8th Class Mathematics Mensuration

MENSURATION

FUNDAMENTALS

• Cuboid:- A cuboid is a solid bounded by the rectangular plane regions. A cuboid has six faces, 12 edges and 8 vertices.

Total surface Area of the cuboid \[=2\left( lb+bh+hl \right)\] sq. units.

Volume of the cuboid \[={{l}^{2}}\times b\times h\]

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

• Cube:- A cuboid whose length, breadth and height are equal is called a cube.

If length of each edge of a cube is a.

Then, volume of the cube \[={{a}^{3}}\]

Total surface area of the cube \[=6{{a}^{2}}\]

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

• Cylinder:- It is formed by rotating one side of a rectangle about its opposite side.

Volume of the cylinder \[=\pi {{r}^{2}}h\]

Area of the base \[=\pi {{r}^{2}}\]

Area of the curved surface \[=2\pi rh\]

Total surface Area

\[=27\pi rh+2{{\pi }^{2}}h=2\pi r(h+r)\]

• Right Circular Cone:- A right circular cone is a solid generated by rotating a right angled triangle around its height.

Radius = r, Height = h

Slant height = l

Volume of the cone \[\frac{1}{3}=\pi {{r}^{2}}h\]

Area of the Base \[=\pi {{r}^{2}}\]

Area of the curved surface \[=\pi r\sqrt{{{h}^{2}}+{{r}^{2}}}=\pi rl\]

• Sphere:- The set of all points in the space which are equidistant from fixed point is called a sphere.

Radius = r

Volume of a sphere\[~=\frac{4}{3}\pi {{r}^{3}}\]

Surface Area of a sphere \[=4\pi {{r}^{2}}\]

• Hemisphere:- A plane through the centre of the sphere divides the sphere into two equal parts each of which is called a hemisphere.

Radius \[=Ox=r\]

Volume of a Hemisphere\[\frac{2}{3}\pi r3\]

Curved surface area of a Hemisphere \[=27\pi {{r}^{2}}\]

Total surface area of a Hemisphere \[=37\pi {{r}^{2}}\]

• Prism:- Volume of Right prism

\[=Area\text{ }of\text{ }Base\times Height.\]

Lateral surface of prism

\[=Perimeter\text{ }of\text{ }base\times Height\]

• Pyramid:- Surface area of pyramid

\[=\frac{1}{2}\left( perimeter\text{ }of\text{ }base \right)\times Slant\text{ }Height\]

• This formula for Surface area is coming from the fact that Surface area of pyramid is nothing but sum total of areas of all its triangular faces.
• Whole surface = The slant surface + the area of the base
• Volume of pyramid

\[=\frac{1}{3}\left( Area\text{ }of\text{ }base \right)\times height.\]