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**NCERT / CBSE NOTES : Chapter Summary**

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__Mensuration__

### Methods to Find Area of Quadrilaterals and Polygons:

We can find the area of the triangle using the formula

To find the area of a quadrilateral we will divide the quadrilateral into two triangles and add the areas of the two triangles.

#### Triangulation

The method of dividing a quadrilateral into two triangles to find out its area is known as triangulation.

Let

**ABCD**is a quadrilateral. Then
Area of quadrilateral ABCD

= (Area of ∆ ABD) + (Area of ∆ BCD)

= ½ d (), where d is the diagonal and and are the heights of the quadrilateral.

Area of quadrilateral ABCD= ½ d ()

In a parallelogram the diagonal divide it into two triangles. Now

**Area of a Parallelogram = base × height**

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**Rhombus**

In a

**rhombus**diagonals are perpendicular to each other, so we can use the method of triangulation to find the area of a rhombus.**Area of a Rhombus**= (), where and are the lengths of the diagonals.

Area of a Rhombus = (), where and are the lengths of the diagonals.

### Trapezium

#### A trapezium has a pair of parallel sides.

Area of trapezium =

Where h = perpendicular distance between the parallel sides of a trapezium and a and b are the lengths of the parallel sides.

Area of trapezium =

### Polygon

A polygon is a closed shape that has at least three sides. It is a closed shape that has at least three sides. Triangles, quadrilaterals, rectangles and squares are all types of polygons. Moreover, a polygon can be of any shape and can have any number of sides. There is no specific formula for calculating the area of a polygon. The best way is to split the polygon into shapes whose area can be calculated individually.

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**Surface Area of Solids**

Cube: A

The

Cuboid: A cuboid is a solid made up of six rectangles, called faces. The total area of a cuboid is the sum of the areas of these faces.

The lateral surface area (LSA) of a cuboid is 2h (l + b), whereas its total surface area (TSA) is (lb + bh + hl).

A cylinder is a solid composed of two congruent circles in parallel planes.

The curved surface area (CSA) of a cylinderical pillar is 2πrh , whereas its total surface area (TSA) is 2πr(r+ h).

**cube**is a three-dimensional figure, a solid made up of six equal squares called**faces**.The

**lateral surface area**(**LSA**) of a cube is equal to 4l2, whereas its total**surface area**(TSA) is 6l2.Cuboid: A cuboid is a solid made up of six rectangles, called faces. The total area of a cuboid is the sum of the areas of these faces.

The lateral surface area (LSA) of a cuboid is 2h (l + b), whereas its total surface area (TSA) is (lb + bh + hl).

**Cylinder:**A cylinder is a solid composed of two congruent circles in parallel planes.

The curved surface area (CSA) of a cylinderical pillar is 2πrh , whereas its total surface area (TSA) is 2πr(r+ h).

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**Volume of Solids**

Bodies that occupy space are called solids.

Solid bodies occur in various shapes, such as cuboid, cube, cylinder and cone.

The space occupied by a solid body is called its volume.

The units for volume are cubic

*centimetre(cm*etc.^{3}), cubic metre(m^{3})
A cuboid is a solid bounded by six rectangular plane faces.

Consider a cuboid of length, breadth and height , and , respectively. Then:

- Volume of cuboid =
*lbh cubic units* - Total surface area of cuboid =
*2(lb + bh + lh)sq.units* - Lateral surface area of cuboid = Area of 4 walls = Area of 4 walls =
*2h(l + b)sq.units*

A cube whose length, breadth and height are all equal is called a cube.

Consider a cube of edge

*a units*. Then:- Volume of cube =
*a*^{3}cubic units - Total surface area of cube =
*6a*^{2}sq.units - Lateral surface area of cube =
*4a*^{2}sq.units

A solid bounded by a cylindrical surface and two parallel circular bases at the top and bottom is called a cylinder.

Consider a cylinder of radius

*r units*and height*h units*. Then:- Volume of cylinder
- Total surface area of cylinder
- Lateral surface area of cylinder

The volume of a cylinder is also sometimes known as its capacity.

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