1. Basic Mensuration Formulas
Mensuration is the branch of mathematics that deals with the measurement of geometrical figures. In competitive examinations, mensuration questions mainly involve perimeter, area, surface area and volume.
Before learning the formulas for circles, triangles, squares, cubes, cuboids, cylinders, cones and other shapes, it is important to understand these basic terms.
What is Mensuration?
Mensuration helps us calculate the size or measurement of a geometrical figure. Depending on the figure, we may need to find its boundary, occupied region, outer surface or the space inside it.
For example, if a farmer wants to put a fence around a rectangular field, we need to calculate its perimeter. If he wants to know how much land the field covers, we calculate its area.
1. Perimeter
The perimeter is the total distance around the boundary of a two-dimensional figure.
Perimeter = Total length around the figure
Example
A rectangular field has a length of 20 m and a breadth of 10 m. Find its perimeter.
= 2 × (20 + 10)
= 2 × 30
= 60 m
2. Area
The area represents the amount of surface or region occupied by a two-dimensional figure.
Area is always expressed in square units, such as cm², m² or km².
Example
A rectangular room is 8 m long and 5 m wide. Find its area.
= 8 × 5
= 40 m²
3. Surface Area
Surface area is the area of the outer surfaces of a three-dimensional object.
It is generally expressed in square units.
For example, when we calculate the amount of material required to cover the outside of a box, we are dealing with surface area.
Two Important Types
| Type | Meaning |
|---|---|
| Lateral Surface Area (LSA) | Area of the lateral/side surfaces, excluding the required base surfaces. |
| Curved Surface Area (CSA) | Area of the curved outer surface of a solid such as a cylinder or cone. |
| Total Surface Area (TSA) | Area of all relevant outer surfaces of the solid. |
4. Volume
Volume is the amount of three-dimensional space occupied by an object.
Volume is expressed in cubic units, such as cm³, m³ or km³.
Example
A cuboid has a length of 10 cm, breadth of 5 cm and height of 4 cm. Find its volume.
= 10 × 5 × 4
= 200 cm³
5. Radius
The radius is the distance from the centre of a circle to any point on its circumference.
r = Radius
Radius is usually represented by the letter r.
6. Diameter
The diameter is the straight-line distance across a circle passing through its centre.
d = 2r
Example
If the radius of a circle is 7 cm, find its diameter.
d = 2r
d = 2 × 7
d = 14 cm
7. Height
Height generally means the perpendicular distance between a base and the corresponding top or opposite point/surface.
Height is represented by h in many mensuration formulas.
8. Slant Height
Slant height is the inclined or sloping height used in figures such as cones and pyramids.
It is commonly represented by l.
Height (h) → Perpendicular measurement
Slant height (l) → Sloping measurement
9. Important Units in Mensuration
| Measurement | Common Units |
|---|---|
| Length | mm, cm, m, km |
| Perimeter | cm, m, km |
| Area | cm², m², km² |
| Volume | cm³, m³, km³ |
10. Important Unit Conversions
11. Perimeter vs Area vs Volume
| Concept | What it measures | Unit |
|---|---|---|
| Perimeter | Boundary of a 2D figure | cm, m, km |
| Area | Region covered by a 2D figure | cm², m², km² |
| Surface Area | Outer surface of a 3D object | cm², m², km² |
| Volume | Space occupied by a 3D object | cm³, m³, km³ |
12. Quick Exam Revision
Perimeter → Boundary
Area → Surface covered by a 2D figure
Surface Area → Outer surface of a 3D object
Volume → Space occupied by a 3D object
Radius → Centre to circumference
Diameter → 2 × Radius
Height → Perpendicular measurement
Slant Height → Sloping measurement
13. Practice Question
Question: A circular garden has a diameter of 28 m. What is its radius?
We know:
Therefore:
r = d ÷ 2
r = 28 ÷ 2
Answer: 14 m