Mensuration

**Category : **6th Class

**Mensuration**

**Perimeter and Area of Plane Figures**

Perimeter of geometrical figure is the sum of its sides. There are different types of geometrical figures. Figures are classified by their shapes and sizes. Area of a geometrical figure is its total surface area.

**Perimeter and Area of a Triangle**

- Perimeter of a triangle = Sum of the length of all sides.
- Area of a right triangle \[\text{=}\frac{\text{1}}{\text{2}}\times \text{Base}\times \text{Height}\]
- Perimeter of an equilateral triangle \[\text{=3}\times \text{Side}\]
- Area of an equilateral triangle \[\text{=}\frac{\sqrt{\text{3}}}{\text{4}}\times {{\text{(Side)}}^{\text{2}}}\]

**Perimeter and Area of a Parallelogram**

Parallelogram is a quadrilateral whose opposite sides are equal and parallel to each other.

In the given figure ABCD is a Parallelogram in which \[\text{AB}\parallel \text{CD,}\,\,\text{BC}\parallel \text{AD,}\,\,\text{AB}=\text{CD}\] and \[\text{AD=BC}\]

Perimeter of a Parallelogram = 2 (sum of two adjacent sides)

Hence, perimeter of a parallelogram \[\text{ABCD=2(AB+BC)}\]Area of a parallelogram = Base \[\text{ }\!\!\times\!\!\text{ }\] Height

Therefore, the area of a parallelogram \[\text{ABCD=AB }\!\!\times\!\!\text{ CE}\]

**Perimeter and Area of a Rectangle**

A rectangle has four right angles and its opposite sides are equal.

Longer side of a rectangle is called length and shorter side is called width.

Perimeter of rectangle

\[ABCD=AB+BC+CD+DA\]

= length + width + length + width = 2(length + width)

Hence, perimeter of a rectangle = 2(length + width)

Area of a rectangle = length \[\times \]width

**Perimeter and Area of a Rhombus**

A rhombus is a parallelogram with four equal sides.

Therefore, perimeter of rhombus\[=4\times side\]. In the figure given below ABCD is a rhombus.

Perimeter of a rhombus, \[=4\times side\]

Area of a rhombus = base \[\times \] height

Also area of a rhombus \[\text{=}\frac{1}{2}\times \] product of length of diagonals.

**Perimeter and Area of a Square**

A square has four equal sides and each angle of\[90{}^\circ \].

In the picture given below, ABCD is a square because its all sides are equal and each angle is a right angle.

Perimeter of square = side + side + side + side\[=4\times side\]

Area of a Square = side \[\times \]side = \[{{(side)}^{2}}\]

**Perimeter and Area of a Trapezium**

A quadrilateral whose one pair of sides are parallel is called a trapezium. The given figure is a trapezium in which parallel sides are AB and CD and non-parallel sides are AD and BC

Perimeter of a trapezium = Sum of the length of all sides

Area of a trapezium \[=\frac{1}{2}\times \] (Sum of lengths of parallel sides) \[\times \]distance between parallel sides.

**Cirumference and Area of a circle**

A round plane figure whose all points are equidistant from a fixe point is called a circle and the fixed point is called centre of the circle and fixed distance is called radius of the circle.

Diameter = 2 \[\times \]Radius.

Circumference or perimeter of a circle\[=2\pi r=\pi d\].

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

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

Perimeter of a semicircle \[=\frac{2\pi r}{2}+\]\[2r=\pi r+2r\]\[=r(\pi +2)\]

Area of a circular ring = Area of outer circle \[-\] Area of inner circle

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

**Example:** What will be the height of a triangle if the area of triangle is 18 \[c{{m}^{2}}\]and base is 12 cm?

(a) 3 cm (b) 9 cm

(c) 10 cm (d) 4 cm

(e) None of these

** **

Answer (a)

**Explanation:** Area of a triangle = \[\frac{1}{2}\]\[\times \]base \[\times \] height

\[\Rightarrow 18=\frac{1}{2}\times 12\times \] height \[\Rightarrow \]height = 3 cm.

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