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

- Mensuration is a topic in Geometry which is a branch of mathematics.
- 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 180^{0 }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 : **

$=\frac{1}{2}\times base\times height\phantom{\rule{0ex}{0ex}}=\frac{1}{2}bh\phantom{\rule{0ex}{0ex}}$

**AREA EQUALITARAL TRIANGLE : **

$AREAOFEQUALITARAL=\frac{1}{2}\times B\times H\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\times a\times \frac{\sqrt{3}}{2}\times a\phantom{\rule{0ex}{0ex}}=\frac{\sqrt{3}}{4}{a}^{2}\phantom{\rule{0ex}{0ex}}$

$A{C}^{2}=A{D}^{2}+D{C}^{2}\phantom{\rule{0ex}{0ex}}{a}^{2}={h}^{2}+\frac{{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}{a}^{2}\u2013\frac{{a}^{2}}{4}={h}^{2}\phantom{\rule{0ex}{0ex}}{h}^{2}=\frac{4{a}^{2}\u2013{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}{h}^{2}=\frac{3{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}h=\frac{\sqrt{3}}{2}a\phantom{\rule{0ex}{0ex}}thereforetheheightoftheequilateraltriangleish=\frac{\sqrt{3}}{2}a$

**HERONS FORMULA : **

$\sqrt{S\left(s\u2013a\right)}\left(s\u2013b\right)\left(s\u2013c\right)\phantom{\rule{0ex}{0ex}}S=seriperimeter\phantom{\rule{0ex}{0ex}}S=\frac{a+b+c}{2}$

**AREA OF A RIGHT ANGLE ISOSCELES TRIANGLE :**

$=\frac{1}{2}\times a\times a\phantom{\rule{0ex}{0ex}}=\frac{1}{2}{a}^{2}$

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

$areaofquadrilateral:\left[\frac{1}{2}d\left({h}_{1}+{h}_{2}\right)\right]$

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

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

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

$areaofparalle\mathrm{log}ram\left(l{l}^{gm}\right)=bh$

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

$areaofrec\mathrm{tan}gle=l\times b\phantom{\rule{0ex}{0ex}}perimeterofrec\mathrm{tan}gle=2\left(l+b\right)$

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

${d}_{1,}{d}_{2}arelengthsofthediagonals\phantom{\rule{0ex}{0ex}}=areaofrhombus=\frac{{d}_{1}{d}_{2}}{2}$

**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.

$areaofthesquare={a}^{2}\phantom{\rule{0ex}{0ex}}perimeterofthesquare=4a$

- 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 : **

$Areaofcircle={\mathrm{\pi r}}^{2}\mathrm{or}\frac{{\mathrm{\pi d}}^{2}}{4}\phantom{\rule{0ex}{0ex}}\mathrm{circumference}\mathrm{of}\mathrm{a}\mathrm{circle}=2\mathrm{\pi r}\mathrm{or}\mathrm{\pi d}\phantom{\rule{0ex}{0ex}}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{circle}=\mathrm{d}\u20132\mathrm{r}\phantom{\rule{0ex}{0ex}}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{semicircle}=\frac{{\mathrm{\pi r}}^{2}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{circumference}\mathrm{of}\mathrm{semicircle}=\mathrm{\pi r}\phantom{\rule{0ex}{0ex}}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{sector}=\frac{\mathrm{lr}}{2}(\mathrm{here}\mathrm{l}=\mathrm{lrength}\mathrm{of}\mathrm{the}\mathrm{arc},\mathrm{r}=\mathrm{radius})\mathrm{or}\frac{{\mathrm{x}}^{0}}{{360}^{0}}\times {\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{arc}=\mathrm{l}=\frac{{\mathrm{x}}^{0}}{{360}^{0}}\times 2\mathrm{\pi r}\phantom{\rule{0ex}{0ex}}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{rectangular}\mathrm{path}=\mathrm{area}\left(\mathrm{outerpath}\right)\u2013\mathrm{area}\left(\mathrm{innerpath}\right)\phantom{\rule{0ex}{0ex}}\mathrm{path}\mathrm{of}\mathrm{the}\mathrm{square}=\mathrm{area}(\mathrm{outer}\mathrm{square})\u2013\mathrm{area}(\mathrm{inner}\mathrm{space})\phantom{\rule{0ex}{0ex}}\mathrm{path}\mathrm{of}\mathrm{the}\mathrm{circle}=\mathrm{area}\left(\mathrm{outercircle}\right)\u2013\mathrm{area}(\mathrm{inner}\mathrm{circle})$

**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

$\mathit{c}\mathit{u}\mathit{b}\mathit{e}\mathbf{}\mathbf{:}\mathbf{}\phantom{\rule{0ex}{0ex}}lateralsurfaceareaofcube:4{a}^{2}\phantom{\rule{0ex}{0ex}}totalsurfaceareaofcube:b{a}^{2}\phantom{\rule{0ex}{0ex}}volumeofcube:{a}^{3}$

$\mathit{c}\mathit{u}\mathit{b}\mathit{o}\mathit{i}\mathit{d}\mathbf{}\mathbf{:}\mathbf{}\phantom{\rule{0ex}{0ex}}lateralsurfaceareaofcuboid:2h\left(l+b\right)\phantom{\rule{0ex}{0ex}}totalsurfaceareaofcuboid:2\left(lb+bh+hl\right)\phantom{\rule{0ex}{0ex}}volumeofcuboid:lbh\phantom{\rule{0ex}{0ex}}lengthofthediagnoalofcuboid:\sqrt{{l}^{2}+{b}^{2}+{h}^{2}}\phantom{\rule{0ex}{0ex}}$

$\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r}\mathbf{}\mathbf{:}\mathbf{}\phantom{\rule{0ex}{0ex}}curvedsurfaceareaofcylinder:2\mathrm{\pi rh}\phantom{\rule{0ex}{0ex}}\mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cylinder}:2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right)\phantom{\rule{0ex}{0ex}}\mathrm{volume}\mathrm{of}\mathrm{cylinder}:{\mathrm{\pi r}}^{2}\mathrm{h}$

$\mathit{c}\mathit{o}\mathit{n}\mathit{e}\mathbf{}\mathbf{:}\phantom{\rule{0ex}{0ex}}\mathit{r}\mathbf{}\mathbf{:}\mathbf{}radiusofcone\phantom{\rule{0ex}{0ex}}h:heightofcone\phantom{\rule{0ex}{0ex}}l:slantheigjhtofcone\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}curvedsurfaceareaofcone:\mathrm{\pi rl}\phantom{\rule{0ex}{0ex}}\mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cone}:\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)\phantom{\rule{0ex}{0ex}}\mathrm{volume}\mathrm{of}\mathrm{cone}:\frac{1}{2}{\mathrm{\pi r}}^{2}\mathrm{h}$

$\mathit{s}\mathit{p}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}\mathbf{:}\mathbf{}\phantom{\rule{0ex}{0ex}}curvedsurfaceareaofsphere:4{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{sphere}:4{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{volume}\mathrm{of}\mathrm{sphere}:\frac{4}{3}{\mathrm{\pi r}}^{3}\phantom{\rule{0ex}{0ex}}$

$\mathit{h}\mathit{e}\mathit{m}\mathit{i}\mathit{s}\mathit{p}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}\mathbf{:}\mathbf{}\phantom{\rule{0ex}{0ex}}curvedsurfaceareaofhemisphere:2{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{hemisphere}:3{\mathrm{\pi r}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{volume}\mathrm{of}\mathrm{hemisphere}:\frac{2}{3}{\mathrm{\pi r}}^{3}$