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

Surface Area and Volume

Category : 9th Class

Surface Area and Volume

In this chapter, we will learn about some important formulas related to $2-D$ and $3-D$ geometrical shapes.

Area of a Triangle

• Area of a triangle $=\frac{1}{2}\times$ (Perpendicular) $\times$ Base
• Area of a triangle having lengths of the sides a, b and c is

$=\sqrt{s(s-a)(s-b)(s-c)}$ sq. units,

Where $s=\frac{1}{2}$ (a + b + c) • Area of an equilateral triangle $=\frac{\sqrt{3}}{4}{{a}^{2}}$, where a is the side of the equilateral triangle.

Circle

• Circumference of the circle = $2\pi r$
• Area of the circle = $\pi {{r}^{2}}$
• Area of the semicircle = $\frac{1}{2}\pi {{r}^{2}}$
• Perimeter of the semicircle =$\pi r+2r$ Length of Arc and Area of a Sector

Let an arc AB makes an angle $\theta <180{}^\circ$ at the center (O) of a circle of radius r, then we have:

• Length of the arc AB = $\frac{2\pi r\theta }{360{}^\circ }$
• Area of the sector OACB = $\frac{\pi {{r}^{2}}\theta }{360{}^\circ }$ • Area of the minor segments ACBA = area of sector OACB$~-$area of the corresponding triangle AOB
• Area of the major segment ADBA = area of the circle $~-$ area of the minor segment

Perimeter and Area of a Rectangle

Let ABCD be a rectangle in which length AB = l units, breadth BC = b units then we have:

• Area = ($l\times b$) square units
• Length (l) $=\frac{area\,\,(A)}{breadth\,\,(B)}$ units • breadth (b) $=\frac{area\,\,(A)}{length\,\,(l)}$ units
• Diagonal (d) = $\sqrt{{{l}^{2}}+{{b}^{2}}}$units

• Perimeter (p) = 2(l + b) units

Area of Four Walls of a Room

Let l, b and h are respectively the length, breadth and height of a room, then area of four walls of the room

$=\text{ }\left\{ 2\text{ }\left( \text{l }+\text{ }b \right)\text{ }x\text{ }h \right\}$ sq units.

Perimeter and Area of Square

Let ABCD be a square with each side equal to ‘a’ units, then

• Area = ${{a}^{2}}$ sq. units
• Area = $\left( \frac{1}{2}\times {{(Diagonal)}^{2}} \right)$ sq. units • Diagonal = $a\sqrt{2}$ units
• Perimeter =4a units

Area of Some Special Types of Quadrilateral

• Area of a parallelogram = (base $\times$ height) • Area of a rhombus = $\frac{1}{2}\times$ (product of diagonals) • Area of a Trapezium = $\frac{1}{2}$ (Sum of lengths of parallel sides) $\times$ (distance between them)

= $\frac{1}{2}$ (a + b) $\times$h Solids

The objects having definite shape and size are called solids. A solid occupies a definite space.

Cuboid

For a cuboid of length = l, breadth = b and height = h, we have:

• Volume = (l $\times$ b $\times$ h) cubic units
• Total surface area = 2 $(lb+bh+lh)$ sq. units
• Lateral surface area = $[2(l+b)\times h]$ sq. units
• Diagonal of a cuboid = $\sqrt{{{l}^{2}}+{{b}^{2}}+{{h}^{2}}}$

Cube

For a cube having each edge = a units, we have:

• Volume = ${{a}^{3}}$ cubic units
• Total surface area = $6{{a}^{2}}$ sq. units
• Lateral surface area = $4{{a}^{2}}$ sq. units
• Diagonal of a cube = a$\sqrt{3}$

Cylinder

Solids like jar, circular pencils, circular pipes, road rollers, gas cylinders are of cylindrical shape. For a cylinder of base radius = r units and height = h units, we have:

• Volume = $\pi {{r}^{2}}h$ cubic units
• Curved surface area = $2\pi rh$ square units
• Total surface area =

$(2\pi rh+2\pi {{r}^{2}})=2\pi r(h+r)$ sq. units

Cone

Consider a cone in which base radius = r, height = h and slant height (l) =$\sqrt{{{h}^{2}}+{{r}^{2}}}$, then we have:

• Volume of the cone = $\frac{1}{3}\pi {{r}^{2h}}$
• Curved surface area of the cone =$\pi rl$
• Total surface area of the cone = (curved surface area) + (area of the base) = $\pi rl$+$\pi r{{l}^{2}}$=$\pi rl$

(l + r)

Sphere

Objects like a football, a cricket ball, etc. are of spherical shapes. For a sphere of radius r, we have:

• Volume of the sphere =$\frac{4}{3}\pi {{r}^{3}}$
• Surface area of the sphere = $4\pi {{r}^{2}}$

Hemisphere

A plane through the centre of a sphere cuts it into two equal parts, each part is called a hemisphere. For a hemisphere of radius r, we have:

• Volume of the hemisphere = $\frac{2}{3}\pi {{r}^{3}}$
• Curved surface area of the hemisphere = $2\pi {{r}^{2}}$
• Total surface area of the hemisphere = $3\pi {{r}^{2}}$

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