Mensuration

 

What is Mensuration?

Mensuration is a branch of mathematics which mainly deals with the study of different kinds of Geometrical shapes along with its area, length, volume and perimeters. It is completely based on the application of both algebraic equations and geometric calculations. The results obtained by the Mensuration are considered very accurate. There are two types of geometric shapes:

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Volume of Solids

Volume of a 3D Object

Volume is the space occupied by the three dimensional object. It is a three dimensional quantity.

Volume of a Cuboid

Volume of a cuboid $=l\times b\times h$

where, $l$ is the length, $b$ is the breadth and $h$ is the height of the cuboid.

Volume of a Cube

Volume of a cube $=l^3$

Where, $l$ is the length of the each side of the cube.

Volume of a Cylinder

Volume of the cylinder $=\pi r^{2}h$

Where $r$ is the radius of the base and $h$ is the height of the cylinder.
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Basics Revisited

Introduction to Mensuration

Mensuration is the study of geometry that deals with the measurement of length, areas and volumes.

Perimeter is the total length or path of a given shape.
Area is the total region covered by the given shape.
Volume is the total space occupied by the given shape.

Identifying Shapes and Areas of Different Regular Figures

Area of a Rectangle : $length\times breadth$,  perimeter : $2(length+breadth)$

Area of Square : $side\times side$, perimeter : $4\times side$

Area of  Triangle : 1/2 $(base\times height)$, perimeter : $a+b+c\left(sumof3sides\right)$

Area of Parallelogram : $base\times height$ , perimeter : $2(length+breadth)$

Area of Circle : $\pi \times (radius{)}^2$, perimeter : $2\times \pi \times radius$.

Trapezium

Area of Trapezium by Division into Shapes of Known Area

Consider the trapezium where $a$ and $b$ are parallel sides, $h$ is the height. Trapezium is divided into 3 parts : two triangles, one rectangle.

Here $h$ is the height, $a$ and $b$ are 2 parallel sides.

Area of trapezium = Area of 2 triangles + Area of rectangle
$=[12\times c\times h+12\times d\times h]+[a\times h]$
$=[12\times c\times h+12\times d\times h]+[12\times 2a\times h]$  [Multiplying and dividing by 2] $=12\times h(c+d+2a)$
$=12\times h(c+d+a+a)$
$=12\times h(b+a)$    [$\because$ c + d + a = b]

Area of trapezium $=12\times h$ (sum of 2 parallel sides)

Area of Trapezium by Finding the Area of a Triangle of Same Area

The area of the trapezium can be found out by dividing it into a triangle and a polygon.

Consider a trapezium $WXYZ$. Mark a midpoint $A$ for side $XY$ and join $AZ$. Cut the trapezium along $AZ$ and obtain a $\Delta AZY$
Flip the $\Delta AZY$ and place it as shown below. Now the new polygon is a triangle.

We know that,
$Areaofatriangle=12\times base\times height$
Substituting the values we get,
$Areaofatriangle=12\times (a+b)\times h$
But the original polygon is a trapezium. So,

$Areaofatrapezium=12\times (a+b)\times h$

Area of a General Quadrilateral

Consider a quadrilateral ABCD. Draw diagonal AC. From B and D draw perpendiculars $h_1,h_2$ to AC

Area of quadrilateral = Area of triangle ABC + Area of triangle ADC
$=12\times base\times height+\frac{1}{2}\times base\times height$
$=(12\times AC\times h_1)+(12\times AC\times h_2)$  [Where, $h_1$, $h_2$ are the heights, AC is the base] $=12\times AC\times (h_1+h_2)$
$=12\times d\times (h_1+h_2)$  [$\because$ AC is a diagonal]

$\therefore AreaofaQuadrilateral=12\times d\times (h_1+h_2)$

where $d$ is diagonal and $h_1,h_2$ are perpendicular drwan to a diagonal.

Area of Rhombus

Area of rhombus $=12\times d_1\times d_2$,

where $d_1$ and $d_2$ are the diagonals.

Area of Polygons

The area of any given polygon can be found by cutting the polygon into shapes whose area is known and adding the area of these shapes.

Some of the ways to find the area is shown below.

Area of this polygon = area of 2 trapeziums

Area of this polygon = Area of 2 triangles + Area of rectangle.

Area of this polygon = Area of 4 triangles.

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Surface Area of Solids

Solid Shapes

Solid shapes or solid figures are the three dimensional figures which have length, breadth and height. Using these, surface areas and volumes of these figures are found out.

Solids with a Pair or More of Identical Faces

Solids with a pair of identical faces are:

Surface Area of Solid Shapes

The surface area of the object is the total area occupied by the surface of the object.
or  surface area is simply the sum of the areas of the flat surfaces (called faces).

Surface Area of a Cuboid

Total Surface area of cuboid $=2(lb+bh+lh)$

Lateral Surface area of cuboid $=2h(l+b)$

Where, $l$ is the length, $b$ is the breadth and $h$ is the height.

Surface Area of a Cube

Total Surface area of a cube $=6l^2$

Lateral Surface area of a cube  $=4l^2$

Where $l$ is the length of each side of the cube.

Surface Area of a Cylinder

Curved surface area of cylinder (C.S.A) $=2\pi rh$

Total Surface area of cylinder(T.S.A) $=2\pi r(r+h)$

Where, $r$ is the radius of the cylinder and $h$ is the height of the cylinder.

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Summary of Mensuration

Relation between Volume and Capacity

Volume is the total space occupied by an object. Volume is measured in cubic units,
Capacity refers to the maximum measure of an object’s ability to hold a substance, like a solid, a liquid or a gas. Capacity can be measured in almost every other unit, including liters, gallons, pounds, etc.

Eg : A bucket contains 9 litres of water, then its capacity is 9 litres.