount of space occupied by a solid (in cubic units)
- Surface Area: Total area of all surfaces of a 3D shape
- Lateral Surface Area: Area of side faces only (excluding top and bottom)
Quick Revision
- Trapezium area = ½ × (a + b) × h
- Rhombus area = ½ × d₁ × d₂
- Cuboid TSA = 2(lb + bh + hl); Volume = lbh
- Cube TSA = 6a²; Volume = a³
- Cylinder CSA = 2πrh; TSA = 2πr(r+h); Volume = πr²h
- 1 litre = 1000 cm³; 1 m³ = 1000 litres