March 2, 2024

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 formulas list, it will be easy to solve the mensuration problems.

Contents

# Mensuration formulas for 2D -shapes:​

 Name Figure Area Perimeter Rectangle l= length,   b=breadth $l&space;\times&space;b$ $2(l+b)$ 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) $\frac{1}{2}\times&space;b\times&space;h$ (ii)  Heron’s Formula $\sqrt{s(s-a)(s-b)(s-c)}$ a+b+c Equilateral triangle a= side , h= height or  altitude    $h=\frac{\sqrt{3}}{2}&space;\times&space;a$ (i) $\frac{1}{2}\times&space;a\times&space;h$ (ii) $\frac{\sqrt{3}}{4}a^{2}$ 3a Quadrilateral AC=diagonals $h_{1},&space;h_{2}$ altitudes on AC from the vertices D and B respectively. (i)  $\frac{1}{2}\times&space;AC\times&space;(h_{1}+h_{2})$   (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 $\frac{1}{2}\times&space;d_{1}\times&space;d_{2}$ 4a Trapezium a, b are parallel sides h is the perpendicular distance between parallel sides $\left&space;(&space;\frac{a+b}{2}&space;\right&space;)&space;\times&space;h$ AB+BC+CD+DA Circle r=radius , $\pi=\frac{22}{7}$ $\pi&space;r^{2}$ circumference=$2\pi&space;r$ Semi-circle r =radius $\frac{1}{2}\pi&space;r^{2}$ $\pi&space;r+&space;2r$ Sector of a circle o  centre   , r= radius l=length of arc AB,  $\theta$= angle of the sector $l=2\pi&space;r.\frac{\theta&space;}{360^{\circ}}$ (i) $\pi&space;r^{2}\frac{\theta&space;}{360^{\circ}}$ (ii) $\frac{1}{2}r\times&space;l$ 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 $2a^{2}\left&space;(&space;1+\sqrt{2}&space;\right&space;)$ 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 $\pi&space;r^{2}$ $2\pi&space;rh$ $2\pi&space;r(h+r)$ Right triangular prism area of base x height perimeter of base x height lateral surface area+2(area of base) Sphere r=radius $\frac{4}{3}\pi&space;r^{3}$ $4\pi&space;r^{2}$ Hemisphere r= radius $\frac{2}{3}\pi&space;r^{3}$ $2\pi&space;r^{2}$ $3\pi&space;r^{2}$ Pyramid l= slant height $\frac{1}{3}\times$base x height $\frac{1}{2}\times$ perimeter of base x slant height lateral surface area+base area Cone l=slant height,   h =height , r= radius of base $\frac{1}{3}\times$ area of base x height=$\frac{1}{3}\pi&space;r^{2}h$ $\pi&space;rl$ $\pi&space;rl+\pi&space;r^{2}$ Frustum of a cone $\frac{1}{3}\pi&space;h(r^{2}+Rr+R^{2})$ $\pi&space;l(r+R)$ lateral surface area+ $\pi&space;(R^{2}+r^{2})$