**Formulas of Mensuration in Maths: **Mensuration concept from basics are explained below with important *mensuration formulas* and *properties of geometrical figures*.

## What is mensuration?

Mensuration is a branch of which, where measurement of various geometrical shapes. The measurements are area, volume, curved surface area, lateral surface area, total surface area, etc.

## Types of Geometrical Shapes:

There are two types of geometrical shapes in mensuration. They are

- 2-dimensional
- 3-dimensional

### 2-Dimensional Geometrical Shape

It is the shape formed by joining three or more straight lines in a plane. 2D geometrical shapes have length and breadth. Here measurements are area, perimeter only.

**The units of measurements are
Area:** cm², M²,km

^{2}, miles².

**Perimeter:**cm,m,km,miles.

### 3-Dimensional Geometrical Shape:

It is the shape formed by joining the number of planes or surfaces. 3D- geometric shape has length, breadth, and height(or depth).

Here measurements are volume, curved surface area (CSA), lateral surface area (LSA) and total surface area (TSA).

**Some Important Terms in 2D Geometric Shapes **

**Area
**It is the amount of space inside the geometrical figure.

**Perimeter**

It is the sum of lengths of all sides of a geometrical figure.

## Mensuration 2D Formulas in Maths

**Triangle**

Area of triangle =(1/2)×base×height

⇒(1/2)×b×h

Perimeter = a+b+c

**Scalene Triangle**

Here, s=a+b+c/2

Perimeter= a+b+c

**Isosceles Triangle**

Area = ½×b×h (or)

Perimeter= 2a+b

**Right Isosceles Triangle**

area= (½)× b^{2} or ¼(hypotenuse)^{2}

#### Equilateral Triangle

**Right Angled Triangle**

Area=1/2×b×h

Perimeter=b+h+hypotenuse

#### Quadrilateral

Here, d1, d2 are diagonal and s= (a+b+c+d)/2

Perimeter=a+b+c+d

**Square**

Area=s^{2
}^{ ⇒ }d^{2}/2, here d is diagonal

Perimeter = 4s

**Rectangle**

Area=l×b

Perimeter= 2×(l+b)

**Parallelogram**

Area=b×h

Perimeter=2(l+b)

**Rhombus**

Area=1/2×d1×d2

Perimeter=4a

**Trapezium**

Area=1/2×(a+b)×h

Perimeter=a+b+c+d

**Circle**

a = πr^{2} or πd^{2}/4 d=2r

Here d=diameter and r=radius

Perimeter=2πr (or) πd

#### Semi Circle

A=Πr^{2}/2 (or) Πd^{2}/8

p=πr + 2r (or) (36/7)r

### For Equivalent Triangle having Inside Circle or Inscribed Circle

In-radius of inscribed circle=a/2√3

Area of incircle= (π /12)a^{2}

### For Equilateral Triangle thing Circumcircle

circumradius of circum circle=a/√3

Area of circum circle= (__π/3)__a^{2}

## Some important Formulas for Areas of Regular Polygons

No. of sides |
Regular polygon |
Area |

3 | Equilateral triangle | 0.433 s^{2} |

4 | Square | 1s^{2} |

5 | Pentagon | 1.720s^{2} |

6 | Hexagon | 2.598 s^{2} |

7 | Septagon | 3.6434 s^{2} |

8 | Octagon | 4.828 s^{2} |

9 | Nanogon | 6.182 s^{2} |

10 | Decagon | 7.694 s^{2} |

**Sector of Circle**

r=radius

θ=angle in radius

Area of sector=lr/2 (or) (θ/360)×πr^{2
} length of Arc l=θr (or) (θ/360) ×2πr

**Ellipse**

x=semi major axis

y=semi minor axis a= πxy

**Annulus**

r is the inner radius

R is the outer radius

Area A= π(R^{2}-r^{2})

Width=R-r

#### Triangle

If r is in the radius of a triangle then

Area= r x s

If R is circumradius of a triangle

Area=abc/4R

Area of triangle=1/2 absinθ

3D units of Measurements are

volume:cm^{3} (or) m^{3
}LSA: m^{2} (or) cm^{2}

CSA: m^{2} (or) cm^{2}

TSA: m^{2} (or) cm^{2}

## Some important Terms in 3D Geometrical Shapes

**Volume:** It is the total space included inside the 3D shape.

Volume= area of the base x height

LSA (Lateral Surface Area): It is the area of all vertical faces excluding the area of bases of the 3D shape.

Eg: cube, cuboid.

CSA(Curved Surface Area): It is the area of all curved regions of a 3D- shape.

Eg: sphere, cylinder, cone, hemisphere.

TSA(Total Surface Area): It is the sum of areas of all the surfaces of a 3D shape.

Eg: cube, cuboid, cylinder, cone, sphere, hemisphere.

## Mensuration 3D Formulas in Maths

**Cube**

S=side

Volume=s^{3} cubic units

Total surface area (TSA)= 6s^{2} sq. Units

Face diagonal= √2×s

Diagonal of cube= √3×s

LSA=4s^{2}

**Cuboid/Rectangular Solid**

l = length

b= breadth

h= height

Volume= l x bx h

LSA= 2h(l+b)

TSA= 2(lb+bh+lh)

**Cylinder**

r= radius

h=height

volume= πr^{2}h

Curved Surface Area(CSA)= 2πrh

Total Surface Area= 2πr(h+r)

Area of base= area of top= πr^{2}

**Sphere**

r=radius

volume= (4/3)× πr^{3}

Curved Surface Area of Sphere= Total Surface Area of Sphere= 4πr^{2}

**Hemisphere**

r=radius

volume=2/3 πr^{3
}Curved surface area= 2πr^{2
}Total surface area of hemisphere= 3πr^{2}

**Right Circular Cone**

r=radius

h=height

l=slant height

Volume of cone= 3Πr^{2}h

Curved surface area of cone= πrl

Total surface area of cone= πr(r+l)

**Frustum of a cone**

r=top radius

R=base radius

h=height

l=slant height

Volume= (Π/3)(r^{2} +rR+R^{2} )h

The curved surface area of a frustum of a cone= πl(r+R)

The total surface area of a frustum of a cone=πl(r+R)πr^{2}+πR^{2
}= πl(r+R)+π(r^{2}+R^{2})

Slant height of the frustum of cone is

### Mensuration Formulas for Prism in maths

Prism is a polyhedron that is formed by joining lateral faces of two polygonal bases that are parallel to each other. These lateral faces are perpendicular to their polygonal bases.

**Prism Formulas in General**

Surface area of a prism= (Lateral Surface Area)+(2 x Base Area)

Lateral surface area = perimeter of base x slant height

Volume= base area x height

**Rectangular Prism**

Volume of rectangular prism= l×b×h

Total surface area of rectangular prism=2(l×b+b×h+l×h)

**Triangular Prism**

Base area=12ab

olVume= ½ abh

TSA= ab+3bh

Here a= apothem length; b= base ; h=height

**Prism example problem with a solution**

**Question: **The base of a right prism is a triangle and the lengths of its sides are 15cm, 25cm and 30 cm. The height of the prism is 10 cm. Find the volume, lateral surface area, and total surface area of prism?

**Solution:
**Let ‘s’ be the semi-perimeter of the base of the prism s=(15+25+30)/2 = 35 cm

Volume of the prism=(area of base) x height

=187 x 10

=1870 cu. Cm

The lateral surface area of prism= (perimeter of base x slant height)

perimeter of base=(15+25+30)=70

LSA=(70 x 10)= 700

TSA of triangular prism=( LSA + 2 × base area)

=(700 + 2 × 187)

=1074 cu. cm

### Mensuration Formulas for a pyramid

The pyramid is a polyhedron where lateral faces are triangular formed by connecting a polygonal base and a point called the apex.

A pyramid with n sided base has

- Verticles (n+1)
- Faces (n+1)
- Edges 2n

**Pyramid Formulas in General**

Volume=(1/3) × (base area) x height

Lateral surface area= ½ (perimeter of base) x lateral height

lateral height =

a=base of square/rectangle/triangle

h=height

Total surface area of pyramid= lateral surface area+area of base

**Square Pyramid**

a=side

s=slant height

h=height

Volume = (¹⁄3)×a²×h

Total surface area= 2as+a^{2}

**Triangular Pyramid**

a=base length

b=apothem length

s=slant height

h=height

Volume of triangular pyramid=(¹⁄6 )×a×b×h

Total surface area of triangular pyramid= ½ ab + 3/2 as

**Regular Tetrahedron**

a=side

Volume= (1/12)×(√2a³)

Total surface area=√3a²