December 9, 2022

Mensuration Formulas for 2D and 3D Shapes PDF Download

Mensuration is a branch of mathematics that deals with the study of different geometrical shapes, their perimeter,area , surface area, curved surface area,  volume  etc. Basically,  there are two type of geometric shapes (i) 2D shapes   (ii) 3D shapes

2D shapes are : circle, square, rectangle, square , parallelogram, rhombus etc.

3D shapes are : cube , cylinder, cone , cuboid, sphere , prism , pyramid , cone etc.

Now let’s learn all the important mensuration formulas involving 2D and 3D shapes. Using this mensuration formula list, it will be easy to solve the mensuration problems.

Mensuration formulas for 2D -shapes:

Name Figure Area   Perimeter
Rectangle 
 
  l= length,   b=breadth
l \times b 2(l+b)
Square a= side ,   d= diagonal a^{2}

If d is given , then A=\frac{d^{2}}{2}

4 x side= 4a
Triangle

(scalene)

   s=\frac{a+b+c}{2}

b =base,    h=  height or altitude of a triangle

(i) \frac{1}{2}\times b\times h

(ii)  Heron’s Formula

\sqrt{s(s-a)(s-b)(s-c)}

a+b+c
Equilateral triangle  a= side , h= height or  altitude    h=\frac{\sqrt{3}}{2} \times a (i) \frac{1}{2}\times a\times h

(ii) \frac{\sqrt{3}}{4}a^{2}

3a
Quadrilateral AC=diagonals

h_{1}, h_{2} altitudes on AC from the vertices D and B respectively.

   

(i)  \frac{1}{2}\times AC\times (h_{1}+h_{2})

 

(ii) \frac{1}{2}\times product of diagonals x sin of the angle between them

AB+BC+CD+AD
Parallelogram a and b be the lengths of parallel sides and h be the height (i)Area= base x  height

(ii) area= absin\theta  , \theta is the angle between the sides of the parallelogram

2(a+b)
Rhombus       a=each equal sides  ,                                        d_{1}  and d_{2}  are the diagonals \frac{1}{2}\times d_{1}\times d_{2} 4a
Trapezium a, b are parallel sides

h is the perpendicular distance between parallel sides

\left ( \frac{a+b}{2} \right ) \times h AB+BC+CD+DA
Circle    r=radius , \pi=\frac{22}{7} \pi r^{2} circumference=2\pi r
Semi-circle            r =radius \frac{1}{2}\pi r^{2} \pi r+ 2r
Sector of a circle o  centre   , r= radius

l=length of arc AB,  \theta= angle of the sector

l=2\pi r.\frac{\theta }{360^{\circ}}

(i) \pi r^{2}\frac{\theta }{360^{\circ}}

(ii) \frac{1}{2}r\times l

l+2r
Regular hexagon   a= each of the equal side \frac{3\sqrt{3}}{2}a^{2} 6a
Regular octagon a= each of the equal side 2a^{2}\left ( 1+\sqrt{2} \right ) 8a

 

Mensuration formulas for 3D -shapes:

Name Figure Volume Lateral /Curved surface area Total surface Area
Cube    a=side/edge a^{3} 4a^{2} 6a^{2}
cuboid  l=length, b=breadth,  h=height lbh 2(l+b)h 2(lb+bh+hl)
Right circular cylinder r= radius of base 

h=height

\pi r^{2} 2\pi rh 2\pi r(h+r)
Right triangular prism area of base x height perimeter of base x height lateral surface area+2(area of base)
Sphere   r=radius \frac{4}{3}\pi r^{3} 4\pi r^{2}
Hemisphere   r= radius \frac{2}{3}\pi r^{3} 2\pi r^{2} 3\pi r^{2}
Pyramid  

l= slant height

\frac{1}{3}\timesbase x height \frac{1}{2}\timesperimeter of base x slant height lateral surface area+base area
Cone l=slant height, 

 h =height , r= radius of base

\frac{1}{3}\timesarea of base x height=\frac{1}{3}\pi r^{2}h \pi rl \pi rl+\pi r^{2}
Frustum of a cone      \frac{1}{3}\pi h(r^{2}+Rr+R^{2}) \pi l(r+R) lateral surface area+ \pi (R^{2}+r^{2})

Download PDF:  mensuration formulas-pdf

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