9th Class Mathematics Surface Areas and Volumes Surface Area and Volume

SURFACE AREA AND VOLUME

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 = \[l\times b\times h\]

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

• A cuboidal 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 \[=2\pi rh+2\pi {{r}^{2}}h=2\pi r\left( h+r \right)\]

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

Radius = r. Height = h

Slant height = 1

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 j3omt is called a sphere.

OX = 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 \[\frac{2}{3}\pi {{r}^{3}}\]

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

Total surface area of a Hemisphere = \[3\pi {{r}^{2}}\]

• Prism:- Volume of Right prism \[=Area\text{ }of\text{ }Base\times Height.\]

Lateral surface area of a prism \[=perimeter\,of\,base\times Height\]

• Pyramid:- Surface area of pyramid \[=\frac{1}{2}\left( perimeter\text{ }of\text{ }base \right)\times Slant\text{ }Height\]

Whole surface = The slant surface + the area of the base

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