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 \[=\frac{1}{2}\,\,\,\times \text{ }Base\text{ }\times \text{ }Height\]
- Perimeter of an equilateral triangle \[=\text{ }3\text{ }\times \text{ }Side\]
- Area of an equilateral triangle \[=\frac{\sqrt{3}}{4}\times {{\left( Side \right)}^{2}}\]

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 \[AB||\,\,CD,\,\,BC||\,\,\,AD,\text{ }AB\text{ }=\text{ }CD\text{ }and\text{ }AD\text{ }=\text{ }BC\]

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

Hence, perimeter of a parallelogram ABCD = 2(AB + BC)

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

Therefore, the area of a parallelogram \[ABCD=AB\times CE\]

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 \[=\text{ }2\left( length\text{ }+\text{ }width \right)\]

Hence, perimeter of a rectangle \[=\text{ }2\left( length\text{ }+\text{ }width \right)\]

Area of a rectangle \[=\text{ }length\text{ }\times \text{ }width\]

Perimeter and Area of a Rhombus

A rhombus is a parallelogram with four equal sides.

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

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

Area of a rhombus \[=\text{ }base\text{ }\times \text{ }height\]

Also area of a rhombus =\[\frac{1}{2}\] 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 \[=\text{ }side\text{ }+\text{ }side\text{ }+\text{ }side\text{ }+\text{ }side\text{ }=\text{ }4\text{ }\times \text{ }side\]

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

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 alludes

Area of a trapezium\[=\frac{1}{2}\times ~\left( sum,\text{ }of\text{ }lengths\text{ }of\text{ }parallel\text{ }sides \right)\text{ }\times \text{ }distance\text{ }between\text{ }parallel\text{ }sides.\]

Circumference and Area of a circle

A round [lane figure whose all point are equidistant from a fixe point is called a circle and he fixed point is called center of the circle and fixed distance is called radius of the circle

Diameter \[=\text{ }2\text{ }\times \text{ }Radius.\]

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 \[\mathbf{18}\text{ }\mathbf{c}{{\mathbf{m}}^{\mathbf{2}}}\]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 \[=\frac{1}{2}\times \text{ }base\text{ }\times \text{ }height\]

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

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