Mensuration is the branch of mathematics that deals with the measurement of geometric figures like area, volume, and perimeter.
Perimeter and Area of Plane Figures
Perimeter of Simple Shapes
The perimeter is the total length of the boundary of a two-dimensional figure.
Formula: For a rectangle, the perimeter (P) is given by P = 2 × (Length + Breadth).
Example: If the length of a rectangle is 10 cm and the breadth is 5 cm, its perimeter is 2 × (10 + 5) = 30 cm.
Area of Rectangle and Square
The area is the measure of the space enclosed within a plane figure.
Rectangle:Area = Length × Breadth.
Square:Area = Side².
Example: The area of a rectangle with length 8 cm and breadth 6 cm is 8 × 6 = 48 cm².
Area of Special Plane Figures
Area of a Triangle
The area of a triangle can be calculated using the base and height.
Formula:Area = ½ × Base × Height.
Example: If the base of a triangle is 5 cm and its height is 12 cm, the area is ½ × 5 × 12 = 30 cm².
Area of a Parallelogram
The area of a parallelogram is calculated using the base and the corresponding height.
Formula:Area = Base × Height.
Example: For a parallelogram with a base of 7 cm and height of 4 cm, the area is 7 × 4 = 28 cm².
Area of a Trapezium
The area of a trapezium is calculated using the lengths of the parallel sides and the height.
Formula:Area = ½ × (a + b) × Height, where a and b are the lengths of the parallel sides.
Example: If a = 8 cm, b = 5 cm, and height = 6 cm, the area is ½ × (8 + 5) × 6 = 39 cm².
Area of a Polygon
Area Calculation for Regular Polygons
Regular polygons can have their area calculated by dividing them into triangles and using the formula for the area of a triangle.
Example: For a regular hexagon, it can be divided into six equilateral triangles.
Surface Area and Volume of Solids
Surface Area of a Cube
The total surface area (TSA) of a cube with side length (s) is given by:
Formula:TSA = 6 × s².
Example: For a cube with side 4 cm, the surface area is 6 × 4² = 96 cm².
Surface Area of a Cuboid
The total surface area (TSA) of a cuboid with length (l), breadth (b), and height (h) is given by:
Formula:TSA = 2 × (lb + bh + hl).
Example: For a cuboid with dimensions 3 cm by 4 cm by 5 cm, the surface area is 2 × (3 × 4 + 4 × 5 + 5 × 3) = 94 cm².
Surface Area of a Cylinder
The surface area (SA) of a cylinder with radius (r) and height (h) is the sum of the lateral surface area and the area of the two bases.
Formula:SA = 2πrh + 2πr².
Example: For a cylinder with radius 3 cm and height 7 cm, the surface area is 2π × 3 × 7 + 2π × 3² = 188.4 cm².
Volume of Cube and Cuboid
The volume (V) of a cube with side (s) is given by:
Formula:V = s³.
Example: For a cube with side 3 cm, the volume is 3³ = 27 cm³.
The volume (V) of a cuboid with length (l), breadth (b), and height (h) is given by:
Formula:V = l × b × h.
Example: For a cuboid with dimensions 3 cm by 4 cm by 5 cm, the volume is 3 × 4 × 5 = 60 cm³.
Volume of a Cylinder
The volume (V) of a cylinder with radius (r) and height (h) is given by:
Formula:V = πr²h.
Example: For a cylinder with radius 3 cm and height 7 cm, the volume is π × 3² × 7 = 198 cm³ (using π ≈ 3.14).
Capacity
Capacity is the measure of how much a container can hold, often used in context with liquids. It is directly related to volume.
Example: If the volume of a tank is 500 cm³, it has a capacity of 500 ml (assuming the density of the liquid is similar to water).
Scenario-Based Mathematics Problems
Scenario 1: Real Estate and Construction
A rectangular plot of land measures 60 m by 40 m. The owner wants to build a fence around the plot. Calculate the length of the fence needed and the total area of the plot.
Solution:
Perimeter of the plot = 2 × (60 + 40) = 200 m.
Area of the plot = 60 × 40 = 2400 m².
Scenario 2: Packaging and Storage
A company manufactures wooden boxes that are cuboidal in shape with dimensions 2 m by 1.5 m by 1 m. Calculate the total surface area and volume of the box.
Solution:
Surface area = 2 × (2 × 1.5 + 1.5 × 1 + 1 × 2) = 11 m².
Volume = 2 × 1.5 × 1 = 3 m³.
Scenario 3: Manufacturing and Production
A cylindrical tank has a radius of 5 m and a height of 10 m. Calculate the capacity of the tank in liters (1 m³ = 1000 liters).
Solution:
Volume = π × 5² × 10 = 785 m³ (approx).
Capacity = 785 × 1000 = 785000 liters.
Key Terms and Concepts
Perimeter
The continuous line forming the boundary of a closed geometric figure.
Area
The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle or circle.
Surface Area
The total area of the surface of a three-dimensional object.
Volume
The amount of space occupied by a three-dimensional object as measured in cubic units.
Additional Value Addition
Practical Application of Mensuration
Understanding mensuration helps in various real-life applications like construction, packaging, and designing objects.
Example: Calculating the amount of material required to build a fence around a garden or determining the amount of paint needed to cover a wall.
Relation to Other Mathematical Concepts
Mensuration problems often integrate with algebraic expressions, especially when dealing with unknown dimensions.
Example: Using algebraic identities to solve for the dimensions of a shape when given its area or volume.
