MENSURATION

**Category : **6th Class

**Learning Objective**

- To understand the concept of Area and Perimeter of plane figures (triangle, rectangle and square).
- To learn how to calculate area and perimeter of triangle, react angel and square.
- To understand how to calculate area and circumference of circle.
- To learn how to calculate surace area and volume of solid figures (cube and cuboid).

** **

**AREA AND PERIMETER OF PLANE FIGURE**

**PERIMETER**

The perimeter of a plane geometrical figure is the total length of sides (or boundary) enclosing the figure. Units of measuring perimeter can be mm, cm, m, km etc.

**AREA**

The area of any figure is the amount of surface enclosed within its bounding lines. Area is always expressed in square units.

** **

**1. TRIANGLE**

Perimeter of a triangle is equal to the sum of its sides.

For a triangle having sides a, b and c,

Perimeter \[=a+b+c\]

and area \[(A)\,=\sqrt{s(s-a)(s-b)\,(s-c)}\]

where \[s=\frac{a+b+c}{2}\]

area of equilateral triangle with each side a is \[\frac{\sqrt{3}}{4}{{\text{(side)}}^{2}}\].

Observe the given figure

Area of \[\Delta ABC\]\[=\frac{1}{2}\times base\times height\]

\[=\frac{1}{2}\times BC\times AD\]

Observe the given figure,

\[\Delta \,ABC\] is a right angled triangle, right angled at B.

Area of \[\Delta \,ABC=\frac{1}{2}\,BC\times AB\]

**For Example:**

Find the perimeter and area of triangle ABC.

**Sol:** \[P=AB+BC+CA\] = 7 + 4 + 5 = 16 cm Base = BC, Height =AD

Area

\[=\frac{1}{2}\times BC\times AD\]

\[=\frac{1}{2}\times 4\times 3\]

\[=6c{{m}^{2}}\]

2. RECTANGLE

Area of rectangle \[ABCD=AB\times BC=l\times b\]

Perimeter of rectangle = sum of all sides

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

Diagonal (AC) of rectangle \[=\sqrt{{{l}^{2}}+{{b}^{2}}}\]

For example:

For the adjoining rectangle find:

(i) the perimeter;

(ii) the area.

Sol. Perimeter \[=2(l+b)\,=2(6+4)\,=2\times 10=20\,cm\] Area \[=l\times b=6\times 4=24\,c{{m}^{2}}\]

**3. SQUARE**

Perimeter of square ABCD with side \[a=4a\]

Area of square with side \[a={{a}^{2}}\]

Diagonal of square

\[=AC=\sqrt{{{a}^{2}}+{{a}^{2}}}=\,\sqrt{2{{a}^{2}}}\,=a\sqrt{2}\]

\[=\sqrt{2}\times \] side of square

For example

For the adjoining square find:

(i) the perimeter;

(ii) the area

**Sol. **Perimeter \[=4a=4\times 5=20\,cm\]

Area \[={{a}^{2}}=5\times 5=25\,c{{m}^{2}}\]

**AREA AND CIRCUMFERENCE OF CIRCLE **

** **

**CIRCLE**

A circle is a path in a plane travelled by a point which moves in such a way that its distance from a fixed point is always constant.

The fixed point is called centre of circle and fixed distance is called radius of the circle,

**Circumference or perimeter **of circle of radius ‘r’, is

\[C=2\pi r=\pi d\](\[(2r=d,d=\]diameter)

Area of circle of radius \['r'=\pi {{r}^{2}}\]

**SEMI-CIRCLE**

A semicircle is a figure enclosed by a diameter and part of circumference of the circle cut-off by it.

Area of semicircle of radius \['r'=\frac{\pi {{r}^{2}}}{2}\]

Circumference of semicircle of radius \['r'=\pi r\]

**SURFACE AREA AND VOLUME OF SOLIDS**

**CUBOID**

A cuboid is a three dimensional box. It has six rectangular faces. It is defined by the virtue of its length \[(\ell )\], breadth (b) and height (h). It can be visualized as a room. It is also called rectangular parallelepiped.

Total surface area \[(TSA)\,=2\,(lh=lb+bh)\] Sq. units.

Lateral surface area (LSA) is the area of four walls (excluding area of base and top)

= 2h (1+b) Sq. units.

Length of diagonal of cuboid \[=\sqrt{{{1}^{2}}+{{h}^{2}}+{{b}^{2}}}\]

Volume of cuboid = Space occupied by cuboid = Area of base x height \[=(1\times b)\,\times h=1\times b\times h\] cubic units

**For example**

Find the surface area of a room having dimension \[12m\times 8m\times 5m\].

Sol.

Total surface area of room \[=2(lb+bh+lh)\]

\[=2(12\times 8+8\times 5+12\times 5)\]

=2(96 + 40 + 60)

\[=2\times 196\]

\[=392{{m}^{2}}\]

**CUBE**

A cube is a cuboid which has all its edges equal i.e. length = breadth = height = 'a' say.

Area of each face of the cube is \[{{a}^{2}}\] square units.

Total surface area (TSA) of square = Area of 6 square faces of cube TSA \[=6\times {{a}^{2}}=6{{a}^{2}}\,sq\].units.

Lateral surface area of cube (LSA) = Area of four faces (excluding bottom and top face)

\[LSA=4\times {{a}^{2}}\]

\[LSA=4{{a}^{2}}\] sq. units.

Length of diagonal (d) of cube

\[=\sqrt{{{a}^{2}}+{{a}^{2}}+{{a}^{2}}}\,=\sqrt{3{{a}^{2}}}\]

\[=a\sqrt{3}\]

Volume of cube (V) = Base Area x Height

\[V={{a}^{2}}\times a={{a}^{3}}\] cubic units

**For example**

F**o**r the adjoining cube find:

(i) Total surface area

(ii) Lateral surface area

(iii) Volume

**Sol.**

(i) Surface area

\[=6\times {{(side)}^{2}}=6\times {{(5\,cm)}^{2}}\,=150c{{m}^{2}}\]

(ii) Lateral surface area

\[=4\times \,{{(side)}^{2}}=4\times {{(5cm)}^{2}}=100c{{m}^{2}}\]

(iii) Valume \[={{(side)}^{2}}={{(5\,cm)}^{3}}=125c{{m}^{3}}\].

*play_arrow*MENSURATION*play_arrow*Mensuration*play_arrow*Mensuration Basics*play_arrow*Mensuration (Perimeter & Area, Review of Earlier Concepts)*play_arrow*Mensuration

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