Mensuration Formula Guide with Area and Volume Examples
Mensuration deals with measuring geometric figures and solids. It includes perimeter, area, surface area and volume calculations for shapes such as squares, circles, triangles, cuboids, cylinders, cones and spheres. This page covers core mensuration formulas, unit conversions, properties, shortcuts and numerical methods used in quantitative aptitude.
What is Mensuration?
For a plane figure, perimeter measures the boundary and area measures the enclosed region. For a solid, surface area measures the outer covering and volume measures the space occupied. Use square units for area and cubic units for volume. If a circle is involved, use π = 22/7 or 3.14 according to the given values.
Mensuration Video Lessons
Watch short topic-wise lessons for quick revision.
Combined Solids: Volume and Surface Area
Learn how to calculate the volume and surface area of combined three-dimensional solids by identifying component shapes and applying the appropriate mensuration formulas.
Mensuration Formula & Tricks
Important Formulas
Here, a is the side of a square, while l and b are the length and breadth of a rectangle.
The height must be perpendicular to the selected base.
Here, r is the radius and d = 2r is the diameter.
TSA means total surface area. In a cube, all edges have length a.
Here, r is the base radius and h is the perpendicular height.
For a right circular cone, r, h and l form a right triangle.
Quick Tricks
If every length of a figure is multiplied by k, its perimeter is multiplied by k, its area by k² and its volume by k³.
Convert all dimensions to one unit first. For example, 1 m = 100 cm, 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
Since r = d/2, circumference can be calculated as πd. The circle area using diameter d is πd²/4.
Mensuration Concepts
Perimeter and area of plane figures
Add the lengths of all sides to find the perimeter of a polygon. For a rectangle, area is length multiplied by breadth. For a triangle, use half the product of base and perpendicular height. Area is expressed in square units.
Circular measurements
Use circumference = 2πr and area = πr². The diameter is twice the radius. For a sector with central angle θ, arc length = θ/360 × 2πr and sector area = θ/360 × πr², where θ is measured in degrees.
Surface area and volume of solids
A cuboid has volume lbh and total surface area 2(lb + bh + hl). A cylinder has volume πr²h. A sphere has surface area 4πr² and volume 4/3πr³. Use square units for surface area and cubic units for volume.
Composite figures and hollow solids
For a compound area, divide the figure into rectangles, triangles or circles and combine their areas. For a hollow cylinder or pipe, material volume = πh(R² − r²), where R is the outer radius and r is the inner radius. For an open solid, exclude the surface that is not present.
Mensuration Visual Guide & Data
Use the table and chart below to understand the data better.
Common mensuration formulas
Selected formulas for standard two-dimensional figures and solids.
| Figure | Perimeter or surface measure | Area or volume |
|---|---|---|
| Square of side a | Perimeter = 4a | Area = a² |
| Rectangle of length l and breadth b | Perimeter = 2(l + b) | Area = lb |
| Circle of radius r | Circumference = 2πr | Area = πr² |
| Cuboid of dimensions l, b, h | TSA = 2(lb + bh + hl) | Volume = lbh |
| Cylinder of radius r and height h | CSA = 2πrh | Volume = πr²h |
| Sphere of radius r | Surface area = 4πr² | Volume = 4/3πr³ |
Note: TSA means total surface area and CSA means curved surface area. Use consistent units in every row.
Rectangle with length and breadth
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Mensuration revision points
Keep dimensions consistent and select the formula according to the required measurement.
- Perimeter and circumference use linear units.
- Area and surface area use square units; volume uses cubic units.
- For a circle, diameter d = 2r and circumference = πd.
- A triangle’s height must be perpendicular to its base.
- For a cone, slant height l = √(r² + h²).
- When a length scale changes by k, area changes by k² and volume by k³.
- For a hollow solid, subtract the inner volume from the outer volume.
- For reverse calculations, use the correct unit conversion before substituting values.
Mensuration FAQs
What is the difference between surface area and volume?
Surface area measures the outer covering of a solid in square units. Volume measures the space occupied by the solid in cubic units.
What is the formula for the area of a sector?
For central angle θ in degrees, sector area = θ/360 × πr². For example, a 90° sector has area one-fourth of the circle area.
How is the area of a path around a rectangle calculated?
Subtract the area of the inner rectangle from the area of the outer rectangle. If the inner dimensions are l and b and the path has width x, the area is (l + 2x)(b + 2x) − lb.
What is the volume of a hollow cylinder?
The volume of material is πh(R² − r²), where R is the outer radius, r is the inner radius and h is the height.
If the radius of a sphere is doubled, how do its area and volume change?
Surface area becomes 2² = 4 times the original, while volume becomes 2³ = 8 times the original.
What is the total surface area of a closed cylinder?
The total surface area is 2πr(h + r). It includes the curved surface and both circular bases.