Mensuration

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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 180o 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 :

=12×base×height=12bh

AREA EQUALITARAL TRIANGLE :

AREA OF EQUALITARAL = 12×B×H                                                 = 12×a×32×a                                                 =34a2

AC2=AD2+DC2a2=h2+a24a2–a24=h2h2 =4a2–a24h2= 3a24h =32atherefore the height of the equilateral triangle is h=32a

HERONS FORMULA :

Ss–as–bs–cS = seri perimeter S=a+b+c2

AREA OF A RIGHT ANGLE ISOSCELES TRIANGLE :

=12×a×a=12a2

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

area of quadrilateral : 12dh1+h2

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

area of trapezium : 12ha+b

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

area of parallelogram llgm=bh

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

area of rectangle = l×bperimeter of rectangle =2l+b

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

d1 , d2  are lengths of the diagonals= area of rhombus = d1 d22

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.

area of the square =a2perimeter of the square = 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 quadrilateralsno of measurements we need for construction
square1
rhombus2
rectangle3
parallelogram4
trapezium5
quadrilateral6

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 :

Area of circle = πr2  or πd24circumference of a circle = 2πr or πddiameter of the circle = d–2rarea of the semicircle = πr22circumference of semicircle= πrarea of the sector = lr2(here l=lrength of the arc, r= radius) or x03600×πr2length of the arc = l=x03600×2πrarea of the rectangular path = area(outerpath)–area(innerpath)path of the square=area(outer square)–area(inner space)path of the circle=area(outercircle)–area(inner 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

cube : lateral surface area of cube : 4a2total surface area of cube : ba2volume of cube : a3

cuboid : lateral surface area of cuboid : 2hl+btotal surface area of cuboid : 2lb+bh+hlvolume of cuboid : lbhlength of the diagnoal of cuboid : l2+b2+h2

cylinder : curved surface area of cylinder : 2πrhtotal surface area of cylinder : 2πrr+hvolume of cylinder : πr2h

cone :r : radius of coneh: height of cone l : slant heigjht of conecurved surface area of cone : πrltotal surface area of cone : πrl+rvolume of cone : 12πr2h

sphere : curved surface area of sphere : 4πr2total surface area of sphere : 4πr2volume of sphere : 43πr3

hemisphere : curved surface area of hemisphere : 2πr2total surface area of hemisphere : 3πr2volume of hemisphere : 23πr3

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