mensuration

mensuration

• Mensuration is the branch of mathematics that studies the measurement of the geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc.
• Here, the concepts of mensuration are explained and all the important mensuration formulas provided.

• A branch of mathematics that talks about the length, volume, or area of different geometric shapes  is called Mensuration.
• These shapes exist in 2 dimensions or 3 dimensions.

what is mensuration

Here we can discuss about what is mensuration

1. Mensuration is a topic in Geometry which is a branch of mathematics.
2. Mensuration deals with length, area and volume of different kinds of shape- both 2D and 3D. So moving ahead in the introduction to Mensuration.

• 2D shape is a shape that is bounded by three or more straight lines or a closed circular line in a plane. These shapes have no depth or height. They have two dimensions length and breadth. Therefore called 2D figures or shapes. For 2D shapes, we measure Perimeter (P) and Area (A).
• 3D shape is a shape that is bounded by a number of surfaces or planes. These are also referred to as solid shapes. These shapes have height or depth unlike 2D shapes, they have three dimensions Length, Breadth and Height/Depth and therefore they are called 3D figures.  3D shapes are actually made up of a number of 2D shapes. Also, know as solid shapes, for 3D shapes we measure Volume (V), Curved Surface Area (CSA), Lateral Surface Area (LSA) and Total Surface Area (TSA).

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CLOSED FIGURE : the plain figure which have same starting and ending points is called closed figure.

OPEN FIGURE : the plain figure which have different starting and ending points is called open figure.

CONFEX FIGURE : the plain which have all angles than 180^o then is called confex figure.

• In plain figure all diagnols lies inside the confex figure.
• if you take any two points in a figure the line segment joining those two points is also inside that figure then it is called confex figure.

CONCAVE FIGURE : at least one angle is greater than 180⁰ then the figure is called concave figure.

• if you take any two points in a figure the line segment joining those two points is also inside that figure then it is called confex figure.
• if you take any two points in the figure the line segment joining those two points is not belongs to interior of the diagram then it is called concave figure.

POLYGON : a closed and confex figure which has 3 or more sides is called polygon.

TRIANGLE : a closed , confex figure which has 3 sides is called triangle.

PERIMETER : total length of boundary line of the plain figure is called perimeter.

AREA : the amount of surface of enclosed plain figure is called area.

NOTE: the perimeter of a triangle is /AB+/BC+//CA = A+B+C

AREA OF RIGHT ANGLED TRIANGLE :

$$\begin{gathered} =\frac{1}{2}\times base\times height \\ =\frac{1}{2}bh \end{gathered}$$

AREA EQUALITARAL TRIANGLE :

$$\begin{gathered} AREA OF EQUALITARAL = \frac{1}{2}\times B\times H \\ = \frac{1}{2}\times a\times \frac{\sqrt{3}}{2}\times a \\ =\frac{\sqrt{3}}{4}{a}^2 \end{gathered}$$

$$\begin{gathered} A{C}^2=A{D}^2+D{C}^2 \\ {a}^2=h^2+\frac{a^2}{4} \\ {a}^2–\frac{a^2}{4}=h^2 \\ {h}^{2 }=\frac{4a^2–a^2}{4} \\ {h}^2= \frac{3a^2}{4} \\ h =\frac{\sqrt{3}}{2}a \\ therefore the height of the equilateral triangle is h=\frac{\sqrt{3}}{2}a \end{gathered}$$

HERONS FORMULA :

$$\begin{gathered} \sqrt{S\left(s–a\right)}\left(s–b\right)\left(s–c\right) \\ S = seri perimeter \\ S=\frac{a+b+c}{2} \end{gathered}$$

AREA OF A RIGHT ANGLE ISOSCELES TRIANGLE :

$$\begin{gathered} =\frac{1}{2}\times a\times a \\ =\frac{1}{2}{a}^2 \end{gathered}$$

QUADRILATERAL : a closed convex figure which has 4sides is called quadrilateral.

$area of quadrilateral : \left[\frac{1}{2}d\left(h_1+h_2\right)\right]$

TRAPEZIUM : In a quadrilateral one pair of opposite sides are parallel.

$area of trapezium : \left[\frac{1}{2}h\left(a+b\right)\right]$

PARALLELOGRAM : In a quadrilateral opposite sides are parallel the lengths of the diagonals are not equal is called parallelogram.

$area of paralle\log ram \left(l{l}^{gm}\right)=bh$

RECTANGLE : In a parallelogram one angle is 90 degrees then it is called rectangle.

$$\begin{gathered} area of rec\tan gle = l\times b \\ perimeter of rec\tan gle =2\left(l+b\right) \end{gathered}$$

RHOMBUS : In a quadrilateral all sides are equal but lengths of the diagonals are not equal then is called rhombus.

$$\begin{gathered} d_{1 , }{d}_{2 } are lengths of the diagonals \\ = area of rhombus = \frac{d_1 d_2}{2} \end{gathered}$$

SQUARE : In a quadrilateral all sides equal and diagonals are also equal then it is called square. (or)

in a rhombus one angle is 90 degrees then it is called square.

$$\begin{gathered} area of the square =a^2 \\ perimeter of the square = 4a \end{gathered}$$

• all squares are rectangles.
• all squares are rhombus.
• all squares are parallelograms.

• all squares are trapeziums.
• all squares are quadrilaterals.

• some rhombus are squares.
• some rhombus are rectangles.
  
• some rectangles are rhombus.
• some rectangles are squares.
• all rhombus are parallelogram.
• all rhombus are trapeziums.
• all rhombus are quadrilaterals.
  
• all rectangles are parallelograms.
• all rectangles are trapeziums.
• all rectangles are quadrilaterals.
• all parallelograms are trapeziums.
  
• all parallelograms are quadrilaterals.
• all trapeziums are quadrilaterals.
• some quadrilaterals are trapezium.
• some quadrilaterals are parallelograms.
• some quadrilaterals are rectangles.
• some quadrilaterals are rhombus.
• some quadrilaterals are squares.
• some trapeziums are parallelograms.
• some trapeziums are rectangles.
• some trapeziums are rhombus.
• some trapeziums are squares.
• some parallelograms are rectangles.
• some parallelograms are rhombus.
• some parallelograms are squares.

types of quadrilaterals	no of measurements we need for construction
square	1
rhombus	2
rectangle	3
parallelogram	4
trapezium	5
quadrilateral	6

CIRCLE : The locus of the points which are equal distance from the fixed point is called a circle denoted by (o).

the fixed point is called centre of the circle.

RADIUS : The distance between the center of the circle and the point on the circle is called radius.

CHORD : A line segment joining any two points on a circle is called chord.

DIAMETER : The largest chord of the circle is called diameter.

NOTE : All diameters passes through the centre of the circle.

we can draw infinite diameter on the circle. it is denoted by (d).

ARC : The part of the circle is called arc.

SECTOR : The area between two radius OA,OB and ARC AB is called sector.

FORMULAS :

$$\begin{gathered} Area of circle = {\mathrm{\pi r}}^{2 } \mathrm{or} \frac{{\mathrm{\pi d}}^2}{4} \\ \mathrm{circumference} \mathrm{of} \mathrm{a} \mathrm{circle} = 2\mathrm{\pi r} \mathrm{or} \mathrm{\pi d} \\ \mathrm{diameter} \mathrm{of} \mathrm{the} \mathrm{circle} = \mathrm{d}–2\mathrm{r} \\ \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{semicircle} = \frac{{\mathrm{\pi r}}^2}{2} \\ \mathrm{circumference} \mathrm{of} \mathrm{semicircle}= \mathrm{\pi r} \\ \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{sector} = \frac{\mathrm{lr}}{2}(\mathrm{here} \mathrm{l}=\mathrm{lrength} \mathrm{of} \mathrm{the} \mathrm{arc}, \mathrm{r}= \mathrm{radius}) \mathrm{or} \frac{{\mathrm{x}}^0}{{360}^0}\times {\mathrm{\pi r}}^2 \\ \mathrm{length} \mathrm{of} \mathrm{the} \mathrm{arc} = \mathrm{l}=\frac{{\mathrm{x}}^0}{{360}^0}\times 2\mathrm{\pi r} \\ \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{rectangular} \mathrm{path} = \mathrm{area}\left(\mathrm{outerpath}\right)–\mathrm{area}\left(\mathrm{innerpath}\right) \\ \mathrm{path} \mathrm{of} \mathrm{the} \mathrm{square}=\mathrm{area}(\mathrm{outer} \mathrm{square})–\mathrm{area}(\mathrm{inner} \mathrm{space}) \\ \mathrm{path} \mathrm{of} \mathrm{the} \mathrm{circle}=\mathrm{area}\left(\mathrm{outercircle}\right)–\mathrm{area}(\mathrm{inner} \mathrm{circle}) \end{gathered}$$

3-d figures : the plane figures which have 3 measures (length,breadth,height) is called 3-d figures.

eg: cube,cuboid,cylinder,cone,sphere,hemisphere

$$\begin{gathered} \mathit{c}\mathit{u}\mathit{b}\mathit{e}\mathbf{ }\mathbf{:}\mathbf{ } \\ lateral surface area of cube : 4a^2 \\ total surface area of cube : b{a}^2 \\ volume of cube : a^3 \end{gathered}$$

$$\begin{gathered} \mathit{c}\mathit{u}\mathit{b}\mathit{o}\mathit{i}\mathit{d}\mathbf{ }\mathbf{:}\mathbf{ } \\ lateral surface area of cuboid : 2h\left(l+b\right) \\ total surface area of cuboid : 2\left(lb+bh+hl\right) \\ volume of cuboid : lbh \\ length of the diagnoal of cuboid : \sqrt{l^2+b^2+h^2} \end{gathered}$$

$$\begin{gathered} \mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r}\mathbf{ }\mathbf{:}\mathbf{ } \\ curved surface area of cylinder : 2\mathrm{\pi rh} \\ \mathrm{total} \mathrm{surface} \mathrm{area} \mathrm{of} \mathrm{cylinder} : 2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right) \\ \mathrm{volume} \mathrm{of} \mathrm{cylinder} : {\mathrm{\pi r}}^2\mathrm{h} \end{gathered}$$

$$\begin{gathered} \mathit{c}\mathit{o}\mathit{n}\mathit{e}\mathbf{ }\mathbf{:} \\ \mathit{r}\mathbf{ }\mathbf{:}\mathbf{ }radius of cone \\ h: height of cone \\ l : slant heigjht of cone \\ curved surface area of cone : \mathrm{\pi rl} \\ \mathrm{total} \mathrm{surface} \mathrm{area} \mathrm{of} \mathrm{cone} : \mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right) \\ \mathrm{volume} \mathrm{of} \mathrm{cone} : \frac{1}{2}{\mathrm{\pi r}}^2\mathrm{h} \end{gathered}$$

$$\begin{gathered} \mathit{s}\mathit{p}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathbf{:}\mathbf{ } \\ curved surface area of sphere : 4{\mathrm{\pi r}}^2 \\ \mathrm{total} \mathrm{surface} \mathrm{area} \mathrm{of} \mathrm{sphere} : 4{\mathrm{\pi r}}^2 \\ \mathrm{volume} \mathrm{of} \mathrm{sphere} : \frac{4}{3}{\mathrm{\pi r}}^3 \end{gathered}$$

$$\begin{gathered} \mathit{h}\mathit{e}\mathit{m}\mathit{i}\mathit{s}\mathit{p}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathbf{:}\mathbf{ } \\ curved surface area of hemisphere : 2{\mathrm{\pi r}}^2 \\ \mathrm{total} \mathrm{surface} \mathrm{area} \mathrm{of} \mathrm{hemisphere} : 3{\mathrm{\pi r}}^2 \\ \mathrm{volume} \mathrm{of} \mathrm{hemisphere} : \frac{2}{3}{\mathrm{\pi r}}^3 \end{gathered}$$