Mensuration-II

 

Mensuration II (Volume and Surface Area)

 

VOLUME

Space occupied by the 3-D is called 'volume' of that Particular object. It is always measured in cube unit.

 

SURFACE AREA

Surface area of a solid body is the area of all of its surfaces together. Surface area is measured in square unit.

 

CUBE

A solid body having 6 equal faces with equal length, breadth and height is called a cube. In fact, each face of a cube is a square.

·         Volume of the cube\[={{a}^{3}}\]

·         Whole surface of a cube \[=6{{a}^{2}}\]

·         Diagonal of the cube \[=a\sqrt{3}\]

Where, a = Side (or edge) of the cube

 

CUBOID

A rectangular solid body having 6 rectangular faces is called a cuboid.

·         Volume of the cuboid = lbh

·         Whole surface or surface area of the cuboid

\[=2(lb+bh+lh)\]

·         Diagonal of cuboid \[=\sqrt{{{l}^{2}}+{{b}^{2}}+{{h}^{2}}}\]

Where, I = Length, b = Breadth and h = Height

 

Room

A rectangular room has 4 walls (surfaces) and opposite walls have equal areas.

·         Total area of wall \[=2\,(l+b)\times h\]

·         Total volume of the room \[=lbh\]

·         Area of floor or roof \[l\times b\]

Where, / = Length, b = Breadth and h = Height

 

Box

A box has its shape like cube or cuboid.

·         Surface area of an open box

\[=2\,(\text{Lengh}t+\text{Breadth})\times \text{Height}+\text{Length}\times \text{Breadth}\]

\[=2\times (l+b)\times h+l\times b\]

·         Capacity of box \[=(l-2t)(b-2t)(h-2t)\]

Where, t = Thickness of the box

·         Volume of the material of the box

= External volume - Internal volume (or capacity)

\[=lbh-(l\,-2t)(b\,-2t)(h\,-2t)\]

Where, 1= Length, 6= Breadth and h = Height

 

Parallelepiped

A rectangular cuboid, also called as rectangular parallelepiped.

 

SPHERE AND HEMISPHERE

A sphere is a three-dimensional solid figure, which is made up of all points in the space, which lie at a constant distance from a fixed point.

·         Volume of the sphere \[=\frac{4}{3}\pi {{r}^{3}}\]

·         Total surface area \[=4\pi {{r}^{2}}\]Where, r = Radius

 

Hollow Sphere or Spherical Shell

Its both external and internal surfaces are spherical and both the surfaces have a common centre point.

·         Volume of hollow sphere \[=\frac{4}{3}\pi \,({{R}^{3}}-{{r}^{3}})\]

·         Internal surface area \[=4\pi {{r}^{2}}\]

·         External surface area \[=4\pi {{R}^{2}}\]

Where,         R = External radius

and                r = Internal radius

 

Hemisphere

It is the half part of a sphere.

·         Volume of the hemisphere\[=\frac{2}{3}\pi {{r}^{3}}\]

·         Total surface area \[=3\pi {{r}^{3}}\]

·         Curved surface area \[=2\pi {{r}^{2}}\] Where, r= Radius

 

CONE AND FRUSTUM OF CONE

Cone is a solid or hollow body with a round base and pointed top.

·         Volume \[=\frac{1}{3}\times \]Base area \[\times \] Height \[=\frac{1}{3}\pi {{r}^{2}}h\]

·         Slant height \[(l)=\sqrt{{{r}^{2}}+{{h}^{2}}}\]

·         Curved surface area \[\pi rl=\pi r\sqrt{{{r}^{2}}+{{h}^{2}}}\]

·         Total surface area \[=\pi rl+\pi {{r}^{2}}=\pi r\,(l+r)\]

Where,         r = Radius of bases

h = Height and

l = Slant height

Frustum of Cone

If a cone is cut by a plane parallel to the base, so as to divide the cone into two parts upper part and lower part, then the lower part is called frustum.

Slant height \[(l)=\sqrt{{{h}^{2}}+{{(R-r)}^{2}}}\]

Curved surface area \[=\pi \,(R+r)\,l\]

·         Total surface area \[=\pi \{(r+R)\,l+{{r}^{2}}+{{R}^{2}}\}\]

·         Volume \[=\frac{\pi h}{3}({{r}^{2}}+{{R}^{2}}+rR)\]

Where,        r = Radius of top, R = Radius of base

h = Height, l= Slant height

CYLINDER

A cylinder is a solid or hollow body that is formed by keeping circles of equal radii one on another.

·         Volume of cylinder = Area of base \[\times \] Height \[=\pi {{r}^{2}}h\]

·         Curved surface area = Perimeter of base \[\times \] Height

 \[=2\pi rh\]

·         Total surface area = Curved surface area + Area of both the circles \[=2\pi rh+2\pi {{r}^{2}}-2\pi r\,(r+h)\]

Where, r = Radius of base and h = Height

 

Hollow Cylinder            

If cylinder is hollow, then

·         Volume of hollow cylinder \[=\pi h\,({{R}^{2}}-{{r}^{2}})\]

·         Curved surface area \[=2\pi h\,(R+r)\]

·         Total surface area of hollow cylinder

\[=2\pi h\,(R+r)+2\pi \,(R-{{r}^{2}})\]

Where,        R = External radius of base

r = Internal radius of base

      and             h = Height

 

PRISM, CONE AND TETRAHEDRON

Prism   

 

·         Volume = Base area \[\times \] Height

·         Lateral surface area

= Perimeter of the base \[\times \] Height

 

Pyramid

·         Volume \[=\frac{1}{3}\times \]Base area \[\times \] Height

·         Lateral surface area

\[=\frac{1}{2}\times \]Perimeter of the base \[\times \] Slant height

·          Total surface area = Lateral surface area + Base area

 

Tetrahedron

A tetrahedron is a polyhedron composed of four triangular faces, three of which meet at corner of vertex. It has six edges and four vertices.

·         Volume of tetrahedron \[=\frac{{{(Side)}^{3}}}{2\sqrt{2}}\]

·         Surface area of tetrahedron \[=\sqrt{3}\,{{(Side)}^{2}}\]

Quicker One

If three dimensions of a three dimensional body are increases by x%, y% and z% respectively, then its volume increases

\[\left[ x+y+z+\frac{xy+xz+yz}{100}+\frac{xyz}{{{(100)}^{2}}} \right]%\]

Now, for cuboid x = Length, y = Breadth

and z = Height

and for cube, x = y = z = Edge

for cylinder, x = y = Radius and z = Height

for cone,    x = y = Radius and z = Height

for sphere and hemisphere, x = y = z = Radius