Mensuration
What is Mensuration?
The total length of the boundary of a two-dimensional shape.
Key formula / rule: Perimeter of Square
Key points
- Identify and differentiate between various 2D and 3D geometric shapes.
- Accurately recall and apply formulas for perimeter, area, volume, and surface area.
- Solve problems involving unit conversions between different systems.
- Calculate dimensions of composite figures by breaking them into simpler shapes.
Common exam trap
Using the wrong formula for a specific shape or measurement.
Definitions
- Term
Perimeter
- Meaning
The total length of the boundary of a two-dimensional shape.
- Term
Area
- Meaning
The amount of surface enclosed within the boundary of a two-dimensional shape.
- Term
Volume
- Meaning
The amount of three-dimensional space occupied by a solid object.
- Term
Surface Area
- Meaning
The total area of all the surfaces of a three-dimensional object.
- Term
Lateral Surface Area (LSA)
- Meaning
The sum of the areas of the lateral (side) faces of a solid, excluding the top and bottom bases (e.g., for cuboids, prisms).
- Term
Curved Surface Area (CSA)
- Meaning
The area of the curved surface of a solid, excluding the flat bases (e.g., for cylinders, cones).
- Term
Total Surface Area (TSA)
- Meaning
The sum of the areas of all surfaces (lateral/curved and bases) of a three-dimensional object.
Learning objectives
Identify and differentiate between various 2D and 3D geometric shapes.
Accurately recall and apply formulas for perimeter, area, volume, and surface area.
Solve problems involving unit conversions between different systems.
Calculate dimensions of composite figures by breaking them into simpler shapes.
Analyze and solve practical problems related to mensuration in real-world contexts.
Formulae
- Name
Perimeter of Square
- Note
where 'a' is the side length
- Expression
4a
- Name
Area of Square
- Note
where 'a' is the side length
- Expression
a²
- Name
Perimeter of Rectangle
- Note
where 'l' is length and 'b' is breadth
- Expression
2(l+b)
- Name
Area of Rectangle
- Note
where 'l' is length and 'b' is breadth
- Expression
l × b
- Name
Circumference of Circle
- Note
where 'r' is radius
- Expression
2πr
- Name
Area of Circle
- Note
where 'r' is radius
- Expression
πr²
- Name
Area of Triangle (Base and Height)
- Note
where 'b' is base and 'h' is height
- Expression
½ × b × h
- Name
Area of Triangle (Heron's Formula)
- Note
where a, b, c are side lengths and s = (a+b+c)/2 (semi-perimeter)
- Expression
√[s(s-a)(s-b)(s-c)]
- Name
Area of Equilateral Triangle
- Note
where 'a' is the side length
- Expression
(√3/4)a²
- Name
Area of Parallelogram
- Note
where 'b' is base and 'h' is height
- Expression
b × h
- Name
Area of Rhombus
- Note
where d₁ and d₂ are the lengths of the diagonals
- Expression
½ × d₁ × d₂
- Name
Area of Trapezium
- Note
where 'a' and 'b' are lengths of parallel sides and 'h' is height
- Expression
½ × (a+b) × h
- Name
Volume of Cube
- Note
where 'a' is the side length
- Expression
a³
- Name
Lateral Surface Area of Cube
- Note
where 'a' is the side length
- Expression
4a²
- Name
Total Surface Area of Cube
- Note
where 'a' is the side length
- Expression
6a²
- Name
Volume of Cuboid
- Note
where 'l' is length, 'b' is breadth, 'h' is height
- Expression
l × b × h
- Name
Lateral Surface Area of Cuboid
- Note
where 'l' is length, 'b' is breadth, 'h' is height
- Expression
2h(l+b)
- Name
Total Surface Area of Cuboid
- Note
where 'l' is length, 'b' is breadth, 'h' is height
- Expression
2(lb+bh+hl)
- Name
Volume of Cylinder
- Note
where 'r' is radius and 'h' is height
- Expression
πr²h
- Name
Curved Surface Area of Cylinder
- Note
where 'r' is radius and 'h' is height
- Expression
2πrh
- Name
Total Surface Area of Cylinder
- Note
where 'r' is radius and 'h' is height
- Expression
2πr(r+h)
- Name
Volume of Cone
- Note
where 'r' is radius and 'h' is height
- Expression
⅓πr²h
- Name
Curved Surface Area of Cone
- Note
where 'r' is radius and 'l' is slant height (l = √(r²+h²))
- Expression
πrl
- Name
Total Surface Area of Cone
- Note
where 'r' is radius and 'l' is slant height
- Expression
πr(r+l)
- Name
Volume of Sphere
- Note
where 'r' is radius
- Expression
⁴⁄₃πr³
- Name
Surface Area of Sphere
- Note
where 'r' is radius
- Expression
4πr²
- Name
Volume of Hemisphere
- Note
where 'r' is radius
- Expression
²⁄₃πr³
- Name
Curved Surface Area of Hemisphere
- Note
where 'r' is radius
- Expression
2πr²
- Name
Total Surface Area of Hemisphere
- Note
where 'r' is radius
- Expression
3πr²
Prerequisites
Basic arithmetic operations (addition, subtraction, multiplication, division).
Understanding of fractions, decimals, and percentages.
Basic algebraic manipulation (solving simple equations).
Fundamental geometric concepts (points, lines, angles, types of triangles, quadrilaterals).
Knowledge of squares, cubes, and square roots.
Common mistakes
Using the wrong formula for a specific shape or measurement.
Errors in unit conversion (e.g., forgetting to convert cm to m).
Calculation errors, especially with fractions, decimals, or square roots.
Confusing Lateral/Curved Surface Area with Total Surface Area.
Incorrectly applying π (e.g., using 3.14 when 22/7 would simplify calculations).
Not drawing diagrams for complex problems to visualize the shapes.
Keywords
Mensuration
2D shapes
3D shapes
Perimeter
Area
Volume
Surface Area
Lateral Surface Area
Curved Surface Area
Total Surface Area
Formulas
Geometric figures
Units
Practice preview
A cylinder has a radius of 5 cm and a height of 10 cm. What is its total surface area? (Use π = 22/7)…
medium
What is the area of a circle with a radius of 7 cm?…
easy
A rectangular field is 120 meters long and 80 meters wide. What is the cost of ploughing it at the rate of ₹5 per square meter?…
medium